Algebras for Generalized Power Constructions

Vom Fachbereich Mathematik der Technischen Hochschule Darmstadt zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation

von

Dipl.-Math. Andrea Schalk

aus Witten/Ruhr

Referent: Prof. Dr. K. Keimel Korreferent: Prof. Dr. K. H. Hofmann Tag der Einreichung: 1. Juni 1993 Tag der mündlichen Prüfung: 16. Juli 1993

Darmstadt 1993 D 17

Contents

Introduction 7

1 Some basic concepts 13

1.1 Dcpo’s and domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.1 Some order-theoretic notions . . . . . . . . . . . . . . . . . . . 13 1.1.2 Directed complete partial orders . . . . . . . . . . . . . . . . . 15 1.1.3 Algebraic dcpo’s . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1.4 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.1.5 Order-consistent topologies . . . . . . . . . . . . . . . . . . . . 19

1.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.1 The free frame . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.2 Presentations of frames by generators and relations . . . . . . 26

2 Locales 29

2.1 Locales and topological spaces . . . . . . . . . . . . . . . . . . . . . . 30 2.1.1 Points for a locale . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.2 Properties for points . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.3 Sobrification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.4 Spatialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.5 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Products in the category of locales . . . . . . . . . . . . . . . 35 2.2.2 Products of locales and products of topological spaces . . . . . 36

3 Local compactness and continuity 39 3.1 Locally compact spaces and domains . . . . . . . . . . . . . . . . . . 39 3.2 Locally compact locales . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Monads and their Algebras 45 4.1 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Monads on the category of posets . . . . . . . . . . . . . . . . . . . . 51 4.4 Monads for O-categories . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Generalized semilattices 57 5.1 Semilattices in arbitrary categories . . . . . . . . . . . . . . . . . . . 57 5.2 Inflationary and deflationary semilattices . . . . . . . . . . . . . . . . 60 5.3 Examples of generalized semilattices . . . . . . . . . . . . . . . . . . 61

5.3.1 Dcpo’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 62 5.3.3 Locales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Retracts of generalized semilattices . . . . . . . . . . . . . . . . . . . 64 5.5 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 The Hoare power construction 73 6.1 Algebraic domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Directed complete partial orders . . . . . . . . . . . . . . . . . . . . . 76

6.2.1 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.2 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3.1 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . 84 6.3.2 Locally compact spaces . . . . . . . . . . . . . . . . . . . . . . 85 6.3.3 Hoare power space versus Hoare power dcpo . . . . . . . . . . 86

6.4 Locales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.4.1 The monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.4.2 The algebras for the monad . . . . . . . . . . . . . . . . . . . 92 6.4.3 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4.4 Locally compact locales . . . . . . . . . . . . . . . . . . . . . 101

6.4.5 Power space versus power locale . . . . . . . . . . . . . . . . . 103

7 The Smyth power construction 107

7.1 Algebraic domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.1.1 P_{A}^{♯}D as a subset of PD . . . . . . . . . . . . . . . . . . . . . . 108 7.2 Directed complete partial orders . . . . . . . . . . . . . . . . . . . . . 111

7.2.1 A Smyth power construction . . . . . . . . . . . . . . . . . . . 111 7.2.2 The monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.2.3 The algebras for the monad . . . . . . . . . . . . . . . . . . . 114 7.2.4 Why F_{D} does not give the free deflationary semilattice . . . . . 116 7.2.5 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2.6 Another Smyth power construction . . . . . . . . . . . . . . . 120 7.2.7 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2.8 Remarks on the Smyth power dcpo constructions . . . . . . . 123

7.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.3.1 A naïve approach . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3.2 A Smyth power construction . . . . . . . . . . . . . . . . . . . 126 7.3.3 Why K_{T} does not give the free deflationary semilattice . . . . . 131 7.3.4 Locally compact spaces . . . . . . . . . . . . . . . . . . . . . . 132 7.3.5 Another Smyth power construction . . . . . . . . . . . . . . . 136 7.3.6 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . 140

7.4 Locales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.4.1 The monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.4.2 Power space versus power locale . . . . . . . . . . . . . . . . . 144 7.4.3 The algebras for the monad . . . . . . . . . . . . . . . . . . . 146 7.4.4 Locally compact locales . . . . . . . . . . . . . . . . . . . . . 150 7.4.5 Unital semilattices . . . . . . . . . . . . . . . . . . . . . . . . 153

Appendix 155

Bibliography 161 6

Introduction

In order to give a denotational semantics for programming languages, quite a number of settings have been proposed: algebraic dcpo’s, universal domains, topological spaces and locales, just to mention a few. This raises the question what these different approaches have in common and how they can be understood to evolve from the operational (or intended) behaviour of the programming language in question. A suggestion how to attack this problem at a high level of abstraction has recently been made by E. Moggi in [Mog91]. He proposes that each connective of a programming language can be described by a (strong) monad on an underlying category of values. Programs then should be interpreted in the Kleisli category for that monad. This approach, if successful, makes it possible to give a semantics for a programming language in an incremental fashion. Step by step one adds a feature to the language resulting in a series of monads. Since it is not prescribed in which category this is carried out, one can hope to unearth the essential properties of those monads that are to model a particular construct. One of the invariants, as we may call them, ought to be found in the algebraic theories associated with the monads. We really expect them to essentially agree. A second benefit from such a meta-semantical approach can be that we may be able to extract a logic for reasoning about programs which will no longer refer to a particular denotational model. A first attempt in this direction has been made in [Pit91]. In the present work we deal with the classical problem of adding non-determinism to a sequential programming language. For most of the settings mentioned above, concrete power constructions have been introduced in order to capture the possibilities of multiple output of non-deterministic computations. Starting with the work of M. Hennessy and G. Plotkin in [HP79] it is well known that each of these constructions gives rise to a monad. The question that we are concerned with here is whether the equational characterization of power constructions given by Hennessy and Plotkin for algebraic dcpo’s remains valid if the setting is changed. We extract the equational description in the usual way, that is, by characterizing the associated (Eilenberg-Moore) algebras. The three settings that we will study are (arbitrary) dcpo’s, topological spaces and locales. For all of them, power construction have been developed previously [Plo76, Smy78, Smy83, Joh82b, Rob88]. We examine the algebras for two power constructions, modeling non-determinism as angelic respectively demonic. It turns out that in all cases the algebras can be understood as semilattices in the respective categories, sometimes satisfying additional properties. The first to define power constructions in order to model non-determinism were G. Plotkin [Plo76] and M. Smyth [Smy78] who did so for algebraic dcpo’s. Later on, the so-called Hoare power domain was introduced. This construction is folklore and cannot be attributed to a single person. A study of possible generalizations for these (among other) constructions to arbitrary dcpo’s has been carried out by R. Heckmann in [Hec90]. Using his definitions, we examine in how far these generalizations still yield the same algebraic theory as the original power domain functors. The category Top of T_{0}-spaces as a setting for denotational semantics has been proposed by M. Smyth. In [Smy83] he presents the idea that an open set can be seen as a computable property. He illustrates this view by formulating concepts like ‘specification’, ‘predicate transformer’ and ‘non-determinism’ in topological terms. In order to model non-determinism in this setting, one also wants to use set-valued functions. M. Smyth gave definitions of Hoare, Smyth and Plotkin power spaces as subspaces of the power set. The topologies suggested by different views of non-determinism are known as lower, upper and Vietoris topology and therefore connect this approach with the study of power- or hyper-spaces in general topology. Such ‘spaces of subsets’ and multivalued functions have been examined by L. Vietoris [Vie21, Vie22] (see also [Mic51]). Building on Smyth’s definition we obtain monads for the Hoare and Smyth power space constructions on Top. We then proceed to examine the algebras for these monads. By doing this we obtain characterizations of several categories of domains as algebras for monads on subcategories of Top similar to those in [Day75, Wyl81, Tay90, Sch93a]. The idea of understanding open sets as properties of their elements can be further abstracted by using arbitrary frames instead of open set lattices. So one can view the elements of a frame as propositions, interpreting join as conjunction and meet as disjunction. These considerations lead to the logic of finite observations and, finally, to a presentation of denotational semantics in an axiomatic form, as a logic of programs [Abr87]. This approach is tightly related to the use of algebraic dcpo’s in denotational semantics via Stone-type dualities. Rather than the category of frames one considers its opposite, the category of locales. In order to handle non-determinism one needs constructions for locales corresponding to power domains. The first power locale construction, the so-called Vietoris power locale, has been defined by P. Johnstone in [Joh82b]. His definition is based on the Vietoris power space and uses properties of the Vietoris topology. This construction yields a monad whose algebras are shown to carry a semilattice structure in the category of locales (see also [Joh85]). In [Vic], S. Vickers shows that for locally compact locales the reverse is also true, that is, every localic semilattice is an algebra for the monad. The connection between this power locale construction and the axiomatic approach to semantics has been made in two steps. G. Winskel [Win85] described the classical power domains via modal assertions about the behaviour of a program. Carrying this point of view a step further, E. Robinson gave a definition of Hoare and Smyth power locale constructions [Rob88]. Moreover, he pointed out that Johnstone’s Vietoris power locale corresponds to the Plotkin power domain. We start our examinations of monads resulting from power construction by giving an account of some basic results on dcpo’s, topological spaces and frames in Chapter 1. In the second chapter, we collect some results about locales. We then point out the connections between continuous dcpo’s, locally compact spaces and locally compact locales in Chapter 3. The fourth chapter is devoted to introducing some categorical notions, in particular that of monads and their algebras. We further give some general results on power monads for order-enriched categories. In Chapter 5 we direct our attention to the theory of generalized semilattices. In any order-enriched category there are two special cases of semilattices which correspond to ∨- and ∧-semilattices in the category of posets: Any semilattice operation gives rise to an order. On a poset, this order or its dual might coincide with the intrinsic order. We speak of inflationary respective deflationary semilattices in these cases. These concepts are meaningful in every order-enriched category. We examine them for dcpo’s, topological spaces and locales. We are also interested in retracts of (generalized) semilattices since, in most cases we study, the algebras for a monad are exactly the retracts of the free algebras. Since the free algebras can all be shown to carry semilattice structures, these results allow us to conclude that every algebra must be a semilattice. We also define unital generalized semilattices and show that the results about retracts of semilattices can be transferred to this case. In Chapter 6, we study the Hoare power construction for all settings described above. Surprisingly enough, it turns out that in all cases the algebras are exactly the inflationary semilattices (with the slight restriction to sober spaces instead of all T_{0}- spaces). Furthermore, the restriction of the power functors to continuous dcpo’s, locally compact spaces and locally compact locales are also well-behaved.

In the last chapter we give a similar account for the Smyth power construction. Here, however, things become more complicated. First of all, there are two different constructions for both, dcpo’s and topological spaces, that generalize the Smyth power domain. Also, the algebras for the resulting monads are not characterized by being (deflationary) semilattices. For dcpo’s and topological spaces we can indeed give examples of deflationary semilattices that do not have a structure map. In the cases of topological spaces and locales, we can prove that algebras for the monads must have small semilattices. However, we can give an example for a deflationary topological semilattice with small semilattices that fails to be an algebra for the monad. We have not been able to find a corresponding example for the category of locales. Nevertheless, for the subcategories of continuous dcpo’s, locally compact spaces and locally compact locales we do get a nice characterization: the algebras are precisely those deflationary semilattices that have small semilattices. Furthermore, the two different Smyth power constructions coincide. In Chapters 6 and 7 we also give a short description of how to modify the definitions of our power constructions in order to make the algebras unital semilattices. We have not tried to characterize the algebras for the generalized Plotkin (Vietoris) power construction. Hopefully, our results can be applied to solve this problem since this construction can be understood to be built from the respective Hoare and Smyth constructions. The Appendix provides tables containing most of the names for categories and functors we use. In addition, our results are summarized there. Finally, it is my desire to thank those who supported me in the course of this work. First of all I wish to express my gratitude towards my supervisor Prof. Klaus Keimel. His ability to see things from different viewpoints repeatedly led me into promising research directions. He and Prof. Karl Heinrich Hofmann provided many useful suggestions when refereeing this thesis. I would also like to thank Dr. Achim Jung; his encouragement and interest in my work helped me a lot. My colleagues Hermann Puhlmann and Philipp Sünderhauf carefully read a preliminary version of this paper. I benefitted greatly from many discussions with them and other members of the Darmstadt AG Domains. Finally I would like to thank Dr. Steven Vickers who encouraged me to tackle the problem of characterizing the algebras for power locale constructions. During the work on my thesis I have been supported by the Land Hessen.

Chapter 1

Some basic concepts

In this chapter we will introduce the basic notions, notations and results on dcpo’s, topological spaces and frames that we are going to need later. We will not give all proofs. However, we try to develop the main ideas as we go along. If a proof is omitted it can be found in one of the references quoted. Most of the results are well-known with the exception of a few facts which are straightforward to prove and for which we do not know any reference.

1.1 Dcpo’s and domains

All structures we shall study (dcpo’s, topological spaces and locales) carry an order. Let us shortly introduce the relevant order-theoretic concepts. The reader who wishes to consult a more detailed introduction into domain theory and the theory of continuous posets may find [Plo81] or [GHK^{+}80] useful references.

1.1.1 Some order-theoretic notions

The lower closure of a subset S of a poset P is ↓ S := {p ∈ P | ∃s ∈ S . p ≤ s}. A set S that is lower closed (that is, S = ↓ S) is called a lower set. An upper set and the upper closure are defined and denoted dually. A poset P is a ∨-semilattice if for every pair of elements of P the join exists in P. A morphism of ∨-semilattices or ∨-semilattice homomorphism is a function that preserves binary joins. The dual notions are that of ∧-semilattice and ∧-semilattice homomorphism. A poset that at the same time is a ∨- and a ∧-semilattice is called a lattice. If a ∨-semilattice S has a least element then this element is the unit for the binary join. We therefore say that S is a unital ∨-semilattice. A morphism of unital ∨-semilattices is a morphism of ∨-semilattices that preserves the least element. There is a dual notion for ∧-semilattices. A poset that has all joins is a complete ∨-semilattice. A morphism of complete ∨- semilattices is a function that preserves all joins. Complete ∧-semilattices and their morphisms are defined analogously. Note that every complete ∨-semilattice has all meets as well and thus is a complete ∧-semilattice or a complete lattice. We keep these different names, however, in order to make clear which morphisms we consider. Functions between complete lattices that preserve all joins have an interesting property: Such a morphism f: L −^{}M can be used to define a morphism of complete ∧-semilattices g: M −^{}L by setting g(m) := ⋁ f^{−1}(↓{m}). The resulting function g is called the upper adjoint for f (and of course f is the lower adjoint for g). Note that in this case f(l) ≤ m iff l ≤ g(m) holds for all l ∈ L, m ∈ M. Analogously, given a morphism of complete ∧-semilattices g: M −^{}L we can define a morphism of complete ∨-semilattices f: L −^{}M in the obvious way. Thus the category of complete ∧-semilattices is dual to the category of complete ∨-semilattices. More generally, let P and Q be two posets and P f^{−^{}} ; − ; − ; − g Q be functions where f(p) ≤ q iff p ≤ g(q). Then we say that (f, g) is an adjoint pair where f is the lower adjoint for g. The definite article may be used as there is at most one lower (or upper) adjoint for every given function. Note that every lower adjoint preserves the order and existing joins while every upper adjoint is also order-preserving but preserves existing meets. If P f^{−^{}} ; − ; − ; − g Q are order-preserving functions then (f, g) is an adjoint pair iff f ∘g ≤ id_{Q} and g ∘ f ≥ id_{P}. The proof of the following lemma is straightforward.

Lemma 1.1 Let P f^{−^{}} ; − ; − ; − g Q ^{f′}^{−^{}} ; − ; − ; −_{g_{′}} R be functions between posets where f and f′ are the lower adjoints for g and g′, respectively. Then f′ ∘ f is the lower adjoint for g ∘ g′.

We have already seen one example of adjoints: For any ∨-semilattice P, the binary join function from P × P to P is the lower adjoint for the diagonal P −^{}P × P since p ∨ p′ ≤ p′′ iff ⟨p, p′⟩ ≤ ⟨p′′, p′′⟩. The following lemma states a well-known fact.

Lemma 1.2 Let P be a retract of Q via P e^{−^{}} ; − ; − ; − r Q. Let S be a subset of P. If the join of e(S) exists in Q, then the join of S exists in P and ⋁ S = r(⋁ e(S)).

Proof: Assume that ⋁ e(S) exists in Q for some S ⊆ P. Since for all s ∈ S we have s = r(e(s)) ≤ r(⋁ e(S)), we get that r(⋁ e(S)) is an upper bound for S. If, on the other hand, p ∈ P is any upper bound for S then e(p) is an upper bound for e(S) implying p = r(e(p)) ≥ r(⋁ e(S)). Hence r(⋁ e(S)) is the least upper bound for S.

By considering the dual posets we may deduce that the above lemma also holds if join is replaced by meet.

Corollary 1.3 Let P be a retract of Q via P e^{−^{}} ; − ; − ; − r Q. Let S be a subset of P. If the meet of e(S) exists in Q, then the meet of S exists in P and ⋀ S = r(⋀ e(S)).

Hence a retract of a ∧-semilattice, for example, is a ∧-semilattice.

1.1.2 Directed complete partial orders

For a partially ordered set P we say that S ⊆ P is directed if S is non-empty and if any two of its elements have a common upper bound in S. The dual notion is that of a filtered set.

Most posets we shall deal with have least upper bounds for all their directed subsets; they are called directed complete partial orders or dcpo’s for short. Note that unlike other authors we do not assume a dcpo to have a least element. We may think of the join ⋁^{↑}S of a directed subset S (where the arrow is to indicate that S is supposed to be directed) as being approximated by the elements of S. This becomes precise if we consider the Scott topology on a dcpo D. A subset U of D is Scott-open if it has the following properties:

(i) It is upper closed, that is U = ↑ U.

(ii) It is inaccessible by directed joins, that is if S is a directed subset of D with ⋁^{↑}S ∈ U then S intersects U.

Any directed subset S of a dcpo D converges to ⋁^{↑}S with respect to the Scott topology. Furthermore, ⋁^{↑}S is the greatest limit point of the net S with respect to that topology. Since singleton closures are lower closures, the Scott topology of a dcpo is T_{0}. The Scott-continuous functions (that is those that are continuous with respect to the Scott topology) can be characterized order-theoretically: A function f: D −^{}E between dcpo’s is Scott-continuous iff it preserves directed joins. Note that a Scottcontinuous function in particular is order-preserving. We denote the category of dcpo’s with Scott-continuous functions by DCPO. Let T_{σ} denote the functor from DCPO to the category of T_{0}-spaces that arises if we consider a dcpo a topological space via its Scott topology. We set T_{σ}f = f for all morphisms f in DCPO.

1.1.3 Algebraic dcpo’s

The usage of dcpo’s in denotational semantics relies on the above idea of approximation. Of particular interest are dcpo’s where every element is approximated by certain elements below it. The elements which cannot be approximated non-trivially play a special rôle. This concept is captured by the notion of compact elements of a dcpo. An element c of a dcpo D is compact if for every directed subset S of D with c ≤ ⋁^{↑}S there is s ∈ S such that c ≤ s. That means that a compact element c of D which is approximated by a directed set S is already reached by an element of S. Let the set of compact elements of a dcpo D be denoted by CD. A dcpo D is algebraic if, for every element d of D, the set of compact elements below d is directed with join d, that is d = ⋁^{↑}(CD ∩ ↓{d}) for all d ∈ D. For an introduction into the use of algebraic dcpo’s in denotational semantics see [Plo81]. Algebraic dcpo’s have some nice properties. The Scott topology of an algebraic dcpo D, for example, has a basis of sets of the form ↑{c} where c is a compact element of D. We can also recover an algebraic dcpo D from its set of compact elements CD. A subset of a poset is an ideal if it is directed and lower closed. The dual notion of an ideal is a filter. We will denote the set of all ideals of a poset P ordered by inclusion by Idl P. For any poset P, we thus get an algebraic dcpo Idl P. The compact elements of Idl P are the ideals of the form ↓{p} where p ∈ P and are called principal ideals. Hence P ≅ C Idl P. If D is an algebraic dcpo, then D ≅ Idl CD. If P is a pre-ordered set (rather than a partially ordered one) then C Idl P is isomorphic to the quotient P/_{≡} where ≡ is the equivalence relation defined by p ≡ p′ iff p ≤ p′ and p′ ≤ p. A Scott-continuous function f from an algebraic dcpo D to any dcpo E is uniquely determined by its values on the compact elements of D. More precisely, any orderpreserving function f: CD −^{}E can be uniquely extended to a Scott-continuous function f̂: D −^{}E by setting f̂(d) := ⋁^{↑}(CD ∩ ↓{d}). We denote the category of posets with order-preserving functions by Pos. Let AlgD be the full subcategory of DCPO whose objects are the algebraic dcpo’s. There is an obvious way to extend Idl to a functor from the category Pos to the category AlgD. The functor thus defined is left adjoint to the forgetful functor from the category of posets to the category of algebraic domains.

1.1.4 Domains

The category AlgD of algebraic dcpo’s with Scott-continuous functions, however, is not closed with respect to all constructions that one would wish to have. An example for such a construction is that of a retract. Hence we weaken the notion of algebraicity. For elements d and d′ of a dcpo D we say that d is way below d′ if for every directed subset S of D with d′ ≤ ⋁^{↑}S there is s ∈ S such that d ≤ s. In this situation, we write d ≪ d′. One can understand the way below relation to be a notion of compactness relative to an element of the dcpo: An element d ∈ D is compact iff it is way below itself. So if d ≪ d′ we also say that d is relatively compact in d′. The set of all elements way below some d ∈ D is denoted by ↓ ↓{d}. If d, d′ ≪ d′′ holds and if d ∨ d′ exists in D then d ∨ d′ ≪ d′′. A dcpo D is continuous if for all d ∈ D the set ↓ ↓{d} is directed with join d. Continuous dcpo’s have the property that every element is approximated by the elements which are relatively compact in that element. We use the generic term domain for them. In an algebraic dcpo, every element is even approximated by the compact elements below it. Therefore we also speak of algebraic domains. If we talk of continuous posets, then the word ‘continuous’ is to imply that the poset in question has directed joins, that is it is a domain. For a poset to be a continuous lattice we require that it is a complete lattice. If D is a domain then the sets ↑ ↑{d} := {d′ ∈ D | d ≪ d′} where d ∈ D form a basis for the Scott topology. Furthermore, the interpolation property holds in D: Given d ≪ d′ in D there is d′′ ∈ D with d ≪ d′′ ≪ d′. Note that a dcpo D is continuous iff ⋁^{↑}: Idl D −^{}D has a lower adjoint. If it exists, this lower adjoint is d −^{}↓ ↓{d}: D −^{}Idl D. Above we have seen that retracts of posets (via order-preserving functions) inherit many order-theoretic properties. A retract of a dcpo, for example, is a dcpo. Continuity, however, is not such a property. It is only inherited by images of Scott-continuous retractions. So in the category of posets (with order-preserving functions) a retract of a domain need not be domain. In the category of dcpo’s with Scott-continuous functions, however, continuity is known to be a property that retracts inherit.

Lemma 1.4 In DCPO, a retract of a continuous dcpo is continuous.

Note that a retract of an algebraic dcpo in DCPO need not be algebraic.

1.1.5 Order-consistent topologies

The Scott topology of a dcpo allows us to regard any dcpo as a topological space. Sometimes we shall do so without explicitly mentioning the topology. In these cases we always mean the Scott topology. There is also another topology on posets that we will need. The weak topology on a poset P is generated by all sets of the form P \ ↓{p} where p ∈ P. Let D be a dcpo. As the Scott closure of a singleton {d} is its lower closure ↓{d}, the Scott topology is finer than the weak topology on D. Both topologies share the property that the closure of a singleton is its lower closure. Such topologies (on posets) are called order-consistent. Note that in [GHK^{+}80], order consistency is defined differently. The weak topology is the weakest order-consistent topology on any given poset. We have seen above that directed joins play a special rôle for the Scott topology. A connection between joins and order-consistent topologies arises from the following result.

Lemma 1.5 Let P be a poset with an order-consistent topology. For any subset S of P the join of S exists in P iff the join of the closure of S exists in P. In this case ⋁ S = ⋁ cl S. Proof: If ⋁ S exists then because of cl S ⊆ ↓{⋁ S} we know that ⋁ S is an upper bound for cl S. Obviously, it is the least such element of P. If ⋁ cl S exists, then it clearly is an upper bound for S. If p is any upper bound for S then cl S ⊆ ↓{p} which proves that ⋁ cl S is the least upper bound for S.

An up-complete ∨-semilattice is a dcpo that also is a ∨-semilattice, that is a poset that has joins of all non-empty subsets.

Corollary 1.6 Let P and Q be posets with order-consistent topologies and further let f: P −^{}Q be a continuous order-preserving function.

(i) Let S be a subset of P. The join of f(S) exists in Q iff the join of f(cl_{P} S) exists in Q. In this case, ⋁ f(S) = ⋁ f(cl_{P} S).

(ii) If P and Q additionally are up-complete ∨-semilattices then f preserves all non-empty joins iff it preserves joins of non-empty closed sets.

Proof: (i) By the preceeding lemma, ⋁ f(S) exists iff ⋁ cl_{Q} f(S) exists. Furthermore, the two must be equal. As f(S) ⊆ f(cl_{P} S) ⊆ cl_{Q} f(cl_{P} S) = cl_{Q} f(S) by continuity of f, this implies the assertion. (ii) If f preserves joins of non-empty closed sets then ⋁ f(cl_{P} S) = f(⋁ cl_{P} S) for any non-empty subset S of P. By (i) and Lemma 1.5 we get ⋁ f(S) = ⋁ f(cl_{P} S) = f(⋁ cl_{P} S) = f(⋁ S) for all non-empty S in P.

1.2 Topological spaces

Another setting (besides dcpo’s) where we study power constructions is that of topological spaces. A source for most result we gather here is [Joh82a]. We denote the category of T_{0}-spaces with continuous functions by Top. As we have seen before, we may equip a poset with a topology. Here we want to go in the opposite direction. The specialization order on a T_{0}-space X is defined by setting x ≤ x′ iff x ∈ cl{x′}. (If X is not T_{0} this only defines a pre-order.) Clearly the closure of {x′} is the lower closure of {x′} with respect to the specialization order on X. Hence the topology given on X is consistent with the specialization order on X.

If P is a poset with an order-consistent topology then the intrinsic order on P and the specialization order on P coincide. Hence an order-consistent topology on a poset contains all the information about the order. We will think of any T_{0}-space as being equipped with its specialization order and thus refer to order-theoretic concepts as upper set, ideal etc. Every open subset of a T_{0}-space is an upper set (and thus of course every closed subset is a lower set). Continuous functions are well-behaved with respect to the specialization order in the sense that they are order-preserving. Hence Top can be understood as a subcategory of Pos. In order to ensure that the specialization order makes a given T_{0}-space not just a poset but a dcpo we introduce the following concept. A non-empty closed subset of a topological space is irreducible if it is not the union of any two of its closed proper subsets. Clearly, every singleton closure is irreducible. If the sets of this form are the only irreducible closed subsets of a T_{0}-space X then we say that X is sober. Hence a T_{0}-space X is sober iff for any non-empty irreducible closed subset A of X there is x ∈ X with A = cl{x}. An alternative way to understand the concept of sobriety is the following: Let ΩX denote the complete lattice of open subsets of X (ordered by inclusion). Every element x of X gives rise to a filter in ΩX, namely to the filter of its open neighbourhoods N (x) := {U ∈ ΩX | x ∈ U}. The filters thus arising are inaccessible by arbitrary unions. Such filters are also called completely prime. More formally, a filter F in a complete lattice L is completely prime iff for any subset S of L with ⋁ S ∈ F there is s ∈ S such that s ∈ F. A T_{0}-space X is sober iff the only completely prime filters in ΩX are the filters of open neighbourhoods of elements of X. There is yet another equivalent way of defining sobriety which we will encounter when we introduce locales. Any domain D, for example, is sober with respect to its Scott topology: We already know that D is T_{0}. Let A be a non-empty irreducible closed subset of D. We show that A has a greatest element. For that consider the set ↓ ↓ A := ⋃_{a∈A} ↓ ↓{a}. This set is directed. Otherwise there are d, d′ ∈ ↓ ↓ A such that ↑ ↑{d} ∩ ↑ ↑{d′} ∩ A = ∅. But then A ⊆ (D \ ↑ ↑{d}) ∪ (D \ ↑ ↑{d′}) which by irreducibility of A implies A ⊆ D \ ↑ ↑{d} or A ⊆ D \ ↑ ↑{d′}. But D \ ↑ ↑{d} and D \ ↑ ↑{d′} both contain an element of A which is a contradiction. Hence ⋁^{↑} ↓ ↓ A exists and is an element of A. But every a ∈ A is the join of the elements way below it which proves that ⋁^{↑} ↓ ↓ A is an upper bound for A. Hence A =↓ {⋁^{↑} ↓ ↓ A}. Other examples of sober spaces arise from the following proposition.

Proposition 1.7 Every up-complete ∨-semilattice is sober with respect to the weak topology.

Proof: Let L be an up-complete ∨-semilattice and let A ⊆ L be non-empty, irreducible and weakly closed. If we have A ⊆ ↓{l_{1}} ∪ ⋯ ∪ ↓{l_{n}} for some l_{1}, …, l_{n} ∈ L then by irreducibility of A there is i ∈ {1, …, n} such that A ⊆ ↓{l_{i}}. Since A is closed with respect to the weak topology it is the intersection of the finitely generated lower sets containing it. Hence, by the above consideration, we have A = ⋂{↓{l} | A ⊆ ↓{l}}. Since A is non-empty, the set of upper bounds for A is also non-empty. Therefore,

A = ⋂{↓{l} | A ⊆ ↓{l}} = ↓{⋀{l | l upper bound for A}} = ↓{⋁ A}.

Hence A is the closure of the singleton {⋁ A}. It is obvious that L is T_{0} and thus must be sober with respect to the weak topology.

If X is a sober space then the specialization order makes X a dcpo. In this case the topology given on X is coarser than the Scott topology (and finer than the weak topology). If f: X −^{}Y is a continuous function between sober spaces then f preserves directed joins. Thus any continuous function between sober spaces is Scott-continuous.

Sobriety is a property that is inherited by retracts in Top. This is a rather wellknown fact. However, we do not know of any (published) reference. In order to prove it we need the following property of continuous functions. If f: X −^{}Y is a continuous function and S a subset of X then

cl_{Y} f(S) = cl_{Y} f(cl_{X} S).

Furthermore, we need the following lemma.

Lemma 1.8 Let f: X −^{}Y be a continuous function between topological spaces. If A is an irreducible closed set in X then so is cl_{Y} f(A) in Y. Proof: Let B and B′ be closed subsets of Y such that cl_{Y} f(A) ⊆ B ∪ B′. Then A ⊆ f^{−1}(cl_{Y} f(A)) ⊆ f^{−1}(B)∪f^{−1}(B′) and, by irreducibility of A, either A ⊆ f^{−1}(B) implying cl_{Y} f(A) ⊆ f(f^{−1}(B)) ⊆ B or A ⊆ f^{−1}(B′) and thus cl_{Y} f(A) ⊆ B′.

Proposition 1.9 Retracts of sober spaces are sober. Proof: Let Y be a sober space and assume that X is a retract of Y via X e^{−^{}} ; − ; − ; − r Y. Let A be an irreducible closed non-empty subset of X. We have to show that there is a unique x ∈ X with A = cl_{X}{x}. Because of Lemma 1.8, the closure of the image of A under e is irreducible in Y. By sobriety of Y there is a unique y ∈ Y satisfying cl_{Y} e(A) = cl_{Y} {y}. We will show that r(y) has the desired property. By the above property of continuous functions,

A = r(e(A)) ⊆ r(cl_{Y} e(A)) ⊆ cl_{X} r( cl_{Y} {y}) = cl_{X}{r(y)}.

On the other hand, r(y) ∈ r(cl_{Y} e(A)) ⊆ cl_{X} r(e(A)) = A again by the above property. Thus A = cl_{X}{r(y)}. If we assume that X satisfies the T_{0}-separation property then r(y) is unique with that property. Otherwise, assume cl_{X}{x} = cl_{X}{r(y)} for some x ∈ X. Hence

cl_{Y} {e(x)} = cl_{Y} e({cl_{X}{x}) = cl_{Y} e( cl_{X}{r(y)}) = cl_{Y} {e(r(y))}

is an irreducible closed subset of Y which by sobriety of Y implies e(x) = e(r(y)). Consequently, x = r(e(x)) = r(e(r(y))) = r(y).

1.3 Frames

In this section we shall look for another connection between topological spaces and ordered sets. We have already seen that there is a natural order on the points of a T_{0}-space (which is determined by its topology). Given a topological space X there is also an order on the set of open subsets of X, namely subset inclusion. Recall that we denote this poset by ΩX. It is obvious that ΩX is in fact a complete lattice. We can say more about those complete lattices that can arise as the open set lattice of a topological space. In such a poset finite meets distribute over arbitrary joins. For if U ⊆ ΩX and U′ ∈ ΩX then U′ ∩ ⋃ U = ⋃_{U∈U}(U′ ∩ U) for any topological space X. Definition. A frame is a poset that has all joins and all finite meets where finite meets distribute over arbitrary joins. But note that for any frame of the form ΩX where X is a topological space a much stronger distributivity law holds. Assume that U_{j}^{i} ∈ ΩX for all i ∈ I and all j ∈ {1, …, n_{i}}. Let M be the set of all choice functions f: I −^{}⋃_{i∈I}{1, …, n_{i}},

that is f(i) ∈ {1, …, n_{i}} holds for all i ∈ I. Then ⋃_{i∈I}(U_{1}^{i} ∩ ⋯ ∩ U_{n_{i}}^{i}) = ⋂_{f∈M} ⋃_{i∈I} U_{f(i)}^{i}.

Let us investigate which morphisms fit with frames. For any continuous function f: X −^{}Y between topological spaces we get a function between the corresponding open set lattices if we take the inverse image function f^{−1}: ΩY −^{}ΩX. (Note the contravariance!) It is straightforward that this function preserves arbitrary unions as well as finite intersections. Hence a morphism of frames is a function that preserves arbitrary joins and finite meets. We will denote the category of frames with these morphisms by Frm. The following results can be found in [Joh82a] or [Vic89].

1.3.1 The free frame

Let us first consider finitary algebraic theories, that is theories with finitely many operators of finite arities. Then the forgetful functor from the category of models for that theory to Set has a left adjoint [Man76]. The free unital semilattice, for example, over any given set S is isomorphic to the set of finite subsets of S which we denote by P_{fin}S. In this case, the semilattice operation is union. In general, the process of constructing the free model for a finitary algebraic theory can be described as follows:

(1) For every element of the given set take a generator.

(2) Form all well-formed expressions involving the operators of the theory and the generators from step (1).

(3) From the equations of the theory, deduce when two expressions resulting from step (2) have to be equal and form the corresponding equivalence classes.

If we try to apply this process to theories that are not finitary we notice that it need not work. The free complete lattice on three generators, for example, does not exist, that is it is a proper class (see Section I.4.7 in [Joh82a]). The theory of frames is another example of a non-finitary theory. So it is quite remarkable that the free frame on every given set does indeed exist. The following result is a consequence of Lemma I.4.4 and Theorem II.1.2 in [Joh82a].

Proposition 1.10 The free frame on any set S is given by the set P_{L}(P_{fin}S, ⊇) of lower subsets of (P_{fin}S, ⊇) ordered by inclusion.

Note that we actually build the free frame on any set S in two steps: First we take the free semilattice over S. In the second step, we construct the joins for the frame, considering the semilattice from the first step a ∧-semilattice. Furthermore, we may think of every L ∈ P_{L}(P_{fin}S, ⊇) as the join of finite meets of generators since

L = ⋁ {↓ {{s_{1}}} ∧ ⋯ ∧ ↓ {{s_{n}}} | {s_{1}, …, s_{n}} ∈ L} .

1.3.2 Presentations of frames by generators and relations

Given a set G of generators, we often do not wish to construct the free frame over the set G but rather ‘build in’ some relations between these generators. For example, we might wish to build the free frame over a poset P. In this case we want the generators to keep the given order. In this section we will describe how to do this. We may think of the relations given on the generators as equations of the form

⋁_{i∈I} g_{1}^{i} ∧ ⋯ ∧ g_{n_{i}}^{i} = ⋁_{j∈J} g_{1}^{j} ∧ ⋯ ∧ g_{m_{j}}^{j}

where the g_{k}^{i} and the g_{l}^{j} are generators. So assume we are given a set of generators G and a set R of relations of the above form. As for the free frame, we will proceed in two steps by first constructing finite meets and then adding arbitrary joins. Doing this, the relations from R have to be taken into consideration. As we cannot have expressions like ⋁_{i∈I} g_{1}^{i} ∧ ⋯ ∧ g_{n_{i}}^{i} in a semilattice, we use a trick. For every relation in R, say ⋁_{i∈I} g_{1}^{i} ∧ ⋯ ∧ g_{n_{i}}^{i} = ⋁_{j∈I} g_{1}^{j} ∧ ⋯ ∧ g_{m_{j}}^{j}, we create a new generator, say x. The above relation are replaced by

g_{1}^{i} ∧ ⋯ ∧ g_{n_{i}}^{i}, g_{1}^{j}, …, g_{m_{j}}^{j} ≤ x for all i ∈ I, j ∈ J.

We then construct the free unital semilattice, say L, for these generators G′ and these new relations R′. The frame we want is a subset of P_{L}L. In general we will not take all lower subsets of S because we then would end up with something too big. Once more, we want to regard a lower set of L as the join of its elements. For that we have to make sure that the relations from R are built in properly. To describe those lower sets that we do admit we make the following definition. We say that a subset I of L is a C-ideal for the relations from R′ if the following hold:

(i) I is a lower set.

(ii) For all l ∈ L and all x ∈ G′: If g_{1}∧…∧g_{n}∧l ∈ I for all relations g_{1} ∧ … ∧ g_{n} ≤ x in R′ then x ∧ l ∈ I.

Theorem 1.11 (Theorem 4.4.2 in [Vic89]) The set of all C-ideals for R′ of L ordered by inclusion is a frame A that has the following universal property: Given a map f from the set G of generators to any frame B such that

⋁_{i∈I} f(g_{1}^{i}) ∧ ⋯ ∧ f(g_{n_{i}}^{i}) = ⋁_{j∈J} f(g_{1}^{j}) ∧ ⋯ ∧ f(g_{m_{j}}^{j})

for all equations ⋁_{i∈I} g_{1}^{i} ∧ ⋯ ∧ g_{n_{i}}^{i} = ⋁_{j∈J} g_{1}^{j} ∧ ⋯ ∧ g_{m_{j}}^{j} from R there is a unique frame homomorphism f̂: A −^{}B such that f̂ |_{G}= f.

We denote the frame thus constructed by F r⟨G | R⟩. We will see concrete examples for this construction later when we construct the co-product in Frm.

28

Locales

In the last chapter we have given a contravariant functor from Top to Frm. This functor is defined by X −^{}ΩX and f −^{}f^{−1} for every morphism f: X −^{}Y in Top. But we would rather make this a covariant functor, that is we want to view the function f^{−1} as a morphism from ΩX to ΩY. The reason for this is that we want to understand constructions like products, co-products etc. given for topological spaces in terms of the corresponding open set lattices. We define the category Loc to be Frm^{op}. In order to be able to clearly distinguish between the two we shall refer to the objects of Loc as locales and the morphisms as continuous functions (between locales). Every continuous function f: A −^{}B between locales corresponds to a unique morphism of frames which we shall denote by f′: B −^{}A. We define the functor Ω: Top −^{}Loc by setting (Ωf)′ := f^{−1} for every continuous function f: X −^{}Y. Most of the following results for locales can be found either in [Joh82a] or in [Vic89].

2.1 Locales and topological spaces

2.1.1 Points for a locale

We now try to answer the following question: Given a locale A, how can we define a topological space X such that A ≅ ΩX? The points of a topological space X can be understood as the set of all functions from a given one-point space (that is a terminal object in Top) to X. As all these functions are continuous, we get a frame homomorphism ΩX −^{}2 for each of them. Here 2 denotes the two-point lattice with elements 0_{2} and 1_{2} which is initial in Frm and terminal in Loc. Given a locale A we define the set PtA of points of A to be the set of all continuous functions 2 −^{}A. It remains to give a topology on this set. If x ∈ X and p_{x}: {⋆} −^{}X is the function that maps ⋆ to x, then x ∈ U holds iff p_{x}^{−1}(U) = {⋆} for any open subset U of X. We thus take as open sets of PtA the sets of the form U_{a} := {p ∈ PtA | p′(a) = 1_{2}} where a ∈ A. It is not hard to check that this gives a topology on PtA and we refer to the topological space thus defined as SpA. For every locale A the space SpA is sober. The specialization order on SpA is the pointwise order for the corresponding frame homomorphisms A −^{}2. From now on, by PtA we mean the ordered set of points for A thus obtained. Note that PtA is a dcpo since it has all directed joins. Given a continuous function f: A −^{}B between locales we want to define a continuous function Spf: SpA −^{}SpB between topological spaces. Note that every morphism of locales f: A −^{}B induces a Scott-continuous function Ptf: PtA −^{}PtB which is given by Ptf(p) := f ∘ p for all p ∈ PtA. Since

(Ptf)^{−1}({q ∈ PtB | q′(b) = 1_{2}}) = {p ∈ PtA | p′(f′(a)) = 1_{2}},

for all b ∈ B, this function is also continuous with respect to the topologies on PtA respective PtB as introduced above. Considered as a function from SpA to SpB we denote it by Spf and thus obtain a functor from Loc to Top. Let Sob be the full subcategory of Top whose objects are all sober spaces. Then Pt is the composition of Sp with the forgetful functor Sob −^{}DCPO that maps a sober space X to the set X with the specialization order. Another way to obtain points for a locale arises as follows: The filter of open neighbourhoods N (x) of every x ∈ X is a completely prime filter in ΩX. So we might as well have taken the set of completely prime filters of a locale A to be the set of points of A. As x ∈ U iff U ∈ N (x), the open sets in this setting should be the sets of the form

{F ⊆ A | F completely prime filter and a ∈ F }

where a ∈ A. How does this fit with SpA as defined above? If p is a point of the locale A then {a ∈ A | p′(a) = 1_{2}} is a completely prime filter. If, on the other hand, F is a completely prime filter in A then p′(a) := 1 iff a ∈ F defines a continuous function p: 2 −^{}A. This gives in fact a homeomorphism between the topological space just given and SpA. There is a third description of the points for a locale, namely as the prime elements of the locale. An element p of a ∧-semilattice S is prime if it is not the greatest element of S and if s ∧ t ≤ p implies s ≤ p or t ≤ p for all s, t ∈ S. This corresponds to associating with each point x of a topological space X the set X \ {x} which is prime in the open set lattice of X. If p is a prime element of a locale A then A \ ↓{p} is a completely prime filter. On the other hand, if F ⊆ A is a completely prime filter then ⋁(A \ F) is a prime element. This idea is in fact the oldest of the three. Hence we have several equivalent ways of understanding the space SpA. For reasons of convenience we will freely switch between these points of view.

2.1.2 Properties for points

For a topological space X, we interpret the membership relation x ∈ U to mean that x satisfies the property given by U. If A is a locale and a ∈ A then we say that a point p of A satisfies a if, understood as an element of SpA, it is an element of the open set U_{a} generated by a. In this case we write p ⊧ a. The topology of SpA consists of all sets of the form U_{a} := {p ∈ SpA | p ⊧ a} where a ∈ A. The idea of viewing the elements of a locale as the properties of its points can be traced to M. Smyth. For a more detailed account on this approach, see [Vic89]. In computer science, these ideas give rise to a presentation of denotational semantics in axiomatic form, that is as a logic of programs [Abr87]. Assume we have a mapping m from the points of A to the set of points of some other locale B. Under which conditions can we assure that m(p) satisfies some b ∈ B? Let us consider the set S ⊆ A of all properties a ∈ A with p ⊧ a ⟹ m(p) ⊧ b for all points p of A. Note that if p ⊧ ⋁ S for some p ∈ PtA then there is s ∈ S with p ⊧ s which implies m(p) ⊧ b. Hence ⋁ S is the greatest element of S. This gives us a mapping from B to A which we will denote by \overline{m}. If it is the case that p ⊧ \overline{m}(b) ⟺ m(p) ⊧ b for all p ∈ PtA then \overline{m} is a frame homomorphism. If, on the other hand, there is a morphism of locales f: A −^{}B with p ⊧ f′(b) iff m(p) ⊧ b for all b ∈ B and all p ∈ PtA then f′(b) is the weakest property a such that if p satisfies a then m(p) satisfies b. This approach corresponds to the idea of a predicate having a weakest precondition [Smy83].

2.1.3 Sobrification

If X is a sober space then the filters of the form N (x) where x ∈ X are the only completely prime filters in ΩX. In this case, the mapping x −^{}N (x) is a homeomorphism between X and SpΩX: A subset U′ of SpΩX is open iff there is an open subset U of X such that

U′ = {F ⊆ ΩX | F completely prime filter and U ∈ F } = {N (x) | x ∈ U}.

One can show that a topological space X is sober iff X ≅ SpΩX via x −^{}N (x). Furthermore, the family of these isomorphisms X −^{}SpΩX, where X is a sober space, is a natural isomorphism from the identity functor on Sob to SpΩ. Since the space SpA is sober for any locale A the above ensures that the image of Loc under Sp is Sob. This also gives us the possibility to assign a sober space to any topological space. The sobrification of a topological space X is SpΩX. The open set lattice of SpΩX is isomorphic to that of X. The functor Sp is right adjoint to Ω.

2.1.4 Spatialization

The analogue of sobriety for locales is spatiality: A locale A is called spatial if A ≅ Ω SpA via a −^{}{p ∈ SpA | p′(a) = 1_{2}}. We also say that such a locale has enough points as it has enough points to separate its elements: A locale is spatial iff for every a, b ∈ A with a ≰ b there is a point p of A with p′(a) = 1_{2} and p′(b) = 0_{2}. For every sober space X the locale ΩX is spatial: SpΩX ≅ X as mentioned above. If A is a locale then we refer to Ω SpA as the spatialization of A. A locale A is spatial iff there is a topological space X such that A ≅ ΩX. This answers the above question: We can only expect to find a topological space X with A ≅ ΩX if A is a spatial locale. In this case we may choose X = SpA. Hence the image of Top under Ω is the full subcategory of spatial locales. The family of isomorphisms A −^{}Ω SpA is a natural isomorphism from the identity functor on the category of spatial locales to Ω Sp. Furthermore, the full subcategory Sob of sober spaces is equivalent to the full subcategory of spatial locales. This equivalence has been well-known for a long time. For details of the proofs we omitted see [Joh82a].

2.1.5 Adjunctions

We have already seen that every T_{0}-space is equipped with an order, namely the specialization order. Hence it makes sense to talk about adjoint pairs of continuous functions. But if (f, g) is an adjoint pair of continuous functions where f: X −^{}Y then the frame homomorphisms (g^{−1}, f^{−1}) form an adjoint pair between the frames ΩY and ΩX. (Note that lower and upper adjoint interchanged places!) The reason for this is the following: Let U be an open subset of X. For all x ∈ U we have g(f(x)) ∈ U as g(f(x)) ≥ x. Hence

f^{−1}(g^{−1}(U)) = {x ∈ X | g(f(x)) ∈ U} ⊇ U.

This proves f^{−1} ∘ g^{−1} ≥ id_{X}^{−1} = id′_{ΩX}. The other inequality needed to prove our claim, namely g^{−1} ∘ f^{−1} ≤ id′_{ΩY}, can be shown analogously. Hence we have shown the following lemma.

Lemma 2.1 Let X and Y be topological spaces and let X f^{−^{}} ; − ; − ; − g Y be continuous functions. Furthermore, let f be the lower adjoint for g. Then f^{−1}: ΩY −^{}ΩX is the upper adjoint for g^{−1}.

On the other hand, we may consider pairs of continuous functions A f^{−^{}} ; − ; − ; − g B between locales such that the corresponding frame homomorphisms (g′, f′) form an adjoint pair. This raises the question whether a result corresponding to the last lemma holds for the order-preserving functions Spf: SpA −^{}SpB and Spg: SpB −^{}SpA.

Lemma 2.2 Let A f^{−^{}} ; − ; − ; − g B be continuous functions between locales such that g′ is the lower adjoint for f′. Then Spg: SpB −^{}SpA is the upper adjoint for Spf. Proof: We have to show that Spf ∘ Spg ≤ id_{Sp_{B}} and Spg ∘ Spf ≥ id_{Sp_{A}} with respect to the order or specialization. Think of the points of the locales (and thus the points of the corresponding spaces) as continuous functions from 2 to the respective locale. The order of specialization is just the pointwise order on the corresponding frame homomorphisms. Assume that p: 2 −^{}B is a point of SpB. We calculate

((Spf ∘ Spg)(p))′ = ( Spf(g ∘ p))′  (Definition of Sp)
= (f ∘ g ∘ p)′  (Definition of Sp)
= p′ ∘ g′ ∘ f′
≤ p′  (g′ ∘ f′ ≤ id′_{B}).

In the same manner we can prove Spg ∘ Spf ≥ id_{Sp A}.

2.2 Products

2.2.1 Products in the category of locales

In the first chapter we have seen how to build the frame presented by a given set of generators with relations between them. Now we shall see a concrete example, namely the product in the category of locales (which is the co-product in Frm). Since we think of locales as generalized topological spaces we will first see what we can say about the open set lattice of the product of two topological spaces. Let X and Y be topological spaces. The topology on X × Y is generated by sets of the form U × V where U ∈ ΩX and V ∈ ΩY. Let us examine the equations that have to hold for such open sets in X × Y. If S is a subset of ΩX and V ∈ ΩY then ⋃_{S∈S}(S × V) = ( ⋃ S) × V and the corresponding equation holds for all U ∈ ΩX and

T ⊆ ΩY, namely ⋃_{T ∈T} (U × T) = U × ( ⋃ T ). Also, if U, U′ ∈ ΩX and V, V′ ∈ ΩY

then (U × V) ∩ (U′ × V′) = (U ∩ U′) × (V ∩ V′). Furthermore, X × Y is the greatest element of Ω(X × Y). For locales A and B we define A ⊗ B to be the frame presented by generators a ⊗ b, where a ∈ A, b ∈ B, subject to the following relations:

(i) 1_{A} ⊗ 1_{B} = 1_{A⊗B}

(ii) (a_{1} ⊗ b_{1}) ∧ (a_{2} ⊗ b_{2}) = (a_{1} ∧ a_{2}) ⊗ (b_{1} ∧ b_{2}) (a_{1}, a_{2} ∈ A, b_{1}, b_{2} ∈ B)

(iii) ⋁_{s∈S}(s ⊗ b) = (⋁ S) ⊗ b (S ⊆ A, b ∈ B)

(iv) ⋁_{t∈T}(a ⊗ t) = a ⊗ (⋁ T) (a ∈ A, T ⊆ B)

We may understand the construction of A ⊗ B as follows: Let S(A, B) be the semilattice {a ⊗ b | a ∈ A, b ∈ B} with meets calculated componentwise and with a ⊗ 0_{B} = 0_{A⊗B} = 0_{A} ⊗ b for all a ∈ A, b ∈ B. We say that I ⊆ S(A, B) is a ⊗-ideal if (i) I is a lower set,

(ii) for all S ⊆ A, b ∈ B: If s ⊗ b ∈ I for all s ∈ S then (⋁ S) ⊗ b ∈ I,

(iii) for all a ∈ A, T ⊆ B: If a ⊗ t ∈ I for all t ∈ T then a ⊗ (⋁ T) ∈ I.

A ⊗-ideal is just a special instance of a C-ideal for the above relations as defined in Section 1.3. Hence by Theorem 1.11, A ⊗ B is isomorphic to the set of all ⊗-ideals of S(A, B) ordered by inclusion. We can define continuous functions π_{1}: A ⊗ B −^{}A and π_{2}: A ⊗ B −^{}B by setting π′_{1}(a) := a ⊗ 1_{B} and π′_{2}(b) := 1_{A} ⊗ b.

Proposition 2.3 (Proposition 6.4.2 in [Vic89]) The product of locales A and B is A ⊗ B together with the projections π_{1} and π_{2}. Therefore the category of locales has finite products.

If f: C −^{}A and g: C −^{}B are continuous functions then ⟨f, g⟩: C −^{}A ⊗ B is given by ⟨f, g⟩′(a ⊗ b) := f′(a) ∧ g′(b). The diagonal Δ_{A} = ⟨id_{A}, id_{B}⟩: A −^{}A ⊗ A, for example, is defined by Δ′_{A}(a_{1} ⊗ a_{2}) := a_{1} ∧ a_{2}.

2.2.2 Products of locales and products of topological spaces

Let us examine the connection between products in Loc and Top. Since the product of spatial locales need not be spatial, ΩX ⊗ ΩY is not necessarily isomorphic to Ω(X × Y) for arbitrary topological spaces X and Y. For a counterexample, see Section II.2.14 in [Joh82a]. However, there is a well-known connection between the two (see for example Section II.2 in [Joh82a]).

Proposition 2.4 The functor Sp: Loc −^{}Top preserves products.

Proof: Since Sp has a left adjoint, it preserves all limits.

Hence the poset Pt(A⊗B) is isomorphic to PtA×PtB for arbitrary locales A and B. Furthermore, this proposition allows us to conclude that if ΩX ⊗ ΩY is a spatial locale for sober spaces X and Y then

ΩX ⊗ ΩY = Ω Sp(ΩX ⊗ ΩY) ≅ Ω(SpΩX × SpΩY) ≅ Ω(X × Y).

As an immediate corollary we obtain the following result.

Proposition 2.5 Arbitrary products of sober spaces are sober.

Proof: If (X_{i})_{i∈I} is an arbitrary family of sober spaces then, by the last proposition, we have

∏_{i∈I} X_{i} ≅ ∏_{i∈I} SpΩX_{i} ≅ Sp ( ⨂_{i∈I} ΩX_{i}).

Since SpA is sober for every locale A we are done.

38

Local compactness and continuity

When studying power constructions, we will also direct our attention to domains. Locally compact spaces and locally compact locales correspond to domains in a way we will explain in this chapter.

3.1 Locally compact spaces and domains

The notion of compactness is one of the most important concepts in general topology. A topological space X is compact if, for every family U of open subsets of X with X ⊆ ⋃ U, there are U_{1}, …, U_{n} ∈ U such that X ⊆ U_{1} ∪ ⋯ ∪ U_{n}. Note that we do not require a compact space to satisfy the Hausdorff separation property since most topologies we encounter are non-Hausdorff ones. A subset K of a topological space X is compact if it is compact as a subspace of X. Note that a subset K of a topological space X is compact iff it satisfies the following property: If U is a directed family of open subsets of X such that K ⊆ ⋃_{↑} U then there is U ∈ U with K ⊆ U. The properties of compact sets which we will need later are the following: Finite unions of compact sets are compact as are continuous images of compact sets. Note, however, that the intersection of two compact sets need not be compact. Furthermore, every finite set is compact. Another possibility to get a compact set from a given compact set is to take its saturation: For a subset S of a topological space X let ⋂{U ⊆ X | U open, U ⊇ S} be the saturation of S. For a T_{0}-space X, the saturation of any subset of X is the upper closure with respect to the specialization order. Therefore, we will denote the saturation of some subset S of a T_{0}-space by ↑ S. Note that arbitrary unions and intersections of saturated sets are saturated. A property of topological spaces that is closely associated with continuity of dcpo’s is the following. Definition. A topological space X is locally compact if for every x ∈ X there is a basis for the neighbourhood filter of x consisting of compact subsets of X. In other words, for every open neighbourhood U of some element x of X there is a compact neighbourhood of x that is contained in U. Hence local compactness guarantees the existence of arbitrarily small compact sets with non-empty interior. The full subcategory of Top whose objects are locally compact spaces will be denoted by LCompT. Every algebraic domain D is locally compact with respect to the Scott topology: A basis for the neighbourhood filter of any d ∈ D is given by the sets of the form ↑{c} where c is a compact element below d in D. Therefore a neighbourhood filter for some d ∈ D has a basis consisting of compact open sets. An arbitrary domain is still locally compact with respect to its Scott topology: If U is a Scott-open subset of a domain D then for all d ∈ U there is d′ ∈ U with d′ ≪ d. Consequently, ↑{d′} is a compact neighbourhood of d in U. Therefore we can restrict and co-restrict the functor T_{σ} that maps a dcpo D to the underlying set with the Scott topology to a functor from the category of domains to the category of locally compact T_{0}-spaces.

Another connection between local compactness and continuity is the following: For a topological space X consider the complete lattice ΩX of open subsets of X. We will study the way below relation in ΩX. If U and U′ are open subsets of X such that there is a compact set K with U ⊆ K ⊆ U′ then U ≪ U′ in ΩX. Indeed, if U is a directed subset of ΩX with U′ ⊆ ⋃_{↑} U then, by compactness of K, there is U′′ ∈ U with U ⊆ K ⊆ U′′. If X is a locally compact space then the reverse is also true: If U ≪ U′ in ΩX then there is a compact subset K of X with U ⊆ K ⊆ U′. For if U ≪ U′ in ΩX then, by local compactness of X, there is a compact neighbourhood K_{x} in U′ for every x ∈ U′.

Then U′ = ⋃_{↑}{int K_{x_{1}} ∪ ⋯ ∪ int K_{x_{n}} | x_{1}, …, x_{n} ∈ U}, and since U ≪ U′ we can find

x_{1}, …, x_{n} ∈ U′ with U ⊆ int K_{x_{1}} ∪ ⋯ ∪ int K_{x_{n}}. Consequently, K_{x_{1}} ∪ ⋯ ∪ K_{x_{n}} is a compact superset of U in U′.

Proposition 3.1 (Theorem II.1.7 in [GHK^{+}80]) If X is a locally compact space then ΩX is a continuous lattice. Proof: By the above considerations, int K ≪ U for any compact set K in U. As X is locally compact,

U = ⋃{int K | K compact subset of U} ⊆ ⋃ ↓ ↓{U} ⊆ U.

Since ΩX is a complete lattice, ↓ ↓{U} has to be directed and we are done.

For sober spaces, local compactness is even characterized by continuity of ΩX.

Proposition 3.2 (Theorem V.5.6 in [GHK^{+}80]) A sober space is locally compact iff its open set lattice is continuous.

3.2 Locally compact locales

Definition. A locale is locally compact if, as a complete lattice, it is continuous. The full subcategory of locally compact locales will be denoted by LCompL.

The following two propositions were first proved by Hofmann and Lawson [HL78].

Proposition 3.3 A locally compact locale is spatial.

Proof: Let a ≰ b in a locally compact locale A. First we show that there is a Scott-open filter in A containing a but not b. By continuity of A there is a′ ≪ a with a′ ≰ b. By the interpolation property we can find a sequence (d_{n})_{n∈ℕ} such that

a′ ≪ d_{n+1} ≪ d_{n} ≪ a for all n ∈ ℕ. Then F := ⋃_{n∈ℕ}↑{d_{n}} = ⋃_{n∈ℕ}↑ ↑{d_{n}} is a Scott-open

filter containing a but not b. By Zorn’s Lemma, there is a maximal Scott-open filter G containing F but not b. We prove that G is completely prime. Suppose there are c and c′ in A \ G with c ∨ c′ ∈ G. Then G′ := {d ∈ A | d ∨ c ∈ G} is a Scottopen filter containing G. Since c′ ∈ G′ \ G, we must have b ∈ G′ by maximality of G. This implies b ∨ c ∈ G. Similarly, we can show that b ∨ c′ ∈ G. Therefore, b ∨ (c ∧ c′) = (b ∨ c) ∧ (b ∨ c′) ∈ G. This implies b ∈ G and thus contradicts the definition of G. Consequently, G is a completely prime filter in A containing a but not b. Hence A has enough points and thus is spatial.

Proposition 3.4 (i) If X is a locally compact space then ΩX is a locally compact locale.

(ii) If A is a locally compact locale then SpA is a locally compact space.

Proof: (i) Follows from Proposition 3.2. (ii) We already know that SpA is a sober space with Sp Ω SpA ≅ SpA. Since A is spatial according to the last lemma, Ω SpA ≅ A. Hence SpA is a sober space with continuous open set lattice and therefore is locally compact by Proposition 3.2.

Furthermore, the full subcategory of locally compact locales is equivalent to the category LCompS of locally compact sober spaces. The equivalences between these categories are given by the restrictions and co-restrictions of Ω and Sp.

3.3 Products

Above, we have seen that the category of locales has finite products. The same is true for the categories of dcpo’s and topological spaces. Our study of power constructions for domains, locally compact spaces and locally compact locales makes it necessary to consider finite products for these categories. The following two propositions are well-known.

Proposition 3.5 The product of finitely many continuous dcpo’s is continuous. The category Dom of domains is closed under finite products formed in DCPO.

Proposition 3.6 Finite products of locally compact spaces are locally compact.

There are two ways of proving that the category LCompL of locally compact locales is also closed with respect to finite products. If A and B are locally compact locales, then they are spatial by Proposition 3.3. Hence, because of Proposition 3.4, SpA and SpB are locally compact spaces with A ≅ Ω SpA and B ≅ Ω SpB. It is sufficient to show that A ⊗ B is spatial, as, by Proposition 2.4, Sp(A ⊗ B) ≅ SpA × SpB which implies A ⊗ B ≅ Ω Sp(A ⊗ B) ≅ Ω(SpA × SpB). Since SpA × SpB is a locally compact space by the last proposition, this proves that Ω(SpA × SpB) is a locally compact locale. This is the way our Proposition 3.8 is usually shown (see e.g. II.2.13 in [Joh82a]). Since we are interested in the way below relation for A ⊗ B we will prove the desired result differently.

Lemma 3.7 Let A and B be locales. If a′ ≪ a in A and b′ ≪ b in B then a′ ⊗ b′ is way below a ⊗ b in A ⊗ B.

Proof: Let a ∈ A and b ∈ B and let further R be a directed subset of A ⊗ B with a ⊗ b ≤ ⋁^{↑}R. We set R′ := {a′ ⊗ b′ ∈ A ⊗ B | ∀a′′ ≪ a′, b′′ ≪ b′.∃r ∈ R . a′′ ⊗ b′′ ≤ r}. Obviously, if a′′ ≤ a′ and b′′ ≤ b′ for some a′ ⊗ b′ ∈ R′ then a′′ ⊗ b′′ ∈ R′. Let S be a subset of A and let b′ ∈ B such that s ⊗ b′ ∈ R′ for all s ∈ S. We show that this implies ⋁ S ⊗ b′ ∈ R′. If a′′ ≪ ⋁ S for some a′′ ∈ A then there are s_{1}, …, s_{n} ∈ X with a′′ ≤ s_{1} ∨ ⋯ ∨ s_{n}. If b′′ ≪ b′ then there are r_{1}, …, r_{n} ∈ R with s_{i} ⊗ b′′ ≤ r_{i} for all i ∈ {1, …, n}. Since R is directed, there is an upper bound r for r_{1}, …, r_{n} in R. Hence

a′′ ⊗ b′′ ≤ (s_{1} ∨ ⋯ ∨ s_{n}) ⊗ b′′ = s_{1} ⊗ b′′ ∨ ⋯ ∨ s_{n} ⊗ b′′ ≤ r_{1} ∨ ⋯ ∨ r_{n} ≤ r

which proves ⋁ S ⊗ b′ ∈ R′. Similarly we can show that a′ ⊗ t ∈ R′ for all elements t of some T ⊆ B implies a′ ⊗ ⋁ T ∈ R′. Therefore R′ is a ⊗-ideal as defined in Section 2.2. The join of R′ can be calculated as follows: If r ∈ R then r = ⋁_{i∈I} a_{i} ⊗ b_{i} for some generators a_{i} ⊗ b_{i} of A ⊗ B. Clearly, all a_{i} ⊗ b_{i} are in R′ which proves that the join of R′ is the join of R. Since R′ is a ⊗-ideal, this implies a ⊗ b ∈ R′. Hence if a′ ≪ a in A and b′ ≪ b in B then, by definition of R′, there is r′ ∈ R with a′ ⊗ b′ ≤ r′ which proves a′ ⊗ b′ ≪ a ⊗ b in A ⊗ B.

Proposition 3.8 The category of locally compact locales LCompL is a subcategory of Loc that is closed under finite products.

Proof: Obviously, the two-element locale 2 serves as terminal object in Loc. Let A and B be locally compact locales. We want to show that A ⊗ B is locally compact. Since A ⊗ B is a complete lattice, the set ↓ ↓{c} is directed for all c ∈ A ⊗ B. By the last lemma, {a′ ⊗ b′ | a′ ≪ a, b′ ≪ b} is a subset of ↓ ↓{a ⊗ b} for all a ⊗ b in A ⊗ B. Hence

a ⊗ b ≥ ⋁^{↑} ↓ ↓{a ⊗ b}
≥ ⋁{a′ ⊗ b′ | a′ ≪ a, b′ ≪ b}
= ⋁_{a′≪a} ⋁_{b′≪b} a′ ⊗ b′
= ( ⋁_{a′≪a} a′) ⊗ ( ⋁_{b′≪b} b′)
= a ⊗ b.

Since any element of A ⊗ B is the join of elements of the form a ⊗ b, we are done.

Monads and their Algebras

In this chapter we introduce a notion that is important for our studies of power constructions. A property that all these constructions have in common is that they can be understood as the functor part of a monad. Our main task will be to characterize the algebras for the arising monads. Besides introducing this concept and related notions we will prove some results for monads on categories that, in a way, behave like the category of posets. The sources for most of the material presented in the first two sections of this chapter are [Mac71] and [BW84].

4.1 Monads

Consider a pair of functors F: C −^{}D and G: D −^{}C where F is left adjoint to G. Then there are natural transformations η: id_{C} −^{}GF and ε: F G −^{}id_{D} such that the following diagrams of natural transformations commute:

[figure: G; ηG; GF G; id_{G}; G; Gε; F; F η; F GF; id_{F}; F; εF]

We can define another natural transformation, namely μ :=GεF: (GF)^{2} −^{}GF. It is not difficult to check that the following diagrams are commutative:

[figure: GF; GF η; (GF)^{2}; ηGF; GF; id_{GF}; id_{GF}; GF; μ; (GF)^{3}; μGF; GF; (GF)^{2}; GF μ; μ; GF; μ]

Generalizing this to a new notion we define a monad on a category C to be a triple (T, η, μ) where

(i) T is an endofunctor on C,

(ii) η: id_{C} −^{}T and μ: T^{2} −^{}T are natural transformations such that the following diagrams of natural transformations commute

[figure: T; T η; T^{2}; ηT; T; id_{T}; id_{T}; T; μ; T^{3}; μT; T^{2}; T^{2}; T μ; μ; T; μ]

We have already seen that if (F, G, η, ε) is an adjunction then the triple (GF, η, GεF) is a monad on C. Here is a concrete example of a monad. Define Wd: Set −^{}Set as follows: For any set S, let Wd S be the set of all words over the alphabet S. For every function f: S −^{}T we map the word s_{1} ⋯ s_{n} over S to the word f(s_{1}) ⋯ f(s_{n}) over T. This definition makes Wd: Set −^{}Set an endofunctor. A natural transformation η: id_{Set} −^{}Wd can be defined in the obvious way: An element s of S is mapped to the one-letter-word s over S. In order to get a monad we still need a natural transformation μ: Wd^{2} −^{}Wd. The elements of Wd^{2} S are words over the alphabet Wd S, that is words consisting of words over S which we can understand to be words over S and thus elements of Wd S if we remove whatever separates different words. We shall employ the monad (Wd, η, μ) again in order to illustrate further concepts.

Let us introduce a concept stating when two monads on a category are ‘the same’: Definition. We say that two monads (T, η, μ) and (T′, η′, μ′) on a category C are equivalent if there is a natural isomorphism ι: T −^{}T′ such that η′ = ι • η and such that μ′ • (ι ∘ ι) = ι • μ. Here • denotes the vertical composite while ∘ means the horizontal composite of natural transformations.

Proposition 4.1 Let (T, η, μ) be a monad on C. Further, let T′ be an endofunctor on C such that there is a natural isomorphism ι: T −^{}T′. If η′: id_{C} −^{}T′ and μ: (T′)^{2} −^{}T′ are natural transformations such that η′ = ι • η and μ′ • (ι ∘ ι) = ι • μ then (T′, η′, μ′) is a monad on C which is equivalent to (T, η, μ).

Proof: We just have to show that (T′, η′, μ′) is a monad. Since ι is a natural isomorphism, we have

μ′_{C} = ι_{C} ∘ μ_{C} ∘ T ι_{C}^{−1} ∘ ι_{T′C}^{−1} = ι_{C} ∘ μ_{C} ∘ ι_{T C}^{−1} ∘ T′ι_{C}^{−1}

for any C-object C. Let C be an arbitrary C-object. Then

μ′_{C} ∘ T′η′_{C} = ι_{C} ∘ μ_{C} ∘ ι_{T C}^{−1} ∘ T′ι_{C}^{−1} ∘ T′(ι_{C} ∘ η_{C})
= ι_{C} ∘ μ_{C} ∘ ι_{T C}^{−1} ∘ T′(ι_{C}^{−1} ∘ ι_{C}) ∘ T′η_{C}
= ι_{C} ∘ μ_{C} ∘ T η_{C} ∘ ι_{C}^{−1}  (Naturality of ι^{−1}: T′ −^{}T)
= ι_{C} ∘ ι_{C}^{−1}  (μ_{C} ∘ T η_{C} = id_{T C})
= id_{T′C} .

Further,

μ′_{C} ∘ η′_{T′C} = ι_{C} ∘ μ_{C} ∘ T ι_{C}^{−1} ∘ ι_{T′C}^{−1} ∘ ι_{T′C} ∘ η_{T′C}
= ι_{C} ∘ μ_{C} ∘ T ι_{C}^{−1} ∘ η_{T′C}
= ι_{C} ∘ μ_{C} ∘ η_{T C} ∘ ι_{C}^{−1}  (Naturality of η: id_{C} −^{}T)
= ι_{C} ∘ ι_{C}^{−1}  (μ_{C} ∘ η_{T C} = id_{T C})
= id_{T′C} .

Finally,

μ′_{C} ∘ T′μ′_{C}
= ι_{C} ∘ μ_{C} ∘ ι_{T C}^{−1} ∘ T′ι_{C}^{−1} ∘ T′(ι_{C} ∘ μ_{C} ∘ T ι_{C}^{−1} ∘ ι_{T′C}^{−1})
= ι_{C} ∘ μ_{C} ∘ ι_{T C}^{−1} ∘ T′μ_{C} ∘ T′T ι_{C}^{−1} ∘ T′ι_{T′C}^{−1}
= ι_{C} ∘ μ_{C} ∘ T μ_{C} ∘ ι_{T^{2}C}^{−1} ∘ T′T ι_{C}^{−1} ∘ T′ι_{T′C}^{−1}  (Naturality of ι^{−1}: T′ −^{}T)
= ι_{C} ∘ μ_{C} ∘ μ_{T C} ∘ ι_{T^{2}C}^{−1} ∘ T′T ι_{C}^{−1} ∘ T′ι_{T′C}^{−1}  (μ_{C} ∘ T μ_{C} = μ_{C} ∘ μ_{T C})
= ι_{C} ∘ μ_{C} ∘ μ_{T C} ∘ T^{2}ι_{C}^{−1} ∘ ι_{T T′C}^{−1} ∘ T′ι_{T′C}^{−1}  (Naturality of ι^{−1}: T′ −^{}T)
= ι_{C} ∘ μ_{C} ∘ T ι_{C}^{−1} ∘ μ_{T′C} ∘ ι_{T T′C}^{−1} ∘ T′ι_{T′C}^{−1}  (Naturality of μ: T^{2} −^{}T)
= ι_{C} ∘ μ_{C} ∘ T ι_{C}^{−1} ∘ ι_{T′C}^{−1} ∘ ι_{T′C} ∘ μ_{T′C} ∘ ι_{T T′C}^{−1} ∘ T′ι_{T′C}^{−1}
= μ′_{C} ∘ μ′_{T′C}.

Hence, given a monad (T, η, μ) and an endofunctor T′ on a category C that is naturally isomorphic to T we can define a monad for T′ that is equivalent to that for T.

4.2 Algebras

Every adjunction defines a monad. A natural question is whether every monad arises like that. Suppose we are given a monad (T, η, μ) on C. Let us consider all adjunctions (F, G, η, ε) such that (GF, η, GεF) = (T, η, μ). Note that the unit for all these adjunctions is η while the co-unit might vary. We take these adjunctions to be the objects of a category which we denote by Ad(T). Let (F, G, η, ε) and (F′, G′, η, ε′) be two objects of Ad(T) where F: C −^{}D and F′: C −^{}D′. A morphism from (F, G, η, ε) to (F′, G′, η, ε′) is given by a functor M: D −^{}D′ such that the two diagrams

[figure: C; F; D; F′; D′; M; C; G; D; G′; D′; M]

commute. Note that for every such functor M an equation relating the natural transformations ε and ε′ holds, namely

Mε = ε′M.

We will now describe the terminal object of Ad(T). Definition. We say that a pair (C, ξ) is an algebra for a monad (T, η, μ) on a category C or a T -algebra if ξ: T C −^{}C is a morphism in C such that the following diagrams commute:

[figure: C; η_{C}; T C; id_{C}; C; ξ; T^{2}C; μ_{C}; T C; T C; T ξ; ξ; C; ξ]

In this case we say that ξ is a structure map for C. The category T -Alg of T -algebras (or Eilenberg-Moore category) is defined as follows: Its objects are all T -algebras. A morphism from (C, ξ) to (C′, ξ′) in T -Alg is given by a morphism f: C −^{}C′ in C such that the following diagram is commutative.

[figure: T C; ξ; C; T C′; T f; ξ′; C′; f]

This definition is due to S. Eilenberg and J. C. Moore [EM65] who also were the first to prove that there is an adjunction, say (F^{A}, G^{A}, η, μ^{A}), corresponding to the category of T -algebras that is terminal in Ad(T): F^{A}: C −^{}T -Alg is obtained by setting F^{A}C := (T C, μ_{C}) and F^{A}f := T f. The functor G^{A}: T -Alg −^{}C results from defining G^{A}(C, ξ) := C and G^{A}f := f. In particular, for any monad (T, η, μ) there is indeed an adjunction (F, G, η, ε) with (T, η, μ) = (GF, η, Gεη). Note the similarity of the diagrams defining a monad with those defining an algebra for the monad. In the above example, the category of Wd-algebras is (isomorphic to) the category of monoids. The functor Wd can be decomposed into the functor that maps a set S to the free monoid over S and the forgetful functor to Set. An adjunction (F, G, η, ε), where the target of F is D, is called monadic if the functor M: D −^{}T -Alg that defines the unique morphism from (F, G, η, ε) to (F^{A}, G^{A}, η, ε^{A}) is an equivalence of the categories D and T -Alg. The category of algebras for a monad (T, η, μ) helps to determine in what respect the functor T can be understood to be free. The fact that in the above example the category of Wd-algebras is the category of all monoids means that Wd gives the free monoid over Set. We can also understand the category of T -algebras to be the largest category D for which there is an adjunction (F, G, η, ε) defining the given monad (T, η, μ) such that F: C −^{}D. Since G^{A} is a forgetful functor (that is F^{A} is more or less equal to T), this gives the largest category D for which T can be understood to assign the free D-object to every C-object. To see this, note the following: The universal property for the adjunction (F^{A}, G^{A}, η, ε^{A}) yields that for all C-morphisms f: C −^{}G^{A}A, where A is an object of T -Alg, there is a unique f̂: F^{A}C −^{}A in T -Alg such that G^{A}f̂∘ η_{C} = f. Let (F, G, η, ε) be any adjunction in Ad(T) where the target of F is D. Every object of the form GD where D is a D-object is of the form G^{A}A for the T -algebra A := MD = (GD, G ε_{D}). Hence the ‘for all’ in the universal property for the adjunction (F^{A}, G^{A}, η, ε^{A}) has the greatest possible domain of variation. We have just seen that the algebras of the form (T C, μ_{C}), where C is a C-object, satisfy a universal property. This is the reason that they are also called free algebras for the monad. They are the objects of the subcategory of T -Alg which we obtain if we take the image of F^{A} in T -Alg. For this category we can find an adjunction which is initial in Ad(T). Let us now investigate algebras for equivalent monads.

Proposition 4.2 Let (T, η, μ) and (T′, η′, μ′) be equivalent monads on the category C. Then the category of T -algebras is isomorphic to that of T′-algebras.

Proof: Let the natural isomorphism from T to T′ be ι. Let (C, ξ′) be a T′-algebra. We claim that (C, ξ′ ∘ ι_{C}) is a T -algebra. We calculate (ξ′ ∘ ι_{C}) ∘ η_{C} = ξ′ ∘ η′_{C} = id_{C}. Furthermore,

(ξ′ ∘ ι_{C}) ∘ T (ξ′ ∘ ι_{C}) = ξ′ ∘ ι_{C} ∘ T ξ′ ∘ T ι_{C}
= ξ′ ∘ T′ξ′ ∘ ι_{T′C} ∘ T ι_{C}  (Naturality of ι: T −^{}T′)
= ξ′ ∘ μ′_{C} ∘ ι_{T′C} ∘ T ι_{C}  (ξ′ ∘ T ξ′ = ξ′ ∘ μ′_{C})
= ξ′ ∘ ι_{C} ∘ μ_{C}  (ι • μ = μ′ • (ι ∘ ι))
= ξ′ ∘ μ′_{C}.

If f: (C, ξ′) −^{}(D, ζ′) is a morphism of T′-algebras then it is also a morphism of T - algebras from (C, ξ′ ∘ ι_{C}) to (D, ζ′ ∘ ι_{D}) since f ∘ ξ′ ∘ ι_{C} = ζ′ ∘ T′f ∘ ι_{C} = ζ′ ∘ ι_{D} ∘ T f.

4.3 Monads on the category of posets

Let Pos denote the category of posets and order-preserving functions. Then the Pos-morphisms P −^{}Q also form a poset: They can be ordered pointwise. Assume that C is a subcategory of Pos and that T is an endofunctor on C which is order-preserving on morphisms. This means if f, g: C −^{}C′ in C such that f ≤ g with respect to the pointwise order then T f ≤ T g. Suppose that T is the functor part of a monad (T, η, μ). What can we say about T -algebras in this setting? By definition, every T -algebra C is a retract of T C in the category of posets and order-preserving functions. Hence if all posets of the form T C are ∧-semilattices, for example, then all T -algebras must be ∧-semilattices as well. We will study, among others, monads on DCPO and Top. Since these categories can be considered subcategories of Pos, the following three lemmata can be applied to them. For most monads we consider we can understand the corresponding functor to map any object to a subset of its power set; the order is either subset inclusion or reverse inclusion.

These two cases are dual, therefore we may concentrate on the first of them. Let P_{L}C denote the subposet of PC consisting of all lower subsets of C, that is those subsets S of C where S = ↓ S, ordered by subset inclusion. The set of all upper subsets of C ordered by reverse inclusion will be denoted by P_{U}C.

Lemma 4.3 Let (T, η, μ) be a monad on a subcategory C of Pos. Furthermore, suppose that T C is a subset of P_{L}C and that η_{C}(x) = ↓{x} for all x ∈ C and all C-objects C.

(i) If (C, ξ) is a T -algebra then ξ is the lower adjoint for η_{C} and ξ(S) = ⋁ S holds for all S ∈ T C. Hence in C the joins of all sets belonging to T C exist.

(ii) Let (C, ξ) and (C′, ξ′) be T -algebras. Additionally, let f: C −^{}C′ be a Cmorphism such that ⋁ T f(S) = ⋁ f(S) for all S ∈ T C. Then f is a morphism of the T -algebras (C, ξ) and (C′, ξ′) iff it preserves joins of all sets belonging to T C.

Proof: (i) Let S ∈ T C where (C, ξ) is a T -algebra. As ↓{s} ⊆ S for all s ∈ S we have s = ξ(η_{C}(s)) = ξ(↓{s}) ≤ ξ(S). Thus ξ(S) is an upper bound for S. If, on the other hand, p is an upper bound for S then we have S ⊆ ↓{p} implying ξ(S) ≤ ξ(↓{p}) = ξ(η_{C}(p)) = p. Hence ξ(S) is the least upper bound for S. For every T -algebra (C, ξ) we have ξ ∘ η_{C} = id_{C} by definition. For all S ∈ T C we know that η_{C}(ξ(S)) = η_{C}(⋁ S) = ↓{⋁ S} ⊇ S which proves η_{C} ∘ ξ ≥ id_{T C}. This implies that ξ is the lower adjoint for η_{C}. (ii) According to (i), the structure maps for C and C′ have to be the functions mapping subsets of T C respective T C′ to their joins. If f: C −^{}C′ is a morphism of T -algebras then ⋁ f(S) = ⋁ T f(S) = ξ′(T f(S)) = f(ξ(S)) = f(⋁ S) for all S ∈ T C. If f preserves the joins of all sets belonging to T C then f(ξ(S)) = f(⋁ S) = ⋁ f(S) = ⋁ T f(S) = ξ′(T f(S)).

Lemma 4.4 Let (T, η, μ) be a monad on a subcategory C of Pos. Furthermore, suppose that T C is a subset of P_{U}C and that η_{C}(x) = ↑{x} for all x ∈ C and all C-objects C.

(i) If (C, ξ) is a T -algebra then ξ is the upper adjoint for η_{C} and ξ(S) = ⋀ S holds for all S ∈ T C. Hence in C the meets of all sets belonging to T C exist.

(ii) Let C and C′ be T -algebras and let f: C −^{}C′ be a C-morphism such that ⋀ f(S) = ⋀ T f(S) for all S ∈ T C. Then f is a morphism of T -algebras iff it preserves the meets of all sets belonging to T C.

Proof: Dual to that for Lemma 4.3.

So in the category of posets we can show that in certain cases the structure maps have to be adjoints for the components of the natural transformation η. We were able to deduce this property by only making use of the triangular diagram for algebras. It turns out that in these cases the commutativity of the rectangular diagram already follows from the fact that the triangular diagram commutes.

Lemma 4.5 Let (T, η, μ) be a monad on a subcategory C of Pos such that T is order-preserving on C-morphisms and such that the structure map for every C-object C necessarily is the lower (upper) adjoint for η_{C}. Furthermore, let ξ: T C −^{}C be the lower (upper) adjoint for η_{C}. If ξ ∘ η_{C} = id_{C} then (C, ξ) is an algebra for the monad.

Proof: We shall only prove the assertion for the case of the lower adjoints since the other part can be shown analogously. For that it remains to show that for ξ: T C −^{}C with ξ ∘ η_{C} = id_{C} the following diagram commutes:

[figure: T^{2}C; μ_{C}; T C; T C; T ξ; ξ; C; ξ]

As μ_{C} is a structure map for T C, it has to be the lower adjoint for η_{T C} by assumption. By Lemma 1.1 this implies that ξ ∘ μ_{C} is the lower adjoint for η_{T C} ∘ η_{C}. But ξ ∘ T ξ is the lower adjoint for T η_{C} ∘ η_{C}:

(ξ ∘ T ξ) ∘ (T η_{C} ∘ η_{C}) = ξ ∘ T (ξ ∘ η_{C}) ∘ η_{C}
= ξ ∘ T id_{C} ∘ η_{C}  (ξ ∘ η_{C} = id_{C})
= ξ ∘ id_{T C} ∘ η_{C}
= id_{C}  (ξ ∘ η_{C} = id_{C})

as well as

(T η_{C} ∘ η_{C}) ∘ (ξ ∘ T ξ) = T η_{C} ∘(η_{C} ∘ ξ) ∘ T ξ
≥ T η_{C} ∘ id_{T C} ∘ T ξ  (η_{C} ∘ ξ ≥ id_{T C})
= T (η_{C} ∘ ξ)
≥ T id_{T C}  (Monotonicity of T)
= id_{T^{2}C} .

Since η is a natural transformation id_{C} −^{}T, the diagram

[figure: C; η_{C}; T C; T C; η_{C}; η_{T C}; T^{2}C; T η_{C}]

commutes. Hence both, ξ ∘ μ_{C} and ξ ∘ T ξ, are lower adjoints for T η_{C} ∘ η_{C} = η_{T C} ∘ η_{C} and thus must coincide.

4.4 Monads for O-categories

The proof of the last lemma also works in a more general setting: This proof mainly uses the fact that the hom-sets in the category of posets are themselves posets. Hence we can define a class of categories for which the above result is still true.

Definition. An order-enriched category, or, for short, an O-category is a category where the hom-sets are partially ordered sets such that composition is orderpreserving. An O-functor is a functor between O-categories that preserves the order on hom-sets. This notion is due to M. Wand [Wan79] and has been employed in [HP79]. If C is an O-category and f, g: C −^{}C′ as well as f′, g′: C′ −^{}C′′ are C-morphisms such that f ≤ g and f′ ≤ g′ then f′ ∘ f ≤ g′ ∘ g. For O-categories we still have the concept of an adjoint pair. If C f^{−^{}} ; − ; − ; − g C′ are morphisms in an O-category C then (f, g) is an adjoint pair if f ∘ g ≤ id_{C′} and g ∘ f ≥ id_{C}. In this case we say that f is a lower adjoint for g and that g is an upper adjoint for f. Obviously, DCPO is an O-category. Since T_{0}-spaces carry an order that is preserved by continuous functions, we can order hom-sets in Top pointwise and thus make Top an O-category. The category Loc of locales can also be understood to be an Ocategory. For continuous functions f, g: A −^{}B in Loc we set f ≤ g iff f′ ≤ g′ holds for the corresponding frame homomorphisms. We choose this order on hom-sets of Loc because it makes Sp and Ω O-functors (compare also Lemmata 2.1 and 2.2). Note that with respect to this ordering, (f, g) is an adjoint pair for A f^{−^{}} ; − ; − ; − g B in Loc iff (g′, f′) is an adjoint pair in Frm: g ∘ f ≤ id_{A} iff (g ∘ f)′ ≤ id′_{A} iff f′ ∘ g′ ≤ id′_{A}. The above proof of Lemma 4.5 works in arbitrary O-categories if we know that adjoints are unique. We say that an O-category C has unique adjoints if the following two conditions are satisfied:

(i) If f, f′: C −^{}C′ both are lower adjoints for some g: C′ −^{}C then f = f′.

(ii) If g, g′: C′ −^{}C both are upper adjoints for some f: C −^{}C′ then g = g′.

The three O-categories mentioned above, that is DCPO, Top and Loc, all have unique adjoints.

Lemma 4.6 Let C be an O-category with unique adjoints and let (T, η, μ) be a monad on C where T is an O-functor. Further, suppose that the structure map for every C-object C necessarily is the lower (upper) adjoint for η_{C}. If the lower (upper) adjoint ξ: T C −^{}C for η_{C} satisfies ξ ∘ η_{C} = id_{C} then (C, ξ) is a T -algebra.

Proof: Exactly as that for Lemma 4.5.

The introduction of O-categories saves us the introduction of co-monads. Indeed, in order to get a result like Lemma 4.5 for the category of locales we have to pass to the concrete category of frames. But a monad on Loc yields a co-monad on Frm and so Lemma 4.5 does not apply to this case. In [Koc92b], A. Kock studies what he calls 2-doctrines. This is a concept similar to that of a monad for 2-categories. He then shows a result similar to the last lemma. He also gives a constructive theory for the lift monad on posets [Koc92a] which can be understood to be of the type of the monads considered in Lemma 4.5.

Generalized semilattices

5.1 Semilattices in arbitrary categories

Until now, we have only considered semilattices in the category Pos of posets and monotone functions. Depending on whether the posets in question have binary joins or binary meets we speak of ∨- or ∧-semilattices. We can carry the concept of semilattices to other categories. For that, let us look at the algebraic characterization of semilattices by means of equations. Let P be a ∨-semilattice and let ∨: P × P −^{}P denote the binary join function. This function has the following properties:

(i) It is idempotent, that is ∨⟨p, p⟩ = p ∨ p = p for all p ∈ P.

(ii) It is symmetric, that is ∨⟨p, p′⟩ = p ∨ p′ = p′ ∨ p = ∨⟨p′, p⟩ for all p, p′ ∈ P.

(iii) It is associative, that is ∨⟨∨⟨p, p′⟩, p′′⟩ = (p ∨ p′) ∨ p′′ = p ∨ (p′ ∨ p′′) = ∨⟨p, ∨⟨p′, p′′⟩⟩ for all p, p′, p′′ ∈ P.

These properties are also satisfied by the binary meet function on a ∧-semilattice. These conditions are actually enough to characterize semilattices: If S is a set and v: S × S −^{}S an idempotent, symmetric and associative function, then s ≤ s′ iff v⟨s, s′⟩ = s′ defines an order on S such that v is the binary join with respect to that order. Note that we could have made S a ∧-semilattice if we had chosen to define the dual order s ≤ s′ iff v⟨s, s′⟩ = s. The above properties make sense in an arbitrary category with binary products. Idempotency for a morphism v: C × C −^{}C, for example, means that v ∘Δ_{C} = id_{C}. Here Δ_{C}: C −^{}C × C denotes the diagonal from C to the product C × C, that is Δ_{C} = ⟨id_{C}, id_{C}⟩. Definition. A semilattice structure for an object C of a category C with binary products is a C-morphism v: C × C −^{}C that renders the following diagrams commutative:

[figure: C; Δ_{C}; C × C; id_{C}; C; v; C × C; ⟨π_{2}, π_{1}⟩; C × C; v; C; v; C × C × C; id_{C} ×v; C × C; C × C; v × id_{C}; v; C; v]
Idempotency  Symmetry  Associativity

In this case we also say that (C, v) is a semilattice in C. Note that we do not distinguish between (C × C) × C and C × (C × C) as the two are naturally isomorphic. Here are some properties of the morphisms from the above diagram that we are going to need later: In a category C with binary products there is an endofunctor on C defined by C −^{}C × C and f −^{}f × f. The diagonal on objects of C is natural from the identity functor on C to this functor, that is for all morphisms f: C −^{}D in C the following diagram commutes:

[figure: C; Δ_{C}; C × C; D; f; Δ_{D}; D × D; f × f]

We can also understand ⟨π_{2}, π_{1}⟩ as a natural transformation: For the category C consider the functor from C × C to C × C that interchanges factors. Then ⟨π_{2}, π_{1}⟩ is natural from the identity on C × C to this functor, that is the following diagram is commutative for all morphisms f × g: C × D −^{}C′ × D′ in C × C:

[figure: C × D; ⟨π_{2}, π_{1}⟩; D × C; C′ × D′; f × g; ⟨π_{2}, π_{1}⟩; D′ × C′; g × f]

Note further that all ⟨π_{2}, π_{1}⟩ are isomorphisms and their own inverses. Hence the family of all ⟨π_{2}, π_{1}⟩ is a natural isomorphism between the two functors from above. Definition. We say that a C-morphism f: C −^{}D, where (C, v) and (D, w) are semilattices in the category C, is a morphism of semilattices if the following diagram commutes:

[figure: C × C; v; C; D × D; f × f; w; D; f]

Let C^{SLat} denote the category of semilattices with semilattice morphisms as a subcategory of C. Let ℕ denote the set of positive integers. From the ‘binary’ semilattice structure v: C × C −^{}C we can get an ‘n-ary’ semilattice structure v_{n} from the n-fold product of C to C for any n ∈ ℕ by the following inductive definition:

\begin{matrix}v_{1} := id_{C}, \\ v_{n+1} := v ∘ (v_{n} × id_{C}) for n ≥ 1. \end{matrix}

By associativity of v we get v_{n+1} = v∘(id_{C} ×v_{n}). We can also show that v_{n}∘Δ_{C}^{n} = id_{C} for all n ∈ ℕ, where Δ_{C}^{n} is the n-ary diagonal ⟨id_{C}, …, id_{C}⟩: C −^{}C × ⋯ × C: For n = 1 the claim is trivial. By induction we get

v_{n+1} ∘ Δ_{C}^{n+1} = v ∘ (v_{n} × id_{C}) ∘ ⟨id_{C}, …, id_{C}⟩
= v ∘ ⟨v_{n} ∘ ⟨id_{C}, …, id_{C}⟩, id_{C}⟩
= v ∘ ⟨id_{C}, id_{C}⟩
= v ∘ Δ_{C}
= id_{C} .

Furthermore, v_{n} is invariant under permutations of factors in C × ⋯ × C. Also, if f: C −^{}D is a morphism of the semilattices (C, v) and (D, w) then the nary product f × ⋯ × f: C × ⋯ × C −^{}D × ⋯ × D is a morphism of the ‘n-ary’ semilattices (C, v_{n}) and (D, w_{n}), that is w_{n} ∘ (f × ⋯ × f) = f ∘ v_{n}. The proof is by induction on n:

w_{n+1} ∘ (f × ⋯ × f) = w ∘ (w_{n} ∘ id_{C}) ∘ (f × ⋯ × f)
= w ∘ ((w_{n} ∘ (f × ⋯ × f)) × f)
= w ∘ (f × f) ∘ (v_{n} × id_{C})
= f ∘ v ∘ (v_{n} × id_{C})
= f ∘ v_{n+1}.

5.2 Inflationary and deflationary semilattices

In the category Set we may think of semilattices as either ∨- or ∧-semilattices since we may introduce an appropriate order on such a set. In the category of posets, however, every object already carries an order. Hence there are two special cases for a semilattice (P, v) in this category: The order on P that we obtain if we interpret v as join or its opposite might coincide with the intrinsic order on P. In the first case, P is a ∨-semilattice as defined in Section 1.1 and the binary join is given by v. In the second case, P is a ∧-semilattice and the binary meet is given by v. A semilattice structure v: P × P −^{}P makes a poset P a ∨-semilattice (with respect to the intrinsic order on P) iff Δ_{P} ∘v ≥ id_{P ×P}: Obviously, this condition is satisfied if v is the binary join for some poset P. On the other hand, given a poset P with semilattice structure v: P × P −^{}P such that Δ_{P} ∘v ≥ id_{P ×P} we know that v must be the lower adjoint for Δ_{P} because v ∘ Δ_{P} = id_{P} holds by idempotency of v. Hence ⟨p, p′⟩ ≤ Δ_{P} (v⟨p, p′⟩) = ⟨v⟨p, p′⟩, v⟨p, p′⟩⟩ which proves that v⟨p, p′⟩ is an upper bound for p and p′. For all p, p′, p′′ ∈ P we must have v⟨p, p′⟩ ≤ p′′ iff ⟨p, p′⟩ ≤ Δ_{P} (p′′) = ⟨p′′, p′′⟩ iff p, p′ ≤ p′′. Therefore v⟨p, p′⟩ is the least upper bound for p and p′. Analogously, one can show that v makes P a ∧-semilattice iff Δ_{P} ∘v ≤ id_{P ×P} iff v is the upper adjoint for Δ_{P}. So all we need in order to introduce an analogue of the concepts of ∨- and ∧- semilattice to more general settings is a partial order on the hom-sets. Definition. A semilattice (C, v) in an O-category C is inflationary iff (v, Δ_{C}) is an adjoint pair and deflationary iff this is true for (Δ_{C}, v). If C is an arbitrary Ocategory then we denote by C^{∨} the category of inflationary semilattices in C. The name for the category of deflationary semilattices in C is C^{∧}.

5.3 Examples of generalized semilattices

In the category of posets, an inflationary semilattice is a ∨-semilattice and a semilattice morphism between inflationary semilattices is a function that preserves binary joins. Analogous statements hold for deflationary semilattices.

5.3.1 Dcpo’s

In the category DCPO of dcpo’s with Scott-continuous functions we order morphisms pointwise. An inflationary semilattice in DCPO is nothing but a poset that has all non-empty joins: A dcpo D has all directed joins. If it additionally is an inflationary semilattice then it has binary joins as well. Recall that we refer to these posets as up-complete ∨-semilattices. A homomorphism of up-complete ∨-semilattices is a function that preserves non-empty joins. A semilattice homomorphism for inflationary semilattices in DCPO is a morphism of up-complete ∨-semilattices since a function preserves all nonempty joins iff it preserves binary and directed joins. The condition that the semilattice structure ∨: D × D −^{}D be a morphism of the category does not mean anything in this case for it is equivalent to finite joins distributing over directed ones. A deflationary semilattice D in the category of dcpo’s is what we call a ∧-continuous ∧-semilattice. By this we mean a dcpo that has binary meets such that directed joins distribute over binary meets, that is d ∧ ⋁^{↑}S = ⋁_{s∈S}^{↑}(d ∧ s) for all d ∈ D and all directed subsets S of D. Such a poset is also referred to as a preframe in [JV91]. We will not use this name, however, because it does not fit with how we wish to think about it, namely as a dcpo with an additional structure. A morphism of deflationary semilattices in DCPO is a function that preserves binary meets and directed joins.

5.3.2 Topological spaces

In the category Top of T_{0}-spaces, a semilattice is also referred to as topological semilattice. Recall that T_{0}-spaces carry an order, namely the specialization order, and that every continuous function preserves that order. To make this category an O-category, we also take the pointwise order on the hom-sets. An inflationary topological semilattice has binary joins with respect to the specialization order and the binary join function is continuous. Note that an inflationary topological semilattice with a sober topology is an up-complete ∨-semilattice because any sober space has directed joins. A morphism of inflationary topological semilattices is a continuous function that preserves binary joins. A deflationary topological semilattice has binary meets with respect to the specialization order and the binary meet function has to be continuous. Considered a poset, a deflationary topological semilattice with sober topology is a ∧-continuous ∧-semilattice. A morphism of deflationary topological semilattices is a continuous function that preserves binary meets.

5.3.3 Locales

Finally, let us have a look at semilattices in the category of locales. We call them localic semilattices after [Joh85].

Lemma 5.1 If A is a localic semilattice with semilattice structure v: A ⊗ A −^{}A then v induces a semilattice structure on SpA making it a topological semilattice.

Proof: By Lemma 2.4, Sp: Loc −^{}Top is a functor preserving binary products. Hence v: A ⊗ A −^{}A induces a continuous function from SpA × SpA to SpA. From the same lemma we get that this function satisfies the semilattice equations.

We give another motivation for the order on hom-sets of Loc we introduced above to make the category of locales an O-category. Because of the last lemma we know that every semilattice structure on A induces a semilattice structure on SpA. So the question is whether the topological semilattice corresponding to an inflationary localic semilattice is inflationary as well. Recall that for continuous functions f, g: A −^{}B between locales we have f ≤ g iff f′ ≤ g′ holds for the corresponding frame homomorphisms with respect to the pointwise order.

Lemma 5.2 If v: A ⊗ A −^{}A is a semilattice structure making the locale A an inflationary (a deflationary) semilattice then the induced structure makes SpA an inflationary (a deflationary) topological semilattice.

Proof: We will only prove the assertion for the case of inflationary semilattices. If v: A ⊗ A −^{}A makes A an inflationary semilattice then Δ_{A} is the upper adjoint for v which means that Δ′_{A} is the lower adjoint for v′. By Lemma 2.2 this implies that SpΔ_{A} is the upper adjoint for Spv. Since Sp preserves products by Proposition 2.4, the semilattice structure induced by Spv is the upper adjoint for Δ_{Sp A} and therefore we are done.

Note that if (X, v) is a topological semilattice then ΩX need not be a localic semilattice since Ωv is a continuous function from Ω(X × X) to ΩX and therefore does not necessarily define a function from ΩX ⊗ ΩX to ΩX. If (A, v) is an inflationary localic semilattice then the frame homomorphism corresponding to v must be the upper adjoint for Δ′_{A}, the frame homomorphism corresponding to the diagonal on A. Using the way the product A⊗A in Loc is constructed via generators and relations, we can explicitly say what v has to look like: For any a ∈ A we must have

v′(a) := ⋁{b_{1} ⊗ b_{2} ∈ A ⊗ A | b_{1} ∧ b_{2} ≤ a}.

Let us now consider semilattice structures for some deflationary semilattice (A, v) in Loc. For any a ∈ A and any family (b_{1}^{i} ⊗ b_{2}^{i})_{i∈I} of generators of A ⊗ A we have

v′(a) ≤ ⋁_{i∈I} b_{1}^{i} ⊗ b_{2}^{i} iff a ≤ Δ′_{A}( ⋁_{i∈I} b_{1}^{i} ⊗ b_{2}^{i}) = ⋁_{i∈I} b_{1}^{i} ∧ b_{2}^{i}. So we get

\begin{aligned}v′(a) & := & ⋀\{ ⋁_{i∈I} b_{1}^{i} ⊗ b_{2}^{i} | a ≤ ⋁_{i∈I} b_{1}^{i} ∧ b_{2}^{i}\} \\ & = & ⋁\{b_{1} ⊗ b_{2} ∈ A ⊗ A | a ≤ ⋁_{i∈I} b_{1}^{i} ⊗ b_{2}^{i} ⟹ b_{1} ⊗ b_{2} ≤ ⋁_{i∈I} b_{1}^{i} ⊗ b_{2}^{i}\}. \end{aligned}

5.4 Retracts of generalized semilattices

We have already seen several properties that are inherited by retracts. In this section we will explore what can be said about retracts of generalized semilattices. The category of posets, for example, is a category where a retract of an inflationary or a deflationary semilattice carries a corresponding structure. Let C be a category with binary products and let C be an object of C with a semilattice structure, say v: C × C −^{}C. What can we say about a C-object D that is a retract of C via D e^{−^{}} ; − ; − ; − r C? If, from this data, we can define a semilattice structure on D at all then it should be r ∘ v ∘ (e × e): D × D −^{}D. For this is the only morphism of ‘the right type’ that we can build from the given data in a natural way:

[figure: D × D; e × e; C × C; D; r; C; v]

In fact, this is the semilattice structure we used to prove that in Pos a retract of an inflationary semilattice is an inflationary semilattice. So let us see what we can show for this structure. First for idempotency. We calculate

r ∘ v ∘ (e × e) ∘ Δ_{C} = r ∘ v ∘ Δ_{C} ∘e  (Naturality of Δ)
= r ∘ e  (Idempotency of v)
= id_{D} .

Next for symmetry. We get

r ∘ v ∘ (e × e) ∘ ⟨π_{2}, π_{1}⟩ = r ∘ v ∘ ⟨π_{2}, π_{1}⟩ ∘ (e × e)  (Naturality of ⟨π_{2}, π_{1}⟩)
= r ∘ v ∘ (e × e)  (Symmetry of v).

So this worked fine so far. But problems arise when we try to prove associativity

(r ∘ v ∘ (e × e)) ∘ ((r ∘ v ∘ (e × e)) × id_{D} )
= (r ∘ v ∘ (e × e)) ∘ ((r ∘ v ∘ (e × e)) × (r ∘ id_{C} ∘e))
= r ∘ v ∘ (e × e) ∘ (r × r) ∘ (v × id_{C}) ∘ (e × e × e)

for here we get stuck. To see that in general it is not possible to go on, consider the following example: Let S := {a, b, c} and let ⊥, l, r, ⊤ be the elements of the four element lattice L that is not a chain. We consider a ∨-semilattice L with semilattice structure ∨_{L}: L × L −^{}L. We define an embedding of S into L by setting e(a) := ⊥, e(b) := l and e(c) := r. In order to define a retraction s: L −^{}S we just have to specify what the image of ⊤ should be. We set s(⊤) := a thus making S a retract of L in Set. If we try to define a semilattice structure on S as described above, namely by setting ∨_{S} := s∘∨_{L}∘(e×e), then we end up with a function from S × S to S that is not associative for

\begin{aligned}b ∨_{S} (b ∨_{S} c) & = & b ∨_{S} (s(e(b) ∨_{L} e(c))) = b ∨_{S} (s(l ∨_{L} r)) \\ & = & b ∨_{S} s(⊤) = b ∨_{S} a = s(l ∨_{L} ⊥) = s(l) = b \end{aligned}

while

(b ∨_{S} b) ∨_{S} c = b ∨_{S} c = a.

So we need an additional condition to prove that a retract of a semilattice is again a semilattice. If we knew, for example, that

r ∘ v ∘ (e × e) ∘ (r × r) = r ∘ v (∗)

is satisfied then we could continue the above calculations:

(r ∘ v ∘ (e × e)) ∘ ((r ∘ v ∘ (e × e)) × id_{D} )

= r ∘ v ∘ (e × e) ∘ (r × r) ∘ (v × id_{C}) ∘ (e × e × e) (By above calculations)
= r ∘ v ∘ (v × id_{C}) ∘ (e × e × e)  (By (∗))
= r ∘ v ∘ (id_{C} ×v) ∘ (e × e × e)  (Associativity of v)
= r ∘ v ∘ (e × e) ∘ (r × r) ∘ (id_{C} ×v) ∘ (e × e × e) (By (∗))
= (r ∘ v ∘ (e × e)) ∘ ( id_{C} ×(r ∘ v ∘ (e × e)))  (r ∘ e = id_{D}).

Note that if (D, r ∘ v ∘ (e × e)) is a semilattice then (∗) implies that r is a morphism of the semilattices C and D. Thus we obtain the following lemma.

Lemma 5.3 Let C be a category with binary products. Further, let C be a C-object with a semilattice structure v: C × C −^{}C. If D is a retract of C via D e^{−^{}} ; − ; − ; − r C such that r∘v∘(e×e)∘(r×r) = r∘v then r ∘ v ∘ (e × e): D × D −^{}D is a semilattice structure for D. Moreover, r: C −^{}D is a semilattice homomorphism.

In Pos, the retract of an inflationary semilattice is an inflationary semilattice. This suggests that an analogue could hold for semilattices in O-categories. Although we cannot show it in general it is possible to prove a weaker version.

Lemma 5.4 Let C be an O-category with binary products such that the endofunctor on C given by C −^{}C × C, f −^{}f × f is an O-functor. Further, let D be a retract of C in C via D e^{−^{}} ; − ; − ; − r C. Let v: C × C −^{}C make (C, v) an inflationary semilattice.

(i) If (r, e) is an adjoint pair then (D, r ∘ v ∘ (e × e)) is a semilattice.

(ii) If (D, r ∘ v ∘ (e × e)) is a semilattice then it is inflationary as well.

Proof: (i) By Lemma 5.3 it is enough to show that r∘v∘(e×e)∘(r×r) = r∘v. As by assumption e∘r ≥ id_{C} and thus (e∘r)×(e∘r) ≥ id_{C×C}, we have r∘v∘(e×e)∘(r×r) = r ∘ v ∘ ((e ∘ r) × (e ∘ r)) ≥ r ∘ v. On the other hand,

r ∘ v ∘ (e × e) ∘ (r × r) ≤ r ∘ v ∘ (e × e) ∘ (r × r) ∘ Δ_{C} ∘v  (id_{C×C} ≤ Δ_{C} ∘v)
= r ∘ v ∘ Δ_{C} ∘e ∘ r ∘ v  (Naturality of Δ)
= r ∘ e ∘ r ∘ v  (v ∘ Δ_{C} = id_{C})
= r ∘ v  (r ∘ e = id_{D}).

(ii) We have to show that Δ_{D} ∘ r ∘ v ∘ (e × e) ≥ id_{D×D}. Since Δ_{C} ∘v ≥ id_{C×C} we may calculate

Δ_{D} ∘r ∘ v ∘ (e × e) = (r × r) ∘ Δ_{C} ∘v ∘ (e × e) ≥ (r × r) ∘ (e × e)
= (r ∘ e) × (r ∘ e)
= id_{D×D} .

The corresponding result for deflationary semilattices can be shown analogously.

Lemma 5.5 Let C be a category that satisfies the conditions from Lemma 5.4 and let D be a retract of C in C via D e^{−^{}} ; − ; − ; − r C. Further, let v: C × C −^{}C make (C, v) a deflationary semilattice.

(i) If (e, r) is an adjoint pair then (D, r ∘ v ∘ (e × e)) is a semilattice.

(ii) If (D, r ∘ v ∘ (e × e)) is a semilattice then it is deflationary as well.

Note that Lemmata 5.4 and 5.5 can be applied to the categories DCPO, Top and Loc. It is obvious that the product functor mentioned there is indeed an O-functor for DCPO and Top. For the category of locales, this follows from the following considerations: Let f, g: A −^{}B be morphisms in Loc such that f ≤ g. By definition, this is equivalent to f′ ≤ g′. As

(f ⊗ f)′(a ⊗ b) = f′(a) ⊗ f′(b) ≤ g′(a) ⊗ g′(b) = (g ⊗ g)′(a ⊗ b)

for all a ∈ A, b ∈ B, we conclude that f ⊗ f ≤ g ⊗ g. For the categories DCPO and Top we have the following: There is a faithful Ofunctor to Pos that preserves products. For such O-categories we can conclude the desired result from Lemma 1.2.

Proposition 5.6 Let C be an O-category with binary products such that there is a faithful and product-preserving O-functor U: C −^{}Pos. If D is a retract of the inflationary (deflationary) semilattice (C, v) via D e^{−^{}} ; − ; − ; − r C then (D, r ∘ v ∘ (e × e)) is an inflationary (a deflationary) semilattice in C.

Proof: We will only prove the result for the case where C is an inflationary semilattice. For simplicity we assume that U(C × C) = UC × UC. It is not hard to show that (UC, Uv) is an inflationary semilattice in Pos. For example,

Uv∘Δ_{UC} = Uv∘⟨id_{UC}, id_{UC}⟩ = Uv∘⟨U id_{C}, U id_{C}⟩ = Uv∘U Δ_{C} = U(v∘Δ_{C}) = id_{UC} .

Symmetry and associativity of Uv also follow from the fact that U preserves binary products. Hence (UC, Uv) is a ∨-semilattice and Uv is the binary join function. As UD is a retract of UC it must also be a ∨-semilattice by Lemma 1.2. Furthermore, the binary join for UD is given by Ur ∘ Uv ∘ (Ue × Ue) = U(r ∘ v ∘ (e × e)). Hence U(r ∘ v ∘ (e × e)) is idempotent, symmetric and associative. By faithfulness of U we may conclude that r ∘v ∘(e×e) has the same properties: In order to prove symmetry, for example, we need (r ∘ v ∘ (e × e)) ∘ ⟨π_{2}, π_{1}⟩ = r ∘ v ∘ (e × e). By symmetry of U(r ∘ v ∘ (e × e)) we have

U(r ∘ v ∘ (e × e) ∘ ⟨π_{2}, π_{2}⟩) = U(r ∘ v ∘ (e × e)) ∘ U⟨π_{2}, π_{1}⟩
= U(r ∘ v ∘ (e × e)) ∘ ⟨π_{2}, π_{1}⟩
= U(r ∘ v ∘ (e × e)).

Hence r∘v∘(e×e)∘⟨π_{2}, π_{1}⟩ and r∘v∘(e×e) are both mapped to U(r∘v∘(e×e)) and thus by faithfulness of U must be equal. Therefore (D, r ∘ v ∘ (e × e)) is a semilattice in C. By Lemma 5.4 (ii) it must be an inflationary semilattice.

Note that this result does not apply to the category of locales though there are two obvious functors to the category of posets. Neither of these, however, satisfies the above requirements: One of these functors is the contravariant functor to the category of frames. But contravariant functors do not transport notions like idempotency, symmetry etc. The second functor, namely Pt, that maps a locale A to the poset PtA of points of A, is not faithful.

5.5 Unital semilattices

In this section we will study the analogue of semilattices with unit. Recall that a unit for a semilattice (S, v) over Set is an element s_{0} of the set S such that v⟨s, s_{0}⟩ = s = v⟨s_{0}, s⟩ for all s ∈ S. (Note that by symmetry of v it is only necessary to demand one of these equations.) In this case we say that the semilattice is unital. A unital inflationary semilattice in Pos, for example, is a ∨-semilattice with a least element. To generalize the notion of unital semilattices to a category C with finite products, recall that an element of a C-object C is given by a morphism from 1 to C. Here 1 denotes the empty product, that is the terminal object in C. So when can we say that an element o: 1 −^{}C is the unit for a semilattice structure v: C × C −^{}C? If we have another look at the above equation we see that we can reformulate it via the following diagram:

1 × C \overline{o × id_{C}_{}}C × C
[figure: π_{2}]
C^{^{}v}

Note that here π_{2}: 1 × C −^{}C is an isomorphism. Definition. A unital semilattice (C, v, o) in a category C with finite products is given by a semilattice structure v: C × C −^{}C for C and a morphism o: 1 −^{}C such that v ∘(o×id_{C}) = π_{2}. A morphism of unital semilattices (C, v, o) and (D, w, p) is a morphism f: C −^{}D of semilattices such that f ∘ o = p. Note that the analogues of Lemma 5.3 to Proposition 5.6 for unital semilattices are true for categories that have finite products. However, to apply Proposition 5.6 we additionally have to make sure that U preserves the empty product. It is sufficient to prove the next proposition as the other results can easily be deduced from it.

Lemma 5.7 Let C be a category with finite products. Further, let (C, v, o) be a unital semilattice in C. If D is a retract of C via D e^{−^{}} ; − ; − ; − r C such that r∘v∘(e×e)∘(r×r) = r ∘ v then (r ∘ v ∘ (e × e), r ∘ o) is a unital semilattice structure for D. Moreover, r: C −^{}D is a homomorphism of unital semilattices.

Proof: Because of Lemma 5.3 it remains to prove that r∘o is a unit for the semilattice structure r ∘ v ∘ (e × e) on D. We calculate

(r ∘ v ∘ (e × e)) ∘ ((r ∘ o) × id_{D}) = r ∘ v ∘ (e × e) ∘ (r × r) ∘ (o × e)
= r ∘ v ∘ (o × id_{C}) ∘ (id_{1} ×e)
= r ∘ π_{2} ∘ (id_{1} ×e)
= r ∘ e ∘ π_{2}
= π_{2}

which proves the desired equation.

In Pos, DCPO and Top, the unit for an inflationary semilattice has to be the least element with respect to the order given for the respective objects. Hence a unital inflationary topological semilattice, for example, is a ∨-semilattice with least element with respect to the specialization order such that the binary join is a continuous function. The requirement that the unit for the semilattice structure be a morphism of the category is mute for these categories. Analogous statements hold for deflationary semilattices in these categories. In the category of locales, we can also characterize units for inflationary or deflationary semilattices.

Lemma 5.8 Let (A, v) be a semilattice in Loc.

(i) If A is an inflationary semilattice then it is unital iff a continuous function from 2 to A is defined by o′(a) := 1_{2} ⟺ a = 1_{A}. In this case, the unit for (A, v) is given by o.

(ii) If A is a deflationary semilattice then it is unital iff a continuous function from 2 to A is defined by o′(a) := 0_{2} ⟺ a = 0_{A}. In this case, the unit for (A, v) is given by o.

Proof: (i) We have seen that v′(a) = ⋁{b_{1} ⊗ b_{2} ∈ A ⊗ A | b_{1} ∧ b_{2} ≤ a} for all a ∈ A in Section 5.3. So if the above definition makes o′ a frame homomorphism then

(o ⊗ id_{A})′(v′(a)) = (o ⊗ id_{A})′ (⋁{b_{1} ⊗ b_{2} ∈ A ⊗ A | b_{1} ∧ b_{2} ≤ a})
= 1_{2} ⊗ (⋁{b_{2} ∈ A | b_{2} ≤ a})
= 1_{2} ⊗ a
= π′_{2}(a)

for all a ∈ A. Now assume that (A, v, o) is a unital inflationary semilattice in Loc. Then

1_{2} ⊗ a = π′_{2}(a)
= (o ⊗ id_{A})′(v′(a))
= (o ⊗ id_{A})′( ⋁{b_{1} ⊗ b_{2} ∈ A ⊗ A | b_{1} ∧ b_{2} ≤ a})
= ⋁{o′(b_{1}) ⊗ b_{2} ∈ A ⊗ A | b_{1} ∧ b_{2} ≤ a}

for all a ∈ A. Hence o′(a) ⊗ 1_{A} ≤ 1_{2} ⊗ a for all a ∈ A which proves o′(a) = 0_{2} for all a ∈ A \ {1_{A}}. (ii) Assume first that o′ as defined above is indeed a frame homomorphism. Since v′(a) ≤ a ⊗ a we have (o ⊗ id_{A})′(v′(a)) ≤ (o ⊗ id_{A})′(a ⊗ a) ≤ π′_{2}(a) for all a ∈ A. By the above considerations, v′(a) = ⋁{b_{1} ⊗ b_{2} ∈ A ⊗ A | b_{1} ⊗ b_{2} ≤ v′(a)}. We have

a = Δ′_{A}(v′(a))
= Δ′_{A}(⋁{b_{1} ⊗ b_{2} ∈ A ⊗ A | b_{1} ⊗ b_{2} ≤ v′(a)})
= ⋁{b_{1} ∧ b_{2} | b_{1} ⊗ b_{2} ≤ v′(a)}
≤ ⋁{b_{2} ∈ A | ∃b_{1} ∈ A \ {0_{A}} . b_{1} ⊗ b_{2} ≤ v′(a)}

and therefore also

(o ⊗ id_{A})′(v′(a)) = ⋁{o′(b_{1}) ⊗ b_{2} | b_{1} ⊗ b_{2} ≤ v′(a)}
= 1_{2} ⊗ ⋁{b_{2} ∈ A | ∃b_{1} ∈ A \ {0_{A}} . b_{1} ⊗ b_{2} ≤ v′(a)}
≥ 1_{2} ⊗ a
= π′_{2}(a)

for all a ∈ A. On the other hand, let o be a unit for v. Then

1_{2} ⊗ a = π′_{2}(a) = (o ⊗ id_{A})′(v′(a)) ≤ (o ⊗ id_{A})′(a ⊗ a) = o′(a) ⊗ a

for all a ∈ A. Hence if a ∈ A \ {0_{A}} then 1_{2} ≤ o′(a) which proves the assertion.

The idea of carrying algebraic concepts like that of a semilattice to arbitrary categories is well-established. Our concrete results on retracts of semilattices in orderenriched categories appear to be new.

The Hoare power construction

6.1 Algebraic domains

Considering non-deterministic programming languages, there are several possibilities how to think of non-determinism. We shall only deal with bounded non-determinism here, that is at any given time a program can only have finitely many choices. In a deterministic programming language any terminating program (deterministically) produces one final state for each initial state. If we consider non-deterministic programming languages, however, we lose this property. So, starting from one initial state, there typically are several possible final states any given program might end up with. Suppose we already have a denotational semantics for the deterministic part (like variable assignments, loops and conditionals) of such a non-deterministic language in the setting of algebraic domains. A program is denoted by a Scott-continuous function between algebraic domains and the obvious idea is to denote a non-deterministic program by a set-valued function. But should we allow all sets of states and how should we order the sets we do allow? In this chapter we shall consider so-called ‘angelic’ non-determinism, that is we say that a program satisfies a certain condition if at least one of its branches does. We may assume that for any given program p we already know how to denote all of its branches, namely by Scott-continuous functions between the algebraic domains D and E, and we want to denote p by a function from D to some domain that lives in the power set of E. Given another program q that is to be denoted by a function between the same domains, when should the denotation of q be greater than or equal to the denotation of p? Because of the above philosophy we want this to be the case iff for any branch of p there is one of q with greater or equal denotation. Hence for any algebraic domain D we (pre-)order P_{fne}CD, the set of non-empty finite sets of compact elements of D, via F ≤^{♭} F′ iff for all c in F there is c′ in F′ with c ≤ c′. The desired domain is now obtained by ideal completion of the poset P_{fne}CD (see e.g. [Plo81]). Definition. The Hoare power domain P_{A}^{♭}D of an algebraic domain D is the ideal completion of (P_{fne}CD, ≤^{♭}). There is another description of the Hoare power domain, namely as a subposet of the power set. It can be shown that Idl(P_{fne}CD, ≤^{♭}) is isomorphic to A_{D}D, the set of non-empty Scott-closed subsets of D ordered by inclusion: We map every non-empty finite set of compact elements to its lower closure in D. Let i: Idl(P_{fne}CD, ≤^{♭}) −^{}A_{D}D be the unique Scott-continuous extension of this function which is defined by setting i(I) := ⋁_{F ∈I}^{↑} ↓ F = cl( ⋃_{i∈I} ↓ F). One easily sees that A −^{}{F ∈ P_{fne}CD | F ⊆ A} defines a function from A_{D}D to Idl(P_{fne}CD, ≤^{♭}) that is the inverse for i. From what we have said before it follows that P_{A}^{♭}D is an algebraic domain. Furthermore, P_{A}^{♭}D has all non-empty joins. Recall that we called such a poset an up-complete ∨-semilattice. In the algebraic case, we also say that such a poset is an algebraic ∨-semilattice since we assume ‘algebraic’ to imply that the poset in question has directed joins. Algebraic ∨-semilattices are also referred to as Scott domains [Sco82]. Let us now consider a slightly different approach to the problem of giving a denotational semantics for a non-deterministic programming language. How is nondeterminism introduced to a language? Often one considers a deterministic language to which a command is added that is responsible for all non-deterministic effects. Assume once more that we already have an interpretation for a deterministic programming language and that we now want to add the command \underline{or} (with the usual meaning) to it. Hence we have to decide how to denote a program like p \underline{or} q. But because of the above way of interpreting non-determinism we would like that denotation to be greater than those of the programs p and q since everything one of the two can do, p \underline{or} q can do as well, but only just as well: the set of possible final states for p \underline{or} q is the union of the corresponding sets for p and q. Hence we want the denotation of p \underline{or} q to be the least upper bound of those for p and q. Consequently, the domain used to model p and q should have binary joins which of course need not exist in an algebraic domain. Thus we would like to somehow add finite joins to the domains we used to model the deterministic part of our language in a way that allows us to extend functions in a unique way. It turns out that we can also do this using the Hoare power domain. To make this precise, let us extend the Hoare power construction to a functor. For every Scottcontinuous function f: D −^{}E we define a function P_{A}^{♭}f: P_{A}^{♭}D −^{}P_{A}^{♭}E by setting P_{A}^{♭}f(A) := cl f(A) for every non-empty Scott-closed subset A of D. Thus we obtain a functor P_{A}^{♭}from the category AlgD of algebraic domains (with Scott-continuous functions) to the category AlgD^{∨} of algebraic ∨-semilattices with functions that preserve non-empty joins. Recall that we can understand AlgD^{∨} to be the category of inflationary semilattices in AlgD. The following result by M. Hennessy and G. Plotkin [HP79] implies that the two approaches we described in fact lead to the same definition.

Theorem 6.1 The functor P_{A}^{♭}: AlgD −^{}AlgD^{∨} is left adjoint to the forgetful functor AlgD^{∨} −^{}AlgD.

Hence the Hoare power domain gives the free inflationary semilattice for AlgD. Remark. If we wish the power domain to contain the empty set (which serves as its least element), an analogous result holds. The construction we thus obtain is left adjoint to the forgetful functor from the category of algebraic lattices with functions that preserve all joins to AlgD.

6.2 Directed complete partial orders

There is an obvious generalization of the Hoare power domain construction to arbitrary dcpo’s. Definition. The Hoare power dcpo P_{D}^{♭}D for a dcpo D is the set of all non-empty Scott-closed subsets of D ordered by inclusion. In the remainder of this section we will prove a result that is the analogue of Theorem 6.1. First of all, note that, for any dcpo, the Hoare power dcpo is a poset that has all non-empty joins, that is it is an up-complete ∨-semilattice. We define DCPO^{∨} to be the category of up-complete ∨-semilattices with functions that preserve non-empty joins. Note that every up-complete ∨-semilattice becomes a complete ∨-semilattice if a least element is adjoined to it. Alternatively, we can understand DCPO^{∨} to be the category of inflationary semilattices in DCPO. We can make P_{D}^{♭}a functor from DCPO to DCPO^{∨}. Let f: D −^{}E be a Scottcontinuous function between dcpo’s. We define P_{D}^{♭}f to be the function that maps any non-empty Scott-closed subset A of D to the Scott-closure of f(A) in E. In order to see that P_{D}^{♭}f indeed preserves all non-empty joins note that

P_{D}^{♭}f( cl ⋃ S) = cl f( cl ⋃ S) = cl f( ⋃ S) = cl ⋃_{A∈S} f(A) = cl ⋃_{A∈S} cl f(A) = cl ⋃ P_{D}^{♭}f(A)

for any non-empty subset S of P_{D}^{♭}D. Now for functoriality of P_{D}^{♭}. Obviously, P_{D}^{♭} preserves identities. If f: D −^{}E and g: E −^{}F are Scott-continuous functions between dcpo’s then P_{D}^{♭}(g ∘ f)(A) = cl g(f(A)) = cl g( cl f(A)) = P_{D}^{♭}f( P_{D}^{♭}g(A)) for any A ∈ P_{D}^{♭}D. We can embed D into P_{D}^{♭}D via the Scott-continuous function ↓_{D}: d −^{}↓{d}. The following theorem can be easily deduced from Theorem 4.4 in [HM85].

Theorem 6.2 The functor P_{D}^{♭}: DCPO −^{}DCPO^{∨} is left adjoint to the forgetful functor U:DCPO^{∨} −^{}DCPO.

Proof: We have to show that for any dcpo D the Hoare power dcpo P_{D}^{♭}D has the universal property given by the following diagram:

[figure: DCPO; D; ↓_{D}; U P_{D}^{♭}D; ∀f; UL; Uf̂; DCPO^{∨}; P_{D}^{♭}D; L; ∃!f̂]

Let f: D −^{}UL be a Scott-continuous function where D is a dcpo and L an upcomplete ∨-semilattice. Clearly, A = ⋃_{a∈A} ↓{a} = ⋁_{a∈A} ↓_{D}(a) for any non-empty Scottclosed subset A of D. There is at most one f̂: P_{D}^{♭}D −^{}L rendering the above diagram commutative because f̂(A) = ⋁_{a∈A} f(a) has to hold for all A ∈ P_{D}^{♭}D. We use this as the definition of f̂. In order to see that f̂ is a DCPO^{∨}-morphism, let S be a non-empty subset of P_{D}^{♭}D. Then

f̂( cl ⋃ S) = ⋁ f( cl ⋃ S)
= ⋁ f( ⋃ S)  (By Corollary 1.6 (i))
= ⋁ ⋃_{A∈S} f(A)
= ⋁{⋁ f(A) | A ∈ S}
= ⋁{f̂(A) | A ∈ S}.

Note that this is where Scott continuity of f is used. If we had chosen P_{D}^{♭}D to be the non-empty closed subsets of D with respect to another topology then we could not have employed Corollary 1.6. Obviously, f̂∘ ↓_{D} = f is satisfied.

The above adjunction is monadic: Note that the co-unit ⋁_{L}: P_{D}^{♭}U −^{}id_{DCPO}^{∨} of P_{D}^{♭}⊣ U is A −^{}⋁ A for every non-empty Scott-closed subset A of an up-complete ∨-semilattice L. In order to prove our claim we have to show that every algebra for the monad (U P_{D}^{♭}, ↓, U ⋁ P_{D}^{♭}) is an up-complete ∨-semilattice. But since every (U P_{D}^{♭})- algebra (D, ξ) is a retract of the up-complete ∨-semilattice U P_{D}^{♭}D it must be another such. As DCPO^{∨} is the category of inflationary semilattices for DCPO we can understand P_{D}^{♭}to give the free inflationary semilattice for DCPO.

6.2.1 Unital semilattices

Let us now investigate what happens if we allow the empty set as an element of the power dcpo. For any dcpo D we denote the resulting poset by Ṗ_{D}^{♭}D. Obviously, Ṗ_{D}^{♭}D is a complete lattice.

Theorem 6.3 The free complete ∨-semilattice in DCPO is given by Ṗ_{D}^{♭}.

Proof: Let D be a dcpo and L a complete ∨-semilattice. We can embed D into Ṗ_{D}^{♭}D via ↓_{D}. Let U denote the forgetful functor from the category of complete ∨- semilattices to DCPO. For every f: D −^{}UL in DCPO and every A ∈ Ṗ_{D}^{♭}D we set f̃(A) := ⋁ f(A). We have to prove that f̃ is the unique morphism of complete ∨-semilattices satisfying Uf̃∘↓_{D} = f. Since f̃ extends f̂ as defined in the proof of the previous theorem it remains to show that f̂ maps the least element of Ṗ_{D}^{♭}D, that is the empty set, to the least element ⊥_{L} of L. We calculate f̃(∅) = ⋁ f(∅) = ⋁ ∅ = ⊥_{L}.

Alternatively, for any dcpo D we can understand Ṗ_{D}^{♭}to arise from adjoining a least element to P_{D}^{♭}D. Hence P_{D}^{♭}gives the free unital inflationary semilattice for DCPO.

6.2.2 Domains

In this section we want to show that the category of domains Dom is closed with respect to the Hoare power construction for dcpo’s. This allows us to conclude that P_{D}^{♭}also gives the free inflationary semilattice for Dom. For that we have to prove that P_{D}^{♭}D is a continuous ∨-semilattice for every domain D. In order to do this we need the following well-known lemma.

Lemma 6.4 The Scott closure of an arbitrary subset S of a domain D is {⋁^{↑}S′ | S′ directed subset of ↓ S}.

Proof: As every Scott-closed subset is lower closed and also closed with respect to directed joins, we must have cl S ⊇ \{⋁^{↑}S′ | S′ directed subset of ↓ S\} =: \overline{S}. Since S is a subset of \overline{S} it remains to prove that \overline{S} is Scott-closed. First we will show that \overline{S} is lower closed. If d ∈ ↓ \overline{S} then there is a directed subset S′ of ↓ S with d ≤ ⋁^{↑}S′. Hence for all d′ ≪ d there is s′ ∈ S′ with d′ ≤ s′. Therefore ↓ ↓{d} is a directed subset of ↓ S. By definition of \overline{S}, this implies d = ⋁^{↑} ↓ ↓\{d\} ∈ \overline{S}. It remains to show that \overline{S} is closed with respect to directed joins. If T is a directed subset of \overline{S} then ⋃_{↑}\{↓ ↓\{t\} | t ∈ T \} is also directed and has the same join as T. If we can show that ⋃_{↑}{↓ ↓{t} | t ∈ T } is a subset of ↓ S then we are done. Let d ≪ t ∈ T. There is a directed subset S′ of ↓ S such that t ≤ ⋁^{↑}S′. Hence there is s′ ∈ S such that d ≤ s′ which proves d ∈ ↓ S.

The following proposition also follows from Corollary 5.10 in [Law79].

Proposition 6.5 If D is a domain so is P_{D}^{♭}D.

Proof: For d ≪ d′ in D we have ↓{d} ≪ ↓{d′} in P_{D}^{♭}: Let ↓{d′} ⊆ cl ⋃_{i∈I}^{↑}A_{i} for a

directed family (A_{i})_{i∈I} of non-empty Scott-closed subsets of D. By Lemma 6.4,

cl ⋃_{i∈I} A_{i} = {⋁^{↑}S | S directed subset of ↓ ⋃_{i∈I} A_{i}}.

Hence there is a directed subset S of ↓ ⋃_{i∈I} A_{i} = ⋃_{i∈I} A_{i} such that d′ ≤ ⋁^{↑}S. As d is way below d′ there is s ∈ S with d ≤ s. Therefore ↓{d} ⊆ ↓ A_{i} = A_{i} for some i ∈ I. We have to show that for any A ∈ P_{D}^{♭}D the set of elements way below A is directed with join A. As P_{D}^{♭}D has binary joins, it is enough to show that ⋁ ↓ ↓{A} = A holds in P_{D}^{♭}D. By the above, if d ≪ a ∈ A then ↓{d} ≪ A in P_{D}^{♭}D. As

A = cl ⋃_{↑}{↓{d_{1}} ∪ … ∪ ↓{d_{n}} | ∃a_{1}, …, a_{n} ∈ A.∀i ∈ {1, …, n} . d_{i} ≪ a_{i}}

we are done.

From the results of Section 6.2 and the last proposition we immediately get the following result.

Theorem 6.6 The free inflationary semilattice for Dom is given by P_{D}^{♭}.

6.3 Topological spaces

The definition of the Hoare power dcpo in the last section suggests a definition for a Hoare power space, namely the set of non-empty closed subsets of a T_{0}-space. However, it is not quite obvious which topology on this set we should choose. Hence we will split the construction of the Hoare power space into two steps. For a T_{0}- space X let AX be the set of non-empty closed subsets of X ordered by inclusion. This yields an up-complete ∨-semilattice for every T_{0}-space X. We can extend A to a functor from Top to DCPO^{∨} in the way suggested from the Hoare power construction for dcpo’s. For a continuous function f: X −^{}Y between T_{0}-spaces let Af: AX −^{}AY be defined by Af(A) := cl f(A) for every non-empty closed subset A of X. The proof that Af is a morphism in DCPO^{∨} and that A indeed is a functor is exactly the same as for P_{D}^{♭}: DCPO −^{}DCPO^{∨}. Now for the topology on AX. We want the function x −^{}cl{x} to be continuous in order to get a natural transformation as in the preceding section. The idea is to use the topology on X and thus demand that all the sets {A ∈ AX | A ⊆ B} be closed for every closed subset B of X. These sets are those of the form ↓{B} where B ∈ AX if we exclude B = ∅ for a moment. Hence we equip AX with the weak topology which ensures that the closed sets in AX are generated by the sets of the form {A ∈ AX | A ⊆ B} where B is a closed subset of X. For any up-complete ∨-semilattice L let T_{ω}L be L with the weak topology. If we set T_{ω}f(l) := f(l) for any DCPO^{∨}-morphism f: L −^{}M we obtain a continuous function T_{ω}f: T_{ω}L −^{}T_{ω}M: We have (T_{ω}f)^{−1}(↓ m) = {l ∈ L | f(l) ≤ m} for all m ∈ M and this set is either empty or equal to ↓{⋁{l ∈ L | f(l) ≤ m}} and therefore weakly closed in L. Obviously, T_{ω}:DCPO^{∨} −^{}Top is a functor. We can show that cl_{X}: x −^{}cl{x} is a continuous function from X to T_{ω}AX for every T_{0}-space X. If A is an element of T_{ω}AX then

cl_{X}^{−1}(↓{A}) = {x ∈ X | cl{x} ⊆ A} = {x ∈ X | x ∈ A} = A.

Note that in general cl_{X} is not continuous with respect to the Scott topology on AX.

Proposition 6.7 The functor A: Top −^{}DCPO^{∨} is left adjoint to T_{ω}.

Proof: Exactly as for Theorem 6.2.

We therefore formulate the following definition which is due to M. Smyth [Smy83]. Definition. For any T_{0}-space X we define the Hoare power space P_{T}^{♭}X to be the up-complete ∨-semilattice of non-empty closed subsets of X with the weak topology. The Hoare power space functor P_{T}^{♭}is given by T_{ω}A. The co-unit for the adjunction A⊣ T_{ω} is ⋁_{L}: AT_{ω}L −^{}L. It is our aim to characterize the algebras for the monad (P_{T}^{♭}, cl, T_{ω}⋁ A) in the remainder of this section. Note that the above adjunction is not monadic since, for example, every continuous ∨- semilattice equipped with its Scott topology is an algebra for the monad. Hence not every P_{T}^{♭}-algebra is an up-complete ∨-semilattice with its weak topology.

First of all note that for any T_{0}-space X we can apply Lemma 4.3 (i) since we can view X as a poset. Thus we know that if a structure map ξ: P_{T}^{♭}X −^{}X exists at all then it has to be the lower adjoint for η_{X} with respect to the specialization order. Therefore ξ(A) = ⋁ A must hold for every non-empty closed subset A of X. Consequently, for every T_{0}-space X there is at most one structure map for X that we will refer to as ξ_{X} from now on. By Lemma 4.6 we also know that if A −^{}⋁ A: P_{T}^{♭}X −^{}X is a continuous function then it is indeed the structure map for X. This yields a preliminary characterization of P_{T}^{♭}-algebras: A T_{0}-space X has a structure map iff A −^{}⋁ A: P_{T}^{♭}X −^{}X is a continuous function. We can do more than that since we can describe the P_{T}^{♭}-algebras without referring to P_{T}^{♭}X at all. For that purpose, note that every space of the form P_{T}^{♭}X is an inflationary topological semilattice. The semilattice structure on P_{T}^{♭}X is given by binary union ∪: P_{T}^{♭}X × P_{T}^{♭}X −^{}P_{T}^{♭}X. As ∪ is the binary join with respect to the specialization order on P_{T}^{♭}X, that is subset inclusion, it is the lower adjoint for the diagonal Δ_{P_{T}^{♭}X}: P_{T}^{♭}X −^{}P_{T}^{♭}X × P_{T}^{♭}X. In order to show that ∪: P_{T}^{♭}X × P_{T}^{♭}X −^{}P_{T}^{♭}X is continuous we calculate

∪^{−1}( ↓{A}) = {⟨B, B′⟩ ∈ P_{T}^{♭}X × P_{T}^{♭}X | B ∪ B′ ⊆ A} = ↓{A} × ↓{A}

for every non-empty closed subset A of X. Consequently, every P_{T}^{♭}-algebra is an inflationary semilattice by Proposition 5.6. Furthermore, every T_{0}-space of the form P_{T}^{♭}X is sober because of Proposition 1.7. Hence, by Proposition 1.9, we know that every P_{T}^{♭}-algebra, as a retract of a sober space, is sober as well. In order to show that every sober inflationary topological semilattice yields a P_{T}^{♭}-algebra we need the following result.

Lemma 6.8 The function ξ_{X}: A −^{}⋁ A: P_{T}^{♭}X −^{}X is continuous for every sober inflationary topological semilattice X.

Proof: As X is an inflationary topological semilattice it has binary joins with respect to the specialization order. By sobriety, it also has directed joins which proves that ξ_{X} is well-defined. Now for continuity. Let B be a closed subset of X. We have to show that ξ_{X}^{−1}(B) = {A ∈ P_{T}^{♭}X | ⋁ A ∈ B} is closed in P_{T}^{♭}X. Since X is sober, B is Scottclosed implying that ⋁ A = ⋁^{↑}{a_{1} ∨ ⋯ ∨ a_{n} | a_{1}, …, a_{n} ∈ A} ∈ B for some A ∈ P_{T}^{♭}X iff a_{1} ∨ ⋯ ∨ a_{n} ∈ B for all a_{1}, …, a_{n} ∈ A. We show that ξ_{X}^{−1}(B) is weakly closed in the up-complete ∨-semilattice AX: Let A ∈ P_{T}^{♭}X \ ξ_{X}^{−1}(B). By the above, there are a_{1}, …, a_{n} ∈ A such that a_{1}∨⋯∨a_{n} ∈ X\B. By continuity of ∨_{n}: X × ⋯ × X −^{}X we can find open subsets O_{1}, …, O_{n} of X such that ⟨a_{1}, …, a_{n}⟩ ∈ O_{1} × ⋯ × O_{n} ⊆ ∨_{n}^{−1}(X \ B). Consequently,

O_{1} ∩ ⋯ ∩ O_{n} = ∨_{n}(O_{1} × ⋯ × O_{n}) ⊆ X \ B.

Hence B ⊆ (X \ O_{1}) ∪ ⋯ ∪ (X \ O_{n}). We prove that ↓{X \ O_{1}} ∪ ⋯ ∪ ↓{X \ O_{n}} is a (weakly closed) superset of ξ_{X}^{−1}(B) in AX that does not contain A: If A′ is an element of P_{T}^{♭}X such that ⋁ A′ ∈ B then ⋁ A′ is an element of one of the X \ O_{i}. Therefore, A′ ⊆ X \ O_{i} for some i ∈ {1, …, n} and thus A′ ∈ ↓{X \ O_{1}} ∪ … ∪ ↓{X \ O_{n}}. This proves ξ_{X}^{−1}(B) ⊆ ↓{X \ O_{1}} ∪ … ∪ ↓{X \ O_{n}}. On the other hand, we know that for every i ∈ {1, …, n} there is a_{i} ∈ A ∩ O_{i}. So A is not a subset of any of the X \ O_{i}. Consequently, A ∈ K_{T}X \ (↓{X \ O_{1}} ∪ … ∪ ↓{X \ O_{n}}).

Let us now investigate the morphisms of P_{T}^{♭}-algebras. We claim that f: X −^{}Y, where X and Y are inflationary topological semilattices with sober topologies, is a morphism of P_{T}^{♭}-algebras iff it is a morphism of topological semilattices. Because ⋁ P_{T}^{♭}f(A) = ⋁ cl f(A) = ⋁ f(A) for all A ∈ P_{T}^{♭}X by Lemma 1.5, we may apply Lemma 4.3 (ii). Hence f is a morphism of P_{T}^{♭}-algebras iff it preserves joins of closed sets. By Corollary 1.6 (ii) this is true iff f preserves non-empty joins. As X and Y are sober spaces, any continuous function between them preserves directed joins anyway. Hence f is a morphism of P_{T}^{♭}-algebras iff it preserves binary joins iff it is a morphism of inflationary topological semilattices. Therefore we have shown the following result.

Theorem 6.9 The category of P_{T}^{♭}-algebras is the subcategory of inflationary topological semilattices in the category of sober spaces.

Consequently, P_{T}^{♭} gives the free inflationary semilattice for the category of sober spaces. Alternatively, we can understand the category of P_{T}^{♭}-algebras as the category of up-complete ∨-semilattices with an order-consistent topology that is weaker than the Scott topology. The morphisms in this setting are continuous functions that preserve non-empty joins.

6.3.1 Unital semilattices

If we allow the empty set to be an element of the Hoare power space of some T_{0}- space X then we obtain a slightly different power space functor, say Ṗ_{T}^{♭}. Obviously, Ṗ_{T}^{♭}X is a complete lattice with respect to the specialization order for every T_{0}-space X. Therefore, Ṗ_{T}^{♭}X is a unital inflationary semilattice in Top. Furthermore, it is still a sober space. We can make Ṗ_{T}^{♭}an endofunctor on Top if we use the same definition as for P_{T}^{♭}, that is if we set Ṗ_{T}^{♭}f(A) := cl f(A) for every continuous function f: X −^{}Y. Put differently, Ṗ_{T}^{♭}f is the unique extension of P_{T}^{♭}f making it a morphism of unital semilattices in Top. As P_{T}^{♭}X can be embedded into Ṗ_{T}^{♭}X we can re-use the definition of the natural transformation cl from above. Also, T_{ω}⋁ A from above can be extended so that we get a monad, say (Ṗ_{T}^{♭}, η, μ). Every Ṗ_{T}^{♭}-algebra is a unital inflationary topological semilattice with sober topology. It can be shown in the same way as the last theorem that every such topological semilattice is a Ṗ_{T}^{♭}-algebra and that morphisms of such semilattices are exactly the morphisms of Ṗ_{T}^{♭}-algebras.

Theorem 6.10 The category of Ṗ_{T}^{♭}-algebras is the subcategory of unital inflationary semilattices in the category of sober spaces.

This category is isomorphic to the category whose objects are complete lattices with an order consistent topology that is weaker than the Scott topology and whose morphisms are continuous functions that preserve all joins.

6.3.2 Locally compact spaces

We have seen above that, in some way, continuous dcpo’s correspond to locally compact spaces. Let us investigate the restriction of P_{T}^{♭}to the category LCompT of locally compact T_{0}-spaces. We can prove that LCompT is closed with respect to P_{T}^{♭}. Let X be a locally compact T_{0}-space. In order to prove that in this case P_{T}^{♭}X is locally compact as well we have to study the open sets of P_{T}^{♭}X. As the closed sets of P_{T}^{♭}X are generated by all the sets ↓{B} = {A ∈ P_{T}^{♭}X | A ⊆ B}, where B is a non-empty closed subset of X, the open sets in P_{T}^{♭}X are generated by the sets of the form P_{T}^{♭}X \ ↓{B} = {A ∈ P_{T}^{♭}X | A ∩ (X \ B) ≠ ∅}. Hence the topology of P_{T}^{♭}X is generated by the sets ♢U := {A ∈ P_{T}^{♭}X | A ∩ U ≠ ∅} where U is an open subset of X. Note that P_{T}^{♭}X need not be continuous even if X is locally compact.

Proposition 6.11 If X is a locally compact space then so is P_{T}^{♭}X.

Proof: Let A ∈ ♢U_{1} ∩ ⋯ ∩ ♢U_{n} where the U_{i} are open subsets of X. In order to prove P_{T}^{♭}X locally compact we have to find a compact neighbourhood of A in ♢U_{1} ∩ ⋯ ∩ ♢U_{n}. As A ∈ ♢U_{1} ∩ ⋯ ∩ ♢U_{n}, we can find some x_{i} ∈ A ∩ U_{i} for all i ∈ {1, …, n}. Also for all i ∈ {1, …, n} there is an open subset V_{i} of X and a compact subset K_{i} of X such that x ∈ V_{i} ⊆ K_{i} ⊆ U_{i} by local compactness of X. Let K := {B ∈ P_{T}^{♭}X | ∀i ∈ {1, …, n} . B ∩ K_{i} ≠ ∅}. We will show that K is compact in P_{T}^{♭}X and that A ∈ ♢V_{1} ∩ ⋯ ∩ ♢V_{n} ⊆ K ⊆ ♢U_{1} ∩ ⋯ ∩ ♢U_{n}. By choice of the V_{i} we have A ∈ ♢V_{1} ∩ ⋯ ∩ ♢V_{n} ⊆ K. If B ∈ K then B intersects all K_{i} and hence all U_{i}. Consequently, K ⊆ ♢U_{1} ∩ ⋯ ∩ ♢U_{n}. It remains to show compactness of K. Using Alexander’s Subbase Lemma, it is enough to find a finite subcover in any open cover O of K where the elements of O are of the form ♢O for some open subset O of X. So assume we are given such a cover O for K. As cl{y_{1}, …, y_{n}} is in K for any

choice of y_{1} ∈ K_{1}, …, y_{n} ∈ K_{n} we have K_{1} ∪ ⋯ ∪ K_{n} ⊆ ⋃_{♢O∈O} O. By compactness of

K_{1} ∪ ⋯ ∪ K_{n} there are ♢O_{1}, …, ♢O_{m} ∈ O such that K_{1} ∪ … ∪ K_{n} ⊆ O_{1} ∪ ⋯ ∪ O_{m}. Obviously, K ⊆ ♢O_{1} ∪ ⋯ ∪ ♢O_{m}.

Theorem 6.12 The functor P_{T}^{♭}gives the free sober inflationary semilattice for the category of locally compact spaces LCompT.

Remark. These considerations also apply to Ṗ_{T}^{♭}.

6.3.3 Hoare power space versus Hoare power dcpo

What is the connection between the Hoare power space and the Hoare power dcpo? Any dcpo D gives rise to a T_{0}-space T_{σ}D if equipped with its Scott topology. Therefore, we get a Hoare power space and a Hoare power dcpo for D. By definition, the two have the same points and if we regard the Hoare power space P_{T}^{♭}T_{σ}D as a poset via the specialization order, then it is isomorphic to P_{D}^{♭}D. When studying universal properties, however, we considered Scott-continuous functions for P_{D}^{♭}D while we equipped P_{T}^{♭}T_{σ}D with the weak topology. Nevertheless, for domains the two coincide: We already know that the weak topology is coarser than the Scott topology on P_{T}^{♭}T_{σ}D. So let U be a Scott-open subset of P_{T}^{♭}T_{σ}D where D is a domain. Further, let A be an element of U. By continuity of D we have

that A = cl ⋃_{↑}{↓{d_{1}} ∪ ⋯ ∪ ↓{d_{n}} | ∃a_{1}, …, a_{n} ∈ A.∀i ∈ {1, …, n} . d_{i} ≪ a_{i}}. By

Scott-openness of U there are a_{1}, …, a_{n} ∈ A such that for any i ∈ {1, …, n} there is d_{i} ≪ a_{i} with ↓{d_{1}} ∪ ⋯ ∪ ↓{d_{n}} ∈ U. We will show that ♢ ↑ ↑{d_{1}} ∩ ⋯ ∩ ♢ ↑ ↑{d_{n}} is a neighbourhood of A in U. By choice of the d_{i} we know that a_{i} ∈ A ∩ ↑ ↑{d_{i}} for all i ∈ {1, …, n}. Hence A ∈ ♢ ↑ ↑{d_{1}} ∩ ⋯ ∩ ♢ ↑ ↑{d_{n}}. If B ∈ ♢ ↑ ↑{d_{1}} ∩ ⋯ ∩ ♢ ↑ ↑{d_{n}}, on the other hand, then B ⊇ ↓{d_{1}} ∪ ⋯ ∪ ↓{d_{n}} ∈ U implying B ∈ U. Because ♢ ↑ ↑{d_{1}} ∩ ⋯ ∩ ♢ ↑ ↑{d_{n}} is weakly open in AD this proves our claim.

Proposition 6.13 The functors T_{σ}P_{D}^{♭}and P_{T}^{♭}T_{σ} coincide on Dom.

6.4 Locales

We motivated the introduction of the Hoare power domain for (algebraic) domains by considering semantics for non-deterministic programming languages. A similar approach works for locales. The general idea behind the use of locales in semantics is to consider the elements of a locale A to be properties that some program p might have. If a program p has the property a ∈ A we write p ⊧ a and say that p satisfies a. We think of the property a ∧ b, where a and b are elements of A, as the property of satisfying both, a and b. Hence we want p ⊧ a ∧ b to hold iff both, p ⊧ a and p ⊧ b. Similarly, p has the property given by ⋁ S, where S is a subset of A, iff p satisfies (at least) one of the properties from S. This leads to p ⊧ ⋁ S iff p ⊧ s for some s ∈ S. If we think of the greatest element 1_{A} of A to be a property that trivially is satisfied by any program, then p ⊧ 1_{A} for all programs p. Analogously we can understand the least element 0_{A} of A to be a property that no program ever satisfies. Hence every program p gives rise to a point p for the locale A if we set p′(a) := 1 iff p ⊧ a. As we did for domains, we assume that we already know how to model the deterministic part of our language. Suppose that p is a non-deterministic program and that we know how to model the properties of the branches of p via the locale A. The elements of the locale A should give rise to new properties that p might have. For every element a of A we will denote this new property by ⋄a. This yields a new locale P_{L}^{♭}A generated by the ⋄a. But we want to model our view of non-determinism with this locale and therefore require certain relations to be satisfied in P_{L}^{♭}A. We introduced angelic non-determinism by stating that a non-deterministic program satisfies a property iff one of its branches does. So we define p ⊧ ⋄a to mean that at least one of the branches of p satisfies a. When does p ⊧ ⋁_{s∈S} ⋄s hold for some S ⊆ A? This is the case iff there is some s ∈ S with p ⊧ ⋄s. It is equivalent to the existence of a branch of p satisfying s, that is the existence of a branch of p satisfying ⋁ S. Hence we want ⋁_{s∈S} ⋄s = ⋄ ⋁ S to hold in P_{L}^{♭}A for all S ⊆ A. We thought of 1_{A} as being satisfied by every deterministic program modelled by A. Hence p ⊧ ⋄1_{A} holds for any non-deterministic program p that can be modelled using P_{L}^{♭}A. So it seems superfluous to have another element 1_{P_{L}^{♭}A} that is also satisfied by any such program p. Therefore we require ⋄1_{A} = 1_{P_{L}^{♭}A}. These are the only relations we want to hold in P_{L}^{♭}A since it does not make sense to demand similar equations for finite meets: A program p is to satisfy ⋄(a ∧ b) iff one of its branches satisfies both, a and b, while p ⊧ ⋄a ∧ ⋄b iff there are two (possibly different) branches of p, one satisfying a and one satisfying b. In Section 1.3 we described the construction of the free frame via generators and relations, so we can define a locale as follows. Definition. The Hoare power locale P_{L}^{♭}A for a locale A is the frame presented by

F r ⟨⋄a : a ∈ A | ⋄1_{A} = 1_{P_{L}^{♭}A} , ⋁_{s∈S} ⋄s = ⋄ ⋁ S (S ⊆ A)⟩ .

The first power construction for locales, the so-called Vietoris or Plotkin power construction, is due to P. Johnstone [Joh82b]. The connection between power locales and power domains as well as their application in denotational semantics may be traced to G. Winskel and E. Robinson. In [Win85], Winskel introduces two modal operators, ⋄ and □, and shows how the three power domain constructions can be built up from assertions using these operators. Building on this work, Robinson gives the above definition of a Hoare power locale in [Rob88]. Let us once more consider the special case where the only non-deterministic command is \underline{or}. By the above philosophy we want p \underline{or} q to satisfy ⋄a iff either p satisfies ⋄a or q does. But this can be expressed using products in Loc. If we set ⟨p, q⟩′(a⊗b) := 1_{2} iff p ⊧ a and q ⊧ b then we obtain a point for the locale P_{L}^{♭}A ⊗ P_{L}^{♭}A. Hence p \underline{or} q ⊧ ⋄a iff either ⟨p, q⟩ ⊧ ⋄a ⊗ 1_{P_{L}^{♭}A} or ⟨p, q⟩ ⊧ 1_{P_{L}^{♭}A} ⊗ ⋄a which is the case iff ⟨p, q⟩ ⊧ (⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a). This gives us a connection between P_{L}^{♭}A and its product P_{L}^{♭}A ⊗ P_{L}^{♭}A. If p and q are programs then we can map the corresponding point ⟨p, q⟩ to the program p \underline{or} q which we see as a point of P_{L}^{♭}A. In this case, (⋄a⊗1_{A})∨(1_{A}⊗⋄a) is the weakest property such that for every ⟨p, q⟩, where p and q are programs, the program p \underline{or} q must satisfy ⋄a. By our considerations in Section 2.1.1 we would expect ⋄a −^{}1_{A} ⊗ ⋄a ∨ ⋄a ⊗ 1_{A} to define a frame homomorphism from P_{L}^{♭}A to P_{L}^{♭}A⊗P_{L}^{♭}A and we shall see that this is indeed the case for the above definition of P_{L}^{♭}A. Furthermore, we can axiomatize the properties of this frame homomorphism such that we can understand P_{L}^{♭}A to be the free frame for which such a morphism exists. There is another way to motivate this definition for the Hoare power locale, namely from purely mathematical considerations. From the Hoare power space one would like to pass to a Hoare power locale. One can do so by finding connections between the locales ΩX and Ω P_{T}^{♭}X where X is a T_{0}-space. In Section 6.3 we obtained a subbasis for Ω P_{T}^{♭}X, namely the sets of the form ♢U = {A ∈ P_{T}^{♭}X | A ∩ U ≠ ∅} where U is an open subset of X. These sets have the following property: Given an arbitrary family U of open subsets of X we have

⋃_{U∈U} ♢U = ⋃_{U∈U}{A ∈ P_{T}^{♭}X | A ∩ U ≠ ∅} = {A ∈ P_{T}^{♭}X | ∃U ∈ U . A ∩ U ≠ ∅} = ♢ ⋃ U.

Furthermore, ♢X = {A ∈ P_{T}^{♭}X | A ∩ X ≠ ∅} = P_{T}^{♭}X which means that the greatest element in ΩX, namely X, generates the greatest element in Ω P_{T}^{♭}X, that is P_{T}^{♭}X. For U, U′ ∈ ΩX we can only prove that ♢(U ∩ U′) ⊆ ♢U ∩ ♢U′ and it is obvious that in general these two sets are not equal. Hence the above equations are the only relations we can show to hold between the sets of the form ♢U where U ∈ ΩX. This suggests the same definition for a Hoare power locale as the one we introduced above: For every element a of some locale A, take a generator ⋄a. Construct the free frame for these generators subject to the relations ⋄1_{A} = 1_{P_{L}^{♭}A} and ⋁_{s∈S} ⋄s = ⋄ ⋁ S for all S ⊆ A.

6.4.1 The monad

As for dcpo’s and T_{0}-spaces, we can extend the Hoare power construction for locales to a functor. Let f: A −^{}B be a continuous function between locales. In order to define a continuous function P_{L}^{♭}f: P_{L}^{♭}A −^{}P_{L}^{♭}B we have to define a function (P_{L}^{♭}f)′ on the generators ⋄b of P_{L}^{♭}B that preserves the relations from the above definition. If we set (P_{L}^{♭}f)′(⋄b) := ⋄f′(b) then obviously (P_{L}^{♭}f)′(⋄1_{B}) = ⋄f′(1_{B}) = ⋄1_{A} = 1_{P_{L}^{♭}A}. Furthermore, for any subset T of B we have

(P_{L}^{♭}f)′(⋄ ⋁ T) = ⋄f′(⋁ T) = ⋄ ⋁_{t∈T} f′(t) = ⋁_{t∈T} ⋄f′(t) = ⋁_{t∈T}(P_{L}^{♭}f)′(⋄t).

Obviously, P_{L}^{♭}preserves identities. To see that it also preserves composition, note that if f: A −^{}B and g: B −^{}C are continuous functions then

( P_{L}^{♭}(g ∘ f))′(⋄c) = ⋄(g ∘ f)′(c) = ⋄f′(g′(c)) = (P_{L}^{♭}f)′(⋄g′(c)) = (P_{L}^{♭}f)′((P_{L}^{♭}g)′(⋄c))

for all c ∈ C. Hence (P_{L}^{♭}(g ∘ f))′ and (P_{L}^{♭}g ∘ P_{L}^{♭}f)′ are frame homomorphisms that agree on the generators of P_{L}^{♭}C and thus must coincide. The functor P_{L}^{♭}is part of a monad on Loc we are going to define now. For that we require a natural transformation η: id_{Loc} −^{}P_{L}^{♭}. So we need a continuous function η_{A}: A −^{}P_{L}^{♭}A for every locale A. Again, we will give a function on the generators of P_{L}^{♭}A. Set η′_{A}(⋄a) := a for any a ∈ A. It is not difficult to check that η′_{A} respects the relations from the above definition; therefore we omit the proof. In order to see that this assignment makes η the desired natural transformation, let f: A −^{}B be a morphism in Loc. For all b ∈ B we have

(η_{B} ∘f)′(⋄b) = f′(η′_{B}(⋄b)) = f′(b) = η′_{A}(⋄f′(b)) = η′_{A} ((P_{L}^{♭}f)′(⋄b)) = (P_{L}^{♭}f ∘ η_{A})′(⋄b)

which proves the commutativity of the following diagram:

[figure: A; η_{A}; P_{L}^{♭}A; B; f; η_{B}; P_{L}^{♭}B; P_{L}^{♭}f]

Now for the second natural transformation needed, μ: (P_{L}^{♭})^{2} −^{}P_{L}^{♭}. We obtain a continuous function from (P_{L}^{♭})^{2}A to P_{L}^{♭}A by setting μ′_{A}(⋄a) := ⋄ ⋄ a. In order to prove naturality of μ we have to check the commutativity of the following diagram for any continuous function f: A −^{}B:

[figure: (P_{L}^{♭})^{2}A; μ_{A}; P_{L}^{♭}A; (P_{L}^{♭})^{2}B; (P_{L}^{♭})^{2}f; μ_{B}; P_{L}^{♭}B; P_{L}^{♭}f]

But for any such f and all b ∈ B we calculate

\begin{aligned}((P_{L}^{♭})^{2}f)′( μ′_{B}(⋄b)) & = & ((P_{L}^{♭})^{2}f)′(⋄ ⋄ b) = ⋄(P_{L}^{♭}f)′(⋄b) \\ & = & ⋄ ⋄ f′(b) = μ′_{A}(⋄f′(b)) = μ′_{A} ((P_{L}^{♭}f)′(⋄b)). \end{aligned}

Since (μ_{B} ∘(P_{L}^{♭})^{2}f)′ and (P_{L}^{♭}f ∘ μ_{A})′ are frame homomorphisms agreeing on the generators of P_{L}^{♭}B, they must coincide.

Proposition 6.14 The above definition makes (P_{L}^{♭}, η, μ) a monad on Loc.

Proof: It remains to check the commutativity of the diagrams for a monad (see the beginning of Section 4.1). First for the two triangular ones. If A is any locale then for all a ∈ A we have

η′_{P_{L}^{♭}A} ( μ′_{A}(⋄a)) = η′_{P_{L}^{♭}A}(⋄ ⋄ a) = ⋄a = ⋄ η′_{A}(⋄a) = (P_{L}^{♭}η_{A})′(⋄ ⋄ a) = (P_{L}^{♭}η_{A})′( μ′_{A}(⋄a))

which proves μ_{A} ∘ η_{P_{L}^{♭}A} = id_{P_{L}^{♭}A} = μ_{A} ∘ P_{L}^{♭}η_{A}. Now let us have a look at the rectangular diagram for monads. For all a ∈ A we calculate

μ′_{P_{L}^{♭}A} ( μ′_{A}(⋄a)) = μ′_{P_{L}^{♭}A}(⋄ ⋄ a) = ⋄ ⋄ ⋄a = ⋄ μ′_{A}(⋄a) = (P_{L}^{♭}μ_{A})′(⋄ ⋄ a) = (P_{L}^{♭}μ_{A})′( μ′_{A}(⋄a))

which proves our claim.

Note the similarities between the monads arising from the Hoare power construction for locales and spaces. We constructed the Hoare power locale P_{L}^{♭}A for any locale A such that its generators satisfy the equations shown for the generators of Ω P_{T}^{♭}X where X is a T_{0}-space. If f: X −^{}Y is a continuous function then (P_{T}^{♭}f)^{−1}(♢U) = ♢f^{−1}(U) for every U ∈ ΩX. Compare this equation to the definition of P_{L}^{♭}f where f: A −^{}B is a continuous function between locales. Furthermore, η_{X}^{−1}(♢U) = U for any U ∈ ΩX which resembles the equation that defines η_{A}, namely η′_{A}(⋄a) = a. A similar analogy holds for μ. We will further investigate the connections between these two Hoare power constructions in a separate paragraph.

6.4.2 The algebras for the monad

Next we want to characterize the algebras for the monad (P_{L}^{♭}, η, μ). So let (A, ξ) be a P_{L}^{♭}-algebra. First of all we show that (ξ, η_{A}) has to be an adjoint pair in the O-category Loc as defined in Section 4.4. By definition, this is true iff ξ′ is the upper adjoint for η′_{A} in the usual sense. Since ξ is a structure map for A we have ξ ∘ η_{A} = id_{A}, which is equivalent to η′_{A} ∘ ξ′ = id′_{A}. Therefore it is enough to show that ξ′ ∘ η′_{A} ≥ id′_{P_{L}^{♭}A}. For all a ∈ A, let ⋁_{i∈I} ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i} := ξ′(a). For arbitrary a ∈ A this implies

a = η′_{A} (ξ′(a)) = η′_{A} ( ⋁_{i∈I} ⋄a_{1}^{i}∧⋯∧⋄a_{n(i)}^{i}) = ⋁_{i∈I} η′_{A}(⋄a_{1}^{i}∧⋯∧⋄a_{n(i)}^{i}) = ⋁_{i∈I} a_{1}^{i}∧⋯∧a_{n(i)}^{i}.

Consequently, ⋄a = ⋄ ⋁_{i∈I} a_{1}^{i} ∧ ⋯ ∧ a_{n(i)}^{i} = ⋁_{i∈I} ⋄(a_{1}^{i} ∧ ⋯ ∧ a_{n_{i}}^{i}) ≤ ⋁_{i∈I} ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n_{i}}^{i}

which suffices to prove ξ′ ∘ η′_{A} ≥ id′_{P_{L}^{♭}A}. There is at most one structure map for any locale A which we will refer to as ξ_{A} from now on. Also, the conditions for Lemma 4.6 are satisfied; this is a fact we will exploit later. Recall that for DCPO and Top the algebras for the monads resulting from the respective Hoare power constructions are inflationary semilattices for the respective category. The same is true for the category of locales. To define a semilattice structure on locales of the form P_{L}^{♭}A let us have another look at the semilattice structure for P_{T}^{♭}X where X is an arbitrary T_{0}-space. The semilattice structure making P_{T}^{♭}X an inflationary semilattice is binary union. Hence we can hope for an idea how to define a semilattice structure for P_{L}^{♭}A, where A is an arbitrary locale, if we look at the corresponding inverse image function. Let U be an open subset of some T_{0}-space X. Then

∪^{−1}(♢U) = {⟨A, B⟩ ∈ P_{T}^{♭}X × P_{T}^{♭}X | (A ∪ B) ∩ U ≠ ∅}
= {⟨A, B⟩ ∈ P_{T}^{♭}X × P_{T}^{♭}X | A ∩ U ≠ ∅ or B ∩ U ≠ ∅}
= {⟨A, B⟩ ∈ P_{T}^{♭}X × P_{T}^{♭}X | A ∩ U ≠ ∅}
∪{⟨A, B⟩ ∈ P_{T}^{♭}X × P_{T}^{♭}X | B ∩ U ≠ ∅}
= (♢U × P_{T}^{♭}X) ∪ (P_{T}^{♭}X × ♢U).

Therefore we try to define a continuous function v: P_{L}^{♭}A ⊗ P_{L}^{♭}A −^{}P_{L}^{♭}A by setting v′(⋄a) := (⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a) for all a ∈ A. Compare this definition with our motivations for the Hoare power locale. Obviously,

\begin{aligned}v′(⋄1_{A}) & = & (⋄1_{A} ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄1_{A}) \\ & = & (1_{P_{L}^{♭}A} ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ 1_{P_{L}^{♭}A}) = 1_{P_{L}^{♭}A} ⊗ 1_{P_{L}^{♭}A} = 1_{P_{L}^{♭}A⊗P_{L}^{♭}A}. \end{aligned}

For any subset S of A we have

v′(⋄ ⋁ S) = (⋄ ⋁ S ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄ ⋁ S)
= ( ⋁_{s∈S} ⋄s ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋁_{s∈S} ⋄s)
= ⋁_{s∈S}(⋄s ⊗ 1_{P_{L}^{♭}A}) ∨ ⋁_{s∈S}(1_{P_{L}^{♭}A} ⊗ ⋄s)
= ⋁_{s∈S} ((⋄s ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄s))
= ⋁_{s∈S} v′(⋄s)

which proves that the above does define a continuous function as claimed. This function is strongly related to the semilattice structure P. Johnstone introduces for the Vietoris power locale in [Joh82b].

Lemma 6.15 The continuous function v: P_{L}^{♭}A ⊗ P_{L}^{♭}A −^{}P_{L}^{♭}A makes P_{L}^{♭}A an inflationary localic semilattice.

Proof: First for idempotency of v. We calculate

(v ∘ Δ_{P_{L}^{♭}A})′(⋄a) = Δ′_{P_{L}^{♭}A} (v′(⋄a))
= Δ′_{P_{L}^{♭}A} ((⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a))
= (⋄a ∧ 1_{A}) ∨ (1_{A} ∧ ⋄a)
= ⋄a

for any a ∈ A. Next for symmetry. For any a ∈ A we have

(v ∘ ⟨π_{2}, π_{1}⟩)′(⋄a) = ⟨π_{2}, π_{1}⟩′((⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a))
= (1_{P_{L}^{♭}A} ⊗ ⋄a) ∨ (⋄a ⊗ 1_{P_{L}^{♭}A})
= (⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ∧ ⋄a)
= v′(⋄a).

Similarly, we can show associativity since

(v ∘ (v ⊗ id_{P_{L}^{♭}A}))′(⋄a)
= (v ⊗ id_{P_{L}^{♭}A})′(v′(⋄a))
= (v ⊗ id_{P_{L}^{♭}A})′((⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a))
= (((⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a)) ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ 1_{P_{L}^{♭}A} ⊗ ⋄a)
= (⋄a ⊗ 1_{P_{L}^{♭}A} ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ 1_{P_{L}^{♭}A} ⊗ ⋄a)
= (v ∘ (id_{P_{L}^{♭}A} ⊗v))′(⋄a)

holds for all a ∈ A. We also claimed that v makes P_{L}^{♭}A an inflationary semilattice, that is Δ_{P_{L}^{♭}A} ∘v ≥ id_{P_{L}^{♭}A⊗P_{L}^{♭}A} holds with respect to the order we introduced on homsets of Loc in Section 4.4. Hence we have to show that (Δ_{P_{L}^{♭}A} ∘v)′ ≥ id′_{P_{L}^{♭}A⊗P_{L}^{♭}A}. If ⋄a ⊗ ⋄b is a generator of P_{L}^{♭}A ⊗ P_{L}^{♭}A then

(Δ_{P_{L}^{♭}A} ∘v)′(⋄a ⊗ ⋄b) = v′(⋄a ∧ ⋄b)
= v′(⋄a) ∧ v′(⋄b)
= ((⋄a ⊗ 1_{A}) ∨ (1_{A} ⊗ ⋄a)) ∧ ((⋄b ⊗ 1_{A}) ∨ (1_{A} ⊗ ⋄b))
≥ ⋄a ⊗ ⋄b.

Therefore (P_{L}^{♭}A, v) is an inflationary semilattice in Loc.

Proposition 6.16 Every P_{L}^{♭}-algebra is an inflationary localic semilattice.

Proof: By the above, (ξ_{A}, η_{A}) has to be an adjoint pair for any P_{L}^{♭}-algebra (A, ξ_{A}). Hence, because of Lemma 5.4, we may conclude from the last lemma that every P_{L}^{♭}-algebra is an inflationary localic semilattice. Its semilattice structure is given by ξ ∘v ∘(η_{A} ⊗ η_{A}) where v denotes the semilattice structure on P_{L}^{♭}A from Lemma 6.15.

Let us now consider morphisms of P_{L}^{♭}-algebras. For that, assume that f: A −^{}B is a morphism for the P_{L}^{♭}-algebras (A, ξ_{A}) and (B, ξ_{B}), that is ξ_{B} ∘ P_{L}^{♭}f = f ∘ ξ_{A}. By the above we know that A and B inherit their semilattice structure from P_{L}^{♭}A and P_{L}^{♭}B, respectively. Let us denote the semilattice structure making P_{L}^{♭}A an inflationary semilattice by v and the corresponding structure for B by w. Then (A, ξ_{A} ∘v ∘ (η_{A} ⊗ η_{A})) and (B, ξ_{B} ∘w ∘ (η_{B} ⊗ η_{B})) are inflationary localic semilattices by Lemma 5.4.

Lemma 6.17 Every morphism of the P_{L}^{♭}-algebras (A, ξ_{A}) and (B, ξ_{B}) is a morphism of semilattices from (A, ξ_{A} ∘v ∘ (η_{A} ⊗ η_{A})) to (B, ξ_{B} ∘w ∘ (η_{B} ⊗ η_{B})).

Proof: Let f: A −^{}B be a morphism of P_{L}^{♭}-algebras. We first show that P_{L}^{♭}f is a semilattice homomorphism from (P_{L}^{♭}A, v) to (P_{L}^{♭}B, w), that is P_{L}^{♭}f ∘v = w∘(P_{L}^{♭}f ⊗P_{L}^{♭}f). If b ∈ B then

v′((P_{L}^{♭}f)′(⋄b)) = v′(⋄f′(b))
= (⋄f′(b) ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄f′(b))
= ((P_{L}^{♭}f)′(⋄b) ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ (P_{L}^{♭}f)′(⋄b))
= (P_{L}^{♭}f ⊗ P_{L}^{♭}f)′((⋄b ⊗ 1_{P_{L}^{♭}B}) ∨ (1_{P_{L}^{♭}B} ⊗ ⋄b))
= (P_{L}^{♭}f ⊗ P_{L}^{♭}f)′(w′(⋄b)).

Hence we get

f ∘ ξ_{A} ∘v ∘ (η_{A} ⊗ η_{A})
= ξ_{B} ∘ P_{L}^{♭}f ∘ v ∘ (η_{A} ⊗ η_{A})  (f morphism of P_{L}^{♭}-algebras)
= ξ_{B} ∘w ∘ (P_{L}^{♭}f ⊗ P_{L}^{♭}f) ⊗ (η_{A} ⊗ η_{A})  (P_{L}^{♭}f morphism of semilattices)
= ξ_{B} ∘w ∘ (η_{B} ⊗ η_{B}) ∘ (f ⊗ f)  (Naturality of η)

which proves that f is a morphism of semilattices as claimed.

Thus we can understand the category of P_{L}^{♭}-algebras as a subcategory of the category of inflationary localic semilattices with semilattice homomorphisms. We claim that every inflationary semilattice in Loc has a structure map ξ_{A}: P_{L}^{♭}A −^{}A making (A, ξ_{A}) a P_{L}^{♭}-algebra. For that let us first note that, because of Lemma 4.6, it is enough to prove that for every inflationary localic semilattice (A, v) the continuous function η_{A} has a lower adjoint making A a retract of P_{L}^{♭}A in Loc. Once more we will obtain an idea how to prove this from the corresponding considerations for topological spaces. If X is a sober inflationary semilattice in Top then it has a structure map ξ_{X} making it a P_{T}^{♭}-algebra. The inverse image of an open set U in X under ξ_{X} is

ξ_{X}^{−1}(U)
= {A ∈ P_{T}^{♭}X | ⋁ A ∈ U}
= {A ∈ P_{T}^{♭}X | ∃U_{1}, …, U_{n} ∈ ΩX . A ∈ ♢U_{1} ∩ ⋯ ∩ ♢U_{n}
and ∨_{n} (U_{1} × ⋯ × U_{n}) ⊆ U}
= ⋃{♢U_{1} ∩ ⋯ ∩ ♢U_{n} | n ∈ ℕ, U_{1}, …, U_{n} ∈ ΩX and U_{1} × ⋯ × U_{n} ⊆ ∨_{n}^{−1}(U)}.

Let (A, v) be an inflationary localic semilattice. We set

ξ′_{A}(a) := ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | n ∈ ℕ, a_{1}, …, a_{n} ∈ A and a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(a)}.

We have to show that ξ′_{A} is a frame homomorphism, that ξ′_{A} is the upper adjoint for η_{A}′, and that ξ_{A} ∘ η_{A} = id_{A}. We start with the last point. Since v′_{n} is the upper adjoint for Δ′_{n} we have a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(a) iff a_{1} ∧ ⋯ ∧ a_{n} = Δ′_{n}(a_{1} ⊗ ⋯ ⊗ a_{n}) ≤ a for all a, a_{1}, …, a_{n} ∈ A. For all a ∈ A we calculate

η′_{A} ( ξ′_{A}(a)) = η′_{A} (⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ P_{L}^{♭}A | a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(a)})
= ⋁{a_{1} ∧ ⋯ ∧ a_{n} | a_{1} ∧ ⋯ ∧ a_{n} ≤ a}
= a.

Furthermore,

ξ′_{A}(a) = ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ P_{L}^{♭}A | a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(a)}
= ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ P_{L}^{♭}A | a_{1} ∧ ⋯ ∧ a_{n} = Δ′_{n}(a_{1} ⊗ ⋯ ⊗ a_{n}) ≤ a}
= ⋁{ ⋁_{i∈I} ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n_{i}}^{i} ∈ P_{L}^{♭}A | η′_{A}( ⋁_{i∈I} ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n}^{i}) ≤ a}
= ⋁{a ∈ P_{L}^{♭}A | η′_{A}(a) ≤ a}.

Hence ξ′_{A} is indeed the upper adjoint for η′_{A} and thus preserves all meets, in particular finite ones. It remains to prove that ξ′_{A} preserves all joins. For that we need the following result.

Lemma 6.18 Let S and T be two sets of generators of the n-fold product of a locale A where n is an arbitrary positive integer. If ⋁ S = ⋁ T in A ⊗ ⋯ ⊗ A then ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | a_{1} ⊗ ⋯ ⊗ a_{n} ∈ S} = ⋁{⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} | b_{1} ⊗ ⋯ ⊗ b_{n} ∈ T } in P_{L}^{♭}A.

Proof: Consider the mapping m from A ⊗ ⋯ ⊗ A to P_{L}^{♭}A which is defined by setting m(a_{1} ⊗ ⋯ ⊗ a_{n}) := ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}. Note that m is not a morphism, neither in Loc nor in Frm. We want to employ the constructions of P_{L}^{♭}A and A ⊗ ⋯ ⊗ A via C-ideals as in Theorem 1.11. So let I be a set of generators of P_{L}^{♭}A that is closed with respect to finite meets satisfying the following condition: If R ⊆ A and a_{1}, …, a_{n} ∈ A such that ⋄r ∧ ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ I for all r ∈ R then ⋄ ⋁ R ∧ ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ I. Assume that I ⊇ {⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | a_{1} ⊗ ⋯ ⊗ a_{n} ∈ S} = m(S). It is enough to show that I ⊇ {⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} | b_{1} ⊗ ⋯ ⊗ b_{n} ∈ T } = m(T) in this case. For that, note that m^{−1}(I) is a ⊗-ideal in A ⊗ ⋯ ⊗ A containing S: If R is an arbitrary subset of A and if a_{2}, …, a_{n} ∈ A such that r ⊗ a_{2} ⊗ ⋯ ⊗ a_{n} ∈ m^{−1}(I) for all r ∈ R then ⋄r ∧ ⋄a_{2} ∧ ⋯ ∧ ⋄a_{n} ∈ I. Hence ⋄ ⋁ R ∧ ⋄a_{2} ∧ ⋯ ∧ ⋄a_{n} ∈ I implying ⋁ R ⊗ a_{2} ⊗ ⋯ ⊗ a_{n} ∈ m^{−1}(I). Similarly we can show that analogous results hold for the other components. Since ⋁ S = ⋁ T in A ⊗ ⋯ ⊗ A, the ⊗-ideal m^{−1}(I) must contain T. This implies m(T) ⊆ m(m^{−1}(I)) ⊆ I.

To prove that ξ′_{A} preserves all joins let S be an arbitrary subset of A. As v′_{n} preserves all joins,

⋁{a_{1} ⊗ ⋯ ⊗ a_{n} | a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(⋁ S)}
= v′_{n}( ⋁ S)
= ⋁_{s∈S} v′_{n}(s)
= ⋁{a_{1} ⊗ ⋯ ⊗ a_{n} | ∃s ∈ S . a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(s)}

for all n ∈ ℕ. Hence by the last lemma

ξ′_{A}(⋁ S) = ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | n ∈ ℕ, a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(⋁ S)}
= ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | n ∈ ℕ, ∃s ∈ S . a_{1} ⊗ … ⊗ a_{n} ≤ v′_{n}(s)}
= ⋁_{s∈S} ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | n ∈ ℕ, a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(s)}
= ⋁_{s∈S} ξ′_{A}(s).

We have shown the following result.

Proposition 6.19 Every inflationary localic semilattice gives rise to a P_{L}^{♭}-algebra.

It remains to characterize the morphisms of P_{L}^{♭}-algebras. We already know that every morphism of P_{L}^{♭}-algebras is a semilattice homomorphism (see Lemma 6.17). The following lemma gives us the reverse.

Lemma 6.20 If f: (A, v) −^{}(B, w) is a morphism of inflationary localic semilattices then f is a morphism from (A, ξ_{A}) to (B, ξ_{B}) in the category of P_{L}^{♭}-algebras.

Proof: First of all note that f ∘ v_{n} = w_{n} ∘ (f ⊗ ⋯ ⊗ f) for all n ∈ ℕ by the considerations from Section 5.1. Hence for all n ∈ ℕ and all b ∈ B we have

⋁{a_{1} ⊗ ⋯ ⊗ a_{n} | a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(f′(b))}
= v′_{n}(f′(b))
= (f ⊗ ⋯ ⊗ f)′(w′_{n}(b))
= (f ⊗ ⋯ ⊗ f)′ (⋁{b_{1} ⊗ ⋯ ⊗ b_{n} | b_{1} ⊗ ⋯ ⊗ b_{n} ≤ w′_{n}(b)})
= ⋁{f′(b_{1}) ⊗ ⋯ ⊗ f′(b_{n}) | b_{1} ⊗ ⋯ ⊗ b_{n} ≤ w′_{n}(b)}.

Because of Lemma 6.18 this yields

\begin{matrix}⋁\{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(f′(b))\} \\ = ⋁\{⋄f′(b_{1}) ∧ ⋯ ∧ ⋄f′(b_{n}) | b_{1} ⊗ ⋯ ⊗ b_{n} ≤ w′_{n}(b)\} \end{matrix}

for all b ∈ B and all n ∈ ℕ. Consequently, for all b ∈ B we finally get

ξ′_{A} (f′(b)) = ⋁{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | n ∈ ℕ, a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(f′(b))}
= ⋁{⋄f′(b_{1}) ∧ ⋯ ∧ ⋄f′(b_{n}) | n ∈ ℕ, b_{1} ⊗ ⋯ ⊗ b_{n} ≤ w′_{n}(b)}
= ⋁{(P_{L}^{♭}f)′(⋄b_{1}) ∧ ⋯ ∧ (P_{L}^{♭}f)′(⋄b_{n}) | n ∈ ℕ, b_{1} ⊗ ⋯ ⊗ b_{n} ≤ w′_{n}(b)}
= (P_{L}^{♭}f)′ (⋁{⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} | n ∈ ℕ, b_{1} ⊗ ⋯ ⊗ b_{n} ≤ w′_{n}(b)})
= (P_{L}^{♭}f)′( ξ′_{B}(b)).

We have shown the following theorem.

Theorem 6.21 The category of P_{L}^{♭}-algebras is the category of inflationary localic semilattices Loc^{∨}.

6.4.3 Unital semilattices

When we examined the Hoare power space we also asked ourselves what happens if we allow the empty set as one of its elements. It turned out that the resulting Hoare power space construction Ṗ_{T}^{♭}gives the free unital inflationary semilattice for Sob. Thus the question arises whether one can modify the Hoare power locale construction such that it also gives the free unital inflationary semilattice for Loc. In order to find such a construction, we examine ΩṖ_{T}^{♭}X, the open set lattice of Ṗ_{T}^{♭}X, for an arbitrary T_{0}-space X. In Ω P_{T}^{♭}X, the greatest element, that is the whole space P_{T}^{♭}X, is one of the generators of Ω P_{T}^{♭}X, namely ♢X = {A ∈ P_{T}^{♭}X | A∩X ≠ ∅} = P_{T}^{♭}X. For ΩṖ_{T}^{♭}X, however, this is no longer true. The reason for this is that the empty set, which is an element of Ṗ_{T}^{♭}X, is not in ♢X anymore. Therefore ♢1_{ΩX} ≠ 1_{ΩṖ_{T}^{♭}X} in Ṗ_{T}^{♭}X.

Hence, for every locale A, we define

Ṗ_{L}^{♭}A := F r ⟨⋄a : a ∈ A | ⋁_{s∈S} ⋄s = ⋄ ⋁ S (S ⊆ A)⟩.

If f: A −^{}B is a continuous function in Loc we set (Ṗ_{L}^{♭}f)′(⋄b) := ⋄f′(b) for all b ∈ B to obtain a continuous function from Ṗ_{L}^{♭}A to Ṗ_{L}^{♭}B. The proof that this does define a morphism in Loc and that this assignment makes Ṗ_{L}^{♭}an endofunctor on Loc is the same as that for P_{L}^{♭}. We can also define a natural transformations η: id_{Loc} −^{}Ṗ_{L}^{♭}and μ: (Ṗ_{L}^{♭})^{2} −^{}P_{L}^{♭}the same way as we did for P_{L}^{♭}, that is by setting η′_{A}(⋄a) := a and μ′_{A}(⋄a) := ⋄ ⋄ a for all a ∈ A where A is a locale. The proof that (Ṗ_{L}^{♭}, η, μ) is a monad is the same as for Proposition 6.14.

Theorem 6.22 The category of Ṗ_{L}^{♭}-algebras is the category of unital inflationary semilattices in Loc.

Proof: We can show in the same way as for the monad (P_{L}^{♭}, η, μ) that every Ṗ_{L}^{♭}- algebra is an inflationary localic semilattice. To prove that these semilattices are unital let us first show this for locales of the form Ṗ_{L}^{♭}A. We define o: 2 −^{}Ṗ_{L}^{♭}A by setting o′(⋄a) := 0_{2} for all a ∈ A. Obviously, o′ preserves the relations defining Ṗ_{L}^{♭}A and thus defines a morphism in Loc as desired. By Lemma 5.8 (i), o makes Ṗ_{L}^{♭}A a unital inflationary semilattice. As Lemma 5.4 also holds for unital semilattices, every Ṗ_{L}^{♭}-algebra is a unital inflationary semilattice in Loc. Furthermore, every morphism in the category of Ṗ_{L}^{♭}-algebras is a morphism of semilattices. The proof for this is exactly as that for Lemma 6.17. To see that every such f: A −^{}B also preserves the respective units, let (Ṗ_{L}^{♭}A, v, o) and (Ṗ_{L}^{♭}B, w, p) be the unital inflationary semilattices corresponding to the Ṗ_{L}^{♭}-algebras (Ṗ_{L}^{♭}A, μ_{A}) and (Ṗ_{L}^{♭}B, μ_{B}), respectively. First of all we will show that Ṗ_{L}^{♭}f preserves the units of these semilattices. For all b ∈ B we have o′((P_{L}^{♭}f)′(⋄b)) = o′(⋄f′(b)) = 0_{2} = p′(⋄b). The semilattice structures on A and B making them inflationary semilattices must be given by (ξ_{A} ∘v ∘(η_{A} × η_{A}), ξ_{A} ∘o) and (ξ_{B} ∘w ∘(η_{B} × η_{B}), ξ_{B} ∘p), respectively. Consequently, f ∘ξ_{A} ∘o = ξ_{B} ∘ P_{L}^{♭}f ∘o = ξ_{B} ∘p which proves our claim.

It remains to prove that every unital inflationary semilattice in Loc gives rise to a Ṗ_{L}^{♭}-algebra and that every morphism of such semilattices is a morphism of the corresponding Ṗ_{L}^{♭}-algebras. But if (A, v, o) is a unital inflationary semilattice in Loc then, by Lemma 5.8 (i), o′(a) = 1_{2} iff a = 1_{A}. Hence 1_{A} is inaccessible by arbitrary joins in A. This allows us to define a morphism ξ_{A}: Ṗ_{L}^{♭}A −^{}A by setting

\begin{aligned}ξ′_{A}(1_{A}) & := & 1_{Ṗ_{L}^{♭}A}, \\ ξ′_{A}(a) & := & ⋁\{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} | n ∈ ℕ, a_{1} ⊗ ⋯ ⊗ a_{n} ≤ v′_{n}(a)\} & (a ∈ A \ \{1_{A}\}). \end{aligned}

The proof that (A, ξ_{A}) is a Ṗ_{L}^{♭}-algebra works as the corresponding proof for P_{L}^{♭}-algebras. Moreover, to show that every morphism of unital inflationary semilattices in Loc is a morphism of the corresponding Ṗ_{L}^{♭}-algebras we can employ the proof of Lemma 6.20.

6.4.4 Locally compact locales

We already examined the Hoare power construction for domains and locally compact T_{0}-spaces and now do so for locally compact locales. Since locally compact locales correspond to locally compact sober spaces one might expect that we can get similar results for them. The proof of Proposition 6.11 suggests that if b_{1} ≪ a_{1}, …, b_{n} ≪ a_{n} in a locale A then ⋄b_{1} ≪ ⋄a_{1},…,⋄b_{n} ≪ ⋄a_{n} in P_{L}^{♭}A.

Lemma 6.23 If b ≪ a in a locale A then ⋄b ≪ ⋄a in P_{L}^{♭}A.

Proof: Let S be a directed subset of P_{L}^{♭}A such that ⋄a ≤ ⋁^{↑}S. We can replace S by a directed set S′ of elements of the form ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}, where a_{1}, …, a_{n} ∈ A, such that ⋁^{↑}S = ⋁^{↑}S′ and such that for any ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ S′ there is s ∈ S with ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ≤ s: Let S′ := {⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ∈ P_{L}^{♭}A | ∃s ∈ S . ⋄ a_{1} ∧ ⋯ ∧ ⋄a_{n} ≤ s}. It is enough to show that S′ is directed as the other properties we claimed it to have are easily seen to be satisfied. Let ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}, ⋄a′_{1} ∧ ⋯ ∧ ⋄a′_{m} ∈ S′. There are s and s′ in S with ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ≤ s and ⋄a′_{1} ∧ ⋯ ∧ ⋄a′_{m} ≤ s′. By directedness of S there is an upper bound s′′ for s and s′ in S. Therefore,

⋀_{i=1}^{n} ⋀_{j=1}^{m} ⋄(a_{i} ∨ a′_{j}) = ⋀_{i=1}^{n} ⋀_{j=1}^{m}(⋄a_{i} ∨ ⋄a′_{j}) = (⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}) ∨ (⋄a′_{1} ∧ ⋯ ∧ ⋄a′_{m}) ≤ s′′

which proves that ⋀_{i=1}^{n} ⋀_{j=1}^{m} ⋄(a_{i} ∨ a′_{j}) is a common upper bound for ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} and

⋄a′_{1} ∧ ⋯ ∧ ⋄a′_{n} in S′. Hence it is sufficient to prove that if ⋄a ≤ ⋁_{i∈I}^{↑} ⋄ a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}

then there is i ∈ I such that ⋄b ≤ ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}. But if ⋄a ≤ ⋁_{i∈I}^{↑} ⋄ a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}

for a directed subset {⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i} | i ∈ I} of P_{L}^{♭}A then a ≤ ⋁_{i∈I}^{↑} a_{1}^{i} ∧ ⋯ ∧ a_{n(i)}^{i}.

Hence by the assumption there is i ∈ I such that b ≤ a_{1}^{i} ∧ ⋯ ∧ a_{n(i)}^{i}. Consequently, ⋄b ≤ ⋄(a_{1}^{i} ∧ ⋯ ∧ a_{n(i)}^{i}) ≤ ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}.

Lemma 6.24 Let a_{1}, …, a_{n}, b_{1}, …, b_{n} be elements of a locale A. If ⋄b_{1} ≪ ⋄a_{1}, …, ⋄b_{n} ≪ ⋄a_{n} in P_{L}^{♭}A. Then ⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} ≪ ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}.

Proof: By Lemma 6.15, P_{L}^{♭}A is an inflationary localic semilattice where the semilattice structure is given by v′: ⋄a −^{}(⋄a ⊗ 1_{P_{L}^{♭}A}) ∨ (1_{P_{L}^{♭}A} ⊗ ⋄a) for all a ∈ A. Assume that {⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i} | i ∈ I} is a directed subset of P_{L}^{♭}A such that

⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ≤ ⋁_{i∈I}^{↑} ⋄ a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}.

As v′_{n} is the upper adjoint for Δ′_{n}, where Δ_{n}: A −^{}A ⊗ ⋯ ⊗ A is the n-ary diagonal, this is equivalent to

⋄a_{1} ⊗ ⋯ ⊗ ⋄a_{n} ≤ v′_{n}( ⋁_{i∈I}^{↑} ⋄ a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}) = ⋁_{i∈I}^{↑} v′_{n}(⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}).

By Lemma 3.7, ⋄b_{1} ⊗ ⋯ ⊗ ⋄b_{n} ≪ ⋄a_{1} ⊗ ⋯ ⊗ ⋄a_{n}. Hence there is i ∈ I such that ⋄b_{1} ⊗ ⋯ ⊗ ⋄b_{n} ≤ v′_{n}(⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}) which is equivalent to ⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} = Δ′_{n}(⋄b_{1} ⊗ ⋯ ∧ ⋄b_{n}) ≤ ⋄a_{1}^{i} ∧ ⋯ ∧ ⋄a_{n(i)}^{i}.

Proposition 6.25 If A is a locally compact locale then so is P_{L}^{♭}A.

Proof: As P_{L}^{♭}A is a complete lattice, we already know that ↓ ↓{s} is directed for all s ∈ P_{L}^{♭}A. By Lemma 6.23 we know that {⋄b | b ≪ a} ⊆ ↓ ↓{⋄a} for all a ∈ A. Hence because of Lemma 6.24 we get

{⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} | b_{1} ≪ a_{1}, …, b_{n} ≪ a_{n} in A} ⊆ ↓ ↓{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}}

in P_{L}^{♭}A for all a_{1}, …, a_{n} ∈ A. Consequently,

⋄a_{1} ∧ ⋯ ∧ ⋄a_{n} ≥ ⋁^{↑} ↓ ↓{⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}}
≥ ⋁^{↑}{⋄b_{1} ∧ ⋯ ∧ ⋄b_{n} | b_{1} ≪ a_{1}, …, b_{n} ≪ a_{n}}
= ⋁^{↑}{⋄b_{1} ∈ A | b_{1} ≪ a_{1}} ∧ ⋯ ∧ ⋁^{↑}{⋄b_{n} ∈ A | b_{n} ≪ a_{n}}
= ⋄ ⋁^{↑} ↓ ↓{a_{1}} ∧ ⋯ ∧ ⋄ ⋁^{↑} ↓ ↓{a_{n}}
= ⋄a_{1} ∧ ⋯ ∧ ⋄a_{n}.

Therefore P_{L}^{♭}gives the free inflationary localic semilattice for the full subcategory of Loc whose objects are the locally compact locales. Remark. The same considerations apply to Ṗ_{L}^{♭}.

6.4.5 Power space versus power locale

We have already seen a connection between the Hoare power space P_{T}^{♭}X and the Hoare power locale P_{L}^{♭}ΩX for some T_{0}-space X: As the generators for the topology on P_{T}^{♭}X satisfy the equations for the Hoare power locale we get a frame homomorphism from P_{L}^{♭}ΩX to Ω P_{T}^{♭}X from the assignment ⋄U −^{}♢ U. The question is when this defines an isomorphism. Let us first examine the points for P_{L}^{♭}A where A is an arbitrary locale. The points of P_{L}^{♭}A correspond to frame homomorphisms from P_{L}^{♭}A to 2 which in turn correspond to functions from A to 2 that preserve all joins and 1. But these can be understood as being given by the elements of A\{1_{A}}: For any a ∈ A\{1_{A}} the function p_{a}: A −^{}2 that maps b ∈ A to 0_{2} iff b is below a is such a homomorphism. On the other hand, if any such homomorphism is mapped to the largest element mapped to 0_{2} then this assignment gives us the inverse for a −^{}p_{a}. Note that this bijection is orderreversing and hence PtP_{L}^{♭}A with respect to the specialization order is isomorphic to (A \ {1_{A}})^{op} (see also [Vic89]). If A is spatial then (A \ {1_{A}})^{op} can be understood to be the up-complete ∨-semilattice of nonempty closed subsets of SpA ordered by inclusion. Thus for any spatial locale A the points of P_{L}^{♭}A agree with the points of P_{T}^{♭}SpA. Put differently, for every T_{0}-space X the points of P_{L}^{♭}ΩX are just the points of P_{T}^{♭}X, that is the non-empty closed subsets of X. This is a homeomorphism ι_{X}: P_{T}^{♭}X −^{}SpP_{L}^{♭}ΩX: An open subset ♢U = {A ∈ P_{T}^{♭}X | A ∩ U ≠ ∅} is mapped to the set

{p ∈ SpP_{L}^{♭}ΩX | ∃A ∈ ♢U . p′(⋄U′) = 0_{2} iff U′ ⊆ X \ A} = {p ∈ SpP_{L}^{♭}ΩX | p′(⋄U) = 1_{2}} = {p ∈ SpP_{L}^{♭}ΩX | p ⊧ ⋄U}

of points of SpP_{L}^{♭}ΩX. The family of all ι_{X} where X is a T_{0}-space is a natural isomorphism P_{T}^{♭}−^{}SpP_{L}^{♭}Ω. To see this, let f: X −^{}Y be a continuous function. The isomorphism ι_{X} maps a non-empty closed subset A of X to the point 2 −^{}P_{L}^{♭}ΩX in SpP_{L}^{♭}ΩX that is defined by p′_{A}(⋄U) = 0_{2} iff A ∩ U = ∅. We want to show that SpP_{L}^{♭}Ωf ∘ ι_{X} = ι_{Y} ∘ P_{T}^{♭}f. We do this by proving that for all A ∈ P_{T}^{♭}X the resulting continuous functions of locales from 2 to P_{L}^{♭}ΩY coincide. For every A ∈ P_{T}^{♭}X we have

(SpP_{L}^{♭}Ωf)(ι_{X}(A)) = SpP_{L}^{♭}Ωf(p_{A}) = P_{L}^{♭}Ωf ∘ p_{A}.

For all V ∈ ΩY we calculate p′_{A}((P_{L}^{♭}Ωf)′(⋄V)) = p′_{A}(⋄(Ωf)′(V)) = p_{A}(⋄f^{−1}(V)) = 0_{2} iff A ∩ f^{−1}(V) = ∅. Since ι_{Y} (P_{T}^{♭}f(A)) = p_{cl f(A)} and since in addition p_{cl f(A)}(⋄V) = 0_{2} iff cl f(A) ∩ V = ∅ for all V ∈ ΩY we have shown naturality of ι. This gives rise to the idea that the monad (P_{T}^{♭}, cl, ⋃) is equivalent to a monad we can build from (P_{L}^{♭}, η, μ). We cannot take SpηΩ as the first natural transformation since SpηΩ is a natural transformation SpΩ −^{}SpP_{L}^{♭}Ω while we want the first functor to be the identity on Top. But there is a natural isomorphism id_{Sob} −^{}SpΩ which we denote by α (see Section 2.1.3). We thus get the natural transformation \overline{η} := SpηΩ • α: id_{Sob} −^{}SpP_{L}^{♭}Ω. Moreover, \overline{μ} := Sp∘(μ • (P_{L}^{♭}∘β ∘ P_{L}^{♭})) ∘ Ω is a natural transformation (SpP_{L}^{♭}Ω)^{2} −^{}SpP_{L}^{♭}Ω where β: Ω Sp −^{}id_{Loc} is the natural transformation from Section 2.1.4.

Proposition 6.26 There is a natural isomorphism P_{T}^{♭}−^{}SpP_{L}^{♭}Ω for the functors P_{T}^{♭}, SpP_{L}^{♭}Ω: Sob −^{}Sob. Furthermore, the monads (P_{T}^{♭}, cl, ⋃) and (SpP_{L}^{♭}Ω, \overline{η}, \overline{μ}) on Sob are equivalent.

If P_{L}^{♭}ΩX is spatial then P_{L}^{♭}ΩX ≅ Ω SpP_{L}^{♭}ΩX ≅ Ω P_{T}^{♭}X. Since for every locally compact space X we get a locally compact space ΩX, we know by Proposition 6.25 that P_{L}^{♭}ΩX is locally compact as well and therefore a spatial locale. As the above isomorphisms all are natural we obtain the following result.

Proposition 6.27 (i) The functors Ω P_{T}^{♭}: LCompT −^{}LCompL and P_{L}^{♭}Ω are naturally isomorphic.

(ii) The functors P_{T}^{♭}Sp, SpP_{L}^{♭}: LCompL −^{}LCompT are naturally isomorphic.

106

The Smyth power construction

7.1 Algebraic domains

We will now present a different way to view non-determinism. Before, we only asked for one of the branches of a non-deterministic program to give the desired result. But, of course, if we run such a program we cannot be sure that we really end up with this result. In the worst case, the program might even be non-terminating. Now we only consider a non-deterministic program to have a certain property if all its branches satisfy that property. We also speak of ‘demonic’ non-determinism in this case. The denotation of a program whose branches are denoted by Scott-continuous functions with source D and target E is to be a function from D which takes its values in the power set of E. This time, however, we want the denotation of a program p to be greater than or equal to that of q iff for each branch of p there is one of q with lesser (or equal) denotation. For any algebraic domain D we thus get the following preorder on non-empty finite subsets F and F′ of CD: We set F ≤^{♯} F′ iff for all c′ ∈ F′ there is c ∈ F with c ≤ c′. The following definition is due to M. Smyth [Smy78]. Definition. The Smyth power domain P_{A}^{♯}D of an algebraic domain D is the ideal completion of (P_{fne}CD, ≤^{♯}).

As for the Hoare power domain, there is another way of trying to capture nondeterminism. Once again, given a denotational semantics for a deterministic programming language, we would like to add the command \underline{or} to that language. The way we have interpreted non-determinism, we want p \underline{or} q to be denoted by the greatest lower bound of the denotations for p and q. This approach calls for the free deflationary semilattice for AlgD. As for the Hoare power construction, this point of view also leads to the Smyth power domain. Let AlgD^{∧} denote the category of deflationary semilattices in AlgD. We will refer to these posets as algebraic ∧-semilattices. The following result is also due to Hennessy and Plotkin [HP79].

Theorem 7.1 The free deflationary semilattice on AlgD is given by P_{A}^{♯}.

Put differently, this means that the forgetful functor from AlgD^{∧} to AlgD has a left adjoint. Furthermore, this adjunction is monadic.

7.1.1 P_{A}^{♯}D as a subset of PD

In order to define a Smyth power construction for arbitrary dcpo’s, we would like to be able to view P_{A}^{♯}D as a subset of the power set of D. If we go back to how we accomplished the corresponding task for the Hoare power construction, we find that we mapped a non-empty finite subset of CD to its lower closure in D. As P_{fne}CD is ordered differently for the Smyth construction, we map a finite subset of CD to its upper closure in D. But in order to make this an order-preserving function we have to order PD by reverse inclusion. This yields an embedding of (P_{fne}CD, ⊇) into (PD, ⊇). There is a unique Scottcontinuous extension e of this embedding to P_{A}^{♯}D = Idl(P_{fne}CD, ⊇). The image of P_{A}^{♯}D under e is the set of filtered intersections of sets of the form ↑{c_{1}} ∪ ⋯ ∪ ↑{c_{n}} where c_{1}, …, c_{n} ∈ CD. There is a nicer characterization for this subset of PD. First we observe that all sets in the image of P_{A}^{♯}D are upper closed. We claim that the image of P_{A}^{♯}D in PD is the set of non-empty saturated Scott-compact subsets of D. In order to prove this we need some results for sober spaces which can be found in [HM81] where they are shown with the help of spectral theory of complete lattices. A direct proof of Lemma 7.2 and Theorem 7.4 has first been given in [KP93]. Recall that an open subset of a topological space is prime (in the open set lattice of X) if it is not the intersection of two of its proper open supersets. As prime open sets obviously correspond to irreducible closed sets (via complementation), a T_{0}-space X is sober iff the only prime elements of ΩX are complements of singleton closures.

Lemma 7.2 (Proposition 2.19 in [HM81]) Let X be a sober space and let S be a filtered family of compact saturated subsets of X.

(i) If ⋂_{↓} S ⊆ U for some open subset U of X then there is K ∈ S with K ⊆ U.

(ii) The set ⋂_{↓} S is also compact in X. If, in addition, all elements of S are non-empty then ⋂_{↓} S is non-empty as well.

Proof: (i) Let R := {U ∈ ΩX | ⋂_{↓} S ⊆ U, K ⊈ U for all K ∈ S}. Suppose that R is non-empty since otherwise (i) is true. Then R is an inductive system: If C is a chain in R then, by compactness of K, we have K ⊆ ⋃ C iff K ⊆ U for some U ∈ C for all K ∈ S. Hence K ⊈ ⋃ C for all K ∈ S which proves that ⋃ C ∈ R. By Zorn’s Lemma there is a maximal element, say U, in R. We show that U is prime. Let U = V ∩ V′ for open subsets V and V′ of X with U ⊂ V, V′. By maximality of U in R there are K and K′ in S with K ⊆ V and K′ ⊆ V′. As S is filtered, there is K′′ ∈ S with K′′ ⊆ K ∩ K′ ⊆ U which contradicts U ∈ R. By sobriety of X there is x ∈ X with U = X \ cl{x}. Since x ∈ V holds for all open V with U ⊂ V and because all elements of S are saturated we must have x ∈ K for all K ∈ S. As this contradicts ⋂_{↓} S ⊆ U we conclude that R is empty. (ii) Follows from (i).

Obviously, the sets of the form ↑{d_{1}, …, d_{n}}, where d_{1}, …, d_{n} ∈ D, are Scottcompact saturated subsets of D. Hence, according to Lemma 7.2, all sets in the image of P_{A}^{♯}D under e: P_{A}^{♯}D −^{}PD are non-empty, Scott-compact and saturated. Therefore it is sufficient to prove that all non-empty saturated compact subsets of D can be written as filtered intersections of sets of the form ↑{c_{1}, …, c_{n}} where c_{1}, …, c_{n} ∈ CD.

Lemma 7.3 If K is a non-empty Scott-compact saturated subset of a domain D

then K = ⋂_{↓}{↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}}.

Proof: Obviously, K ⊆ ⋂_{↓}{↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}}. On the other hand

we know that every saturated set is the intersection of its open neighbourhoods. By compactness of K this yields

K = ⋂_{↓}{↑ ↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}}.

But if K ⊆ ↑ ↑{d_{1}, …, d_{n}} for some d_{1}, …, d_{n} ∈ D then, by the interpolation property, for all k ∈ K we can find e_{k} ∈ ↑ ↑{d_{1}, …, d_{n}} with e_{k} ≪ k. Again by compactness of K there are d′_{1}, …, d′_{m} ∈ ↑ ↑{d_{1}, …, d_{n}} such that K is a subset of ↑ ↑{d′_{1}, …, d′_{m}}. Hence

K ⊆ ↑ ↑{d′_{1}, …, d′_{m}} ⊆ ↑{d′_{1}, …, d′_{m}} ⊆ ↑ ↑{d_{1}, …, d_{n}}

and we may conclude

⋂_{↓}{↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}}

⊆ ⋂_{↓}{↑ ↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}}
= K.

For a T_{0}-space X let KX denote the set of non-empty compact saturated subsets of X ordered by reverse inclusion. For an algebraic domain D we have just seen that P_{A}^{♯}D is isomorphic to K_{D}D := KT_{σ}D.

7.2 Directed complete partial orders

7.2.1 A Smyth power construction

The observations from the last section allow us to define a Smyth power construction for sober dcpo’s. For a non-sober domain D we cannot guarantee K_{D}D to have directed joins, that is filtered intersections. Hence we search for another way to understand the Smyth power construction. For that, let Ω_{ne} X denote the set of non-empty open subsets of a topological space X. The following result is also known as the Hofmann-Mislove Theorem.

Theorem 7.4 (Theorem 2.16 in [HM81]) If X is a sober space then the set KX of non-empty compact saturated subsets of X ordered by reverse inclusion is isomorphic to the set of Scott-open filters in Ω_{ne} X ordered by inclusion. The isomorphisms are given by

K −^{}F(K) := {U ∈ Ω_{ne} X | K ⊆ U} where K ∈ KX and
φ −^{}⋂_{↓} φ  where φ ⊆ Ω_{ne} X is a Scott-open filter.

For an arbitrary dcpo D let F_{D}D be the set of Scott-open filters in Ω_{ne} D ordered by reverse inclusion. Recall that by default the topology we consider for a dcpo is the Scott topology; hence Ω_{ne} D is to denote the lattice of non-empty Scott-open subsets of D. As directed unions of Scott-open filters in Ω_{ne} D again are Scott-open filters, F_{D}D is indeed a dcpo. Later on it will become obvious that this construction does not satisfy our expectations. For this reason, we introduce another Smyth power dcpo construction. This power construction has been studied by R. Heckmann in [Hec92b]. His goal, however, is different from ours; he does not consider the monad associated with this construction. We can extend F_{D} to an endofunctor on DCPO that yields a monad. From a Scottcontinuous function f: D −^{}E between dcpo’s we obtain another Scott-continuous function F_{D}f: F_{D}D −^{}F_{D}E if we set

F_{D}f(φ) := {V ∈ Ω_{ne} E | f^{−1}(V) ∈ φ}

for all φ ∈ F_{D}D. We show that F_{D}f is well-defined: If ⋃_{↑} V ∈ F_{D}f(φ) for some directed family V in Ω_{ne} E then ⋃_{↑}{f^{−1}(V) | V ∈ V} = f^{−1}(⋃_{↑} V) ∈ φ which, by Scott-openness of φ, implies the existence of some V ∈ V with f^{−1}(V) ∈ φ and thus V ∈ F_{D}f(φ). We now show that F_{D}f is indeed Scott-continuous. So let S be a directed family of Scott-open filters in Ω_{ne} D. Then

F_{D}f( ⋃_{↑} S) = {V ∈ Ω_{ne} E | f^{−1}(V) ∈ ⋃_{↑} S}
= {V ∈ Ω_{ne} E | ∃ φ ∈ S . f^{−1}(V) ∈ φ}
= ⋃_{↑}{F_{D}f(φ) | φ ∈ S}.

Lemma 7.5 The above definition makes F_{D} an endofunctor on DCPO.

Proof: It is obvious that F_{D} preserves identities. Let f: D −^{}E and g: E −^{}F be morphisms in DCPO. For any Scott-open filter φ in Ω_{ne} D we get

( F_{D}g ∘ F_{D}f)(φ) = F_{D}g({V ∈ Ω_{ne} E | f^{−1}(V) ⊆ φ})
= {W ∈ Ω_{ne} F | f^{−1}(g^{−1}(W)) ∈ φ}
= F_{D}(g ∘ f)(φ).

7.2.2 The monad

In order to define a monad we still need two natural transformations. We can embed any dcpo D into F_{D}D if we map any element x of D to the Scott-open filter of its open neighbourhoods N_{D}(x). This defines a function N_{D}: D −^{}F_{D}D which is Scottcontinuous since for any directed subset S of D and any U ∈ Ω_{ne} D we have ⋁^{↑}S ∈ U iff there is s ∈ S with s ∈ U. This assignment makes N a natural transformation from the identity on DCPO to F_{D}. To see this, let f: D −^{}E be a Scott-continuous function. For any d ∈ D we have

F_{D}f( N_{D}(d)) = {V ∈ Ω_{ne} E | f^{−1}(V) ∋ d} = {V ∈ Ω_{ne} E | V ∋ f(d)} = N_{D} (f(d)).

We can also define a natural transformation (F_{D})^{2} −^{}F_{D}. For every dcpo D we set μ_{D}(Φ) := {U ∈ Ω_{ne} D | {φ | U ∈ φ} ∈ Φ} for every Φ in (F_{D})^{2}D. First of all we show that μ_{D} is well-defined, that is {U ∈ Ω_{ne} D | {φ | U ∈ φ} ∈ Φ} is a Scott-open filter in Ω_{ne} D for all Φ ∈ (F_{D})^{2}D.

Lemma 7.6 If U is a Scott-open subset of a dcpo D then □U := {φ ∈ F_{D}D | U ∈ φ} is Scott-open in F_{D}D.

Proof: Obviously, □U is an upper subset of F_{D}D. If S is a directed family in F_{D}D with ⋃_{↑} S ∈ □U then U ∈ φ must hold for some φ ∈ S. Hence there is φ ∈ S with φ ∈ □U which proves Scott-openness of □U.

Obviously, μ_{D}(Φ) = {U ∈ Ω_{ne} D | □U ∈ Φ} is an upper set in F_{D}D. It is closed with respect to binary intersections since □U ∩ □U′ = □(U ∩ U′) for all U, U′ ∈ ΩD. Furthermore, this set is Scott-open since ⋃_{↑}{□U | U ∈ U} = □ ⋃_{↑} U for all directed subsets U of Ω_{ne} D. We also have to show that for all dcpo’s D the function μ_{D} is Scott-continuous. If 𝔖 is a directed family in (F_{D})^{2}D then

μ_{D} ( ⋃_{↑} 𝔖) = {U ∈ Ω_{ne} D | □U ∈ ⋃_{↑} 𝔖}
= {U ∈ Ω_{ne} D | ∃Φ ∈ 𝔖 . □U ∈ Φ}
= ⋃_{↑}{μ_{D}(Φ) | Φ ∈ 𝔖}.

Let f: D −^{}E be a morphism in DCPO. Note that, for any open subset V of E, we get □f^{−1}(V) = {φ ∈ F_{D}D | f^{−1}(V) ∈ φ} = (F_{D}f)^{−1}(□V). Now for all Φ ∈ (F_{D})^{2}D we calculate

μ_{E} ((F_{D})^{2}f(Φ)) = μ_{E} ({V ∈ Ω_{ne} F_{D}E | (F_{D}f)^{−1}(V) ∈ Φ})
= {V ∈ Ω_{ne} E | (F_{D}f)^{−1}(□V) ∈ Φ}
= {V ∈ Ω_{ne} E | □f^{−1}(V) ∈ Φ}
= F_{D}f(μ_{D}(Φ)),

which shows that μ is indeed a natural transformation (F_{D})^{2}−^{}F_{D}.

Proposition 7.7 The triple (F_{D}, N , μ) is a monad on DCPO. Proof: It remains to prove commutativity of the diagrams defining a monad. For any dcpo D and any φ ∈ F_{D}D we have

μ_{D} ( N_{F_{D} D}(φ)) = μ_{D} ({U ∈ Ω_{ne} F_{D}D | φ ∈ U})
= {U ∈ Ω_{ne} D | φ ∈ □U}
= φ
= {U ∈ Ω_{ne} D | N_{D}^{−1}(□U) ∈ φ}
= μ_{D} ({U ∈ Ω_{ne} F_{D}D | N_{D}^{−1}(U) ∈ φ})
= μ_{D} ( F_{D}N_{D}(φ))

which proves the commutativity of the two triangular diagrams. In order to show that the rectangular diagram also commutes, let Φ be an element of (F_{D})^{3}D. Because μ_{D}^{−1}(□U) = □□U for all U ∈ Ω_{ne} D, we have

μ_{D} ( μ_{F_{D} D}(Φ)) = {U ∈ Ω_{ne} D | □□U ∈ Φ}
= {U ∈ Ω_{ne} D | μ_{D}^{−1}(□U) ∈ Φ}
= μ_{D} ( F_{D}μ_{D}(Φ)).

7.2.3 The algebras for the monad

In the remainder of this section we will examine whether F_{D} gives the free deflationary semilattice for DCPO. Recall that a deflationary semilattice in DCPO is a dcpo with binary meets that distribute over directed joins. We call such a poset a ∧- continuous ∧-semilattice. Note that every dcpo of the form F_{D}D is a ∧-semilattice since the intersection of two Scott-open filters is again a Scott-open filter. Also, binary meets distribute over directed joins in F_{D}D. Hence any F_{D}D is a deflationary semilattice in DCPO.

Lemma 7.8 Every F_{D}-algebra is a deflationary semilattice in DCPO. Proof: This follows from Proposition 5.6.

Moreover, every morphism of F_{D}-algebras preserves the semilattice structure, that is binary meets.

Lemma 7.9 If f: D −^{}E is a Scott-continuous function between dcpo’s then the function F_{D}f: F_{D}D −^{}F_{D}E preserves binary meets.

Proof: Let φ and φ′ be elements of F_{D}D. We calculate

F_{D}f(φ) ∩ F_{D}f(φ′) = {V ∈ Ω_{ne} E | f^{−1}(V) ∈ φ} ∩ {V ∈ Ω_{ne} E | f^{−1}(V) ∈ φ′}
= {V ∈ Ω_{ne} E | f^{−1}(V) ∈ φ ∩ φ′}
= F_{D}f(φ ∩ φ′).

Lemma 7.10 Every morphism of F_{D}-algebras preserves binary meets.

Proof: The semilattice structure for an F_{D}-algebra (D, ξ) is ∧ = ξ ∘ ∩ ∘ (N_{D} × N_{D}). Hence for every morphism of F_{D}-algebras f: (D, ξ) −^{}(E, ζ) we have to show that f ∘ ξ ∘ ∩ ∘ (N_{D} × N_{D}) = ζ ∘ ∩ ∘ (N_{E} × N_{E}) ∘ (f × f). We calculate

f ∘ ξ ∘ ∩ ∘ (N_{D} × N_{D})
= ζ ∘ F_{D}f ∘ ∩ ∘ (N_{D} × N_{D})  (f morphism of F_{D}-algebras)
= ζ ∘ ∩ ∘ (F_{D}f × F_{D}f) ∘ (N_{D} × N_{D}) (F_{D}f preserves binary meets)
= ζ ∘ ∩ ∘ (N_{E} × N_{E}) ∘ (f × f)  (Naturality of N).

Thus we achieve the following result.

Proposition 7.11 The category of F_{D}-algebras is a subcategory of the category of deflationary semilattices in DCPO.

However, the reverse is not true. The reason for this is that any F_{D}-algebra has more than just binary meets.

Lemma 7.12 Every F_{D}-algebra has meets of non-empty compact subsets.

Proof: Let (D, ξ) be an F_{D}-algebra. Each non-empty compact subset K of D gives rise to a Scott-open filter in Ω_{ne} D, namely the filter F(K) of its open neighbourhoods. We will show that ξ(F(K)) = ⋀ K. For any k ∈ K the set N_{D}(k) of open neighbourhoods of k contains F(K). Therefore k = ξ(N_{D}(k)) ≥ ξ(F(K)) for all k ∈ K which proves that ξ(F(K)) is a lower bound for K. If d is a lower bound for K then every open neighbourhood of d is a neighbourhood of K. Hence d = ξ(N_{D}(d)) ≤ ξ(F(K)). This proves that ξ(F(K)) is indeed the greatest lower bound for K.

We will now give an example that shows that F_{D} does not give the free deflationary semilattice for DCPO.

7.2.4 Why F_{D} does not give the free deflationary semilattice for DCPO

We construct a dcpo which is a deflationary semilattice and which has lots of compact subsets that do not have a meet. Let D be the dcpo of non-empty subsets of the unit interval [0, 1] which have countable complements, the order on D being subset inclusion. First of all we will show that D is a deflationary semilattice in DCPO. For that purpose we have to prove that the binary meet function from D × D to D is Scott-continuous. This is equivalent to binary meets distributing over directed joins. Since in D these operations are given by binary intersection and directed union, respectively, D is indeed a deflationary semilattice in DCPO. We will show that D contains a non-empty Scott-compact subset that does not have a meet in D. By the last lemma this implies that D is not an F_{D}-algebra. Let K := {[0, 1] \ {x} | x ∈ [0, 1/2]}. To show that K is compact (with respect to the Scott topology), let U be a non-empty Scott-open subset of D. We claim that all but finitely many of the elements of K belong to U. Assume there are infinitely many x ∈ [0, 1/2] such that [0, 1] \ {x} ∈ D \ U. We can therefore find a sequence (z_{n})_{n∈ℕ} in [0, 1/2] with [0, 1] \ {z_{n}} ∈ D \ U for all n ∈ ℕ such that z_{n} = z_{m} iff n = m. But

[0, 1] = [0, 1] \ ∅ = [0, 1] \ ⋂_{m∈ℕ}_{↓}{z_{n} | n ≥ m} = ⋃_{m∈ℕ}^{↑}([0, 1] \ {z_{n} | n ≥ m})

is an element of U. Hence by Scott-openness of U we can find m ∈ ℕ such that [0, 1] \ {z_{n} | n ≥ m} ∈ U. This implies X \ {z_{n}} ∈ U for all n ≥ m which contradicts the choice of the z_{n}. Consequently, any non-empty Scott-open subset of D contains all but finitely many elements of K. This proves the compactness of K in D. But K does not have any lower bound in D and thus we are done. It can also be deduced from R. Heckmann’s results that F_{D} does not give the free deflationary semilattice for DCPO since he shows that this functor does not have the necessary universal property [Hec92a]. In [HP79] it is shown that the forgetful functor from DCPO^{∨} to DCPO does have a left adjoint. However, this left adjoint is not explicitly given. It seems that this functor does not map a dcpo D to a subset of its power set.

7.2.5 Domains

The reason that F_{D} does not give the free deflationary semilattice for DCPO is that it is too big. For if we take all Scott-open filters in Ω_{ne} D then we automatically get the filters of open neighbourhoods of compact subsets of D. This implies that any F_{D}- algebra must have meets of compact sets while we only require meets of non-empty finite sets. We therefore will give an alternative definition for a generalization of the Smyth power construction to arbitrary dcpo’s. As this alternative construction is easier to handle and closer to what we want, we will not examine the monad (F_{D}, N , μ) in this general setting any longer. Instead we will investigate a subcategory of DCPO where the F_{D} does give the free deflationary semilattice. We restrict the monad (F_{D}, N , μ) to the category Dom of domains. As any domain is sober we may consider the set of compact saturated subsets K_{D}D instead of taking Scott-open filters of non-empty open sets. First of all we show that Dom is closed with respect to the application of K_{D}. Because of the Hofmann-Mislove Theorem this result is a consequence of the Lawson duality for domains [Law79].

Proposition 7.13 If D is a continuous dcpo so is K_{D}D.

Proof: By Lemma 7.3, K = ⋂_{↓}{↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}} if K is a non-

empty compact saturated subset of D. Note that any set of the form ↑{d_{1}, …, d_{n}} with ↑ ↑{d_{1}, …, d_{n}} ⊇ K is way below K in K_{D}D: If K ⊇ ⋂_{↓} S for some directed family S in K_{D}D then, by Lemma 7.2 (i), there is K′ ∈ S with

K′ ⊆ ↑ ↑{d_{1}, …, d_{n}} ⊆ ↑{d_{1}, …, d_{n}}

since ↑ ↑{d_{1}, …, d_{n}} is an open neighbourhood for K. Continuity of K_{D}D follows from the fact that {↑{d_{1}, …, d_{n}} | K ⊆ ↑ ↑{d_{1}, …, d_{n}}} is a directed subset of ↓ ↓{K} in K_{D}D with join (that is intersection) K.

We therefore get a monad on Dom that is equivalent to (F_{D}, N , μ): As a functor, take K_{D}. A natural transformation from the identity on Dom to K_{D} is ↑ where ↑_{D}(d) := ↑{d}. For the natural transformation (K_{D})^{2} −^{}K_{D} take ⋃ where the components of ⋃ are defined by _{⋃D}(K) = ⋃ K. Since the resulting monad (K_{D}, ↑, ⋃) is equivalent to (F_{D}, N , μ) on Dom it is enough to characterize the K_{D}-algebras. For this it is sufficient to show that every deflationary semilattice in Dom gives rise to a K_{D}-algebra and that every morphism of such semilattices is a morphism of the corresponding K_{D}-algebras because of Proposition 7.11.

Lemma 7.14 Every deflationary semilattice in Dom has a unique structure map making it a K_{D}-algebra. This structure map maps every non-empty compact and saturated set to its meet.

Proof: Let D be a deflationary semilattice in Dom. This means that D is a domain with binary meets that distribute over directed joins. For every K ∈ K_{D}D set ξ_{D}(K) := ⋁^{↑}{d_{1} ∧ ⋯ ∧ d_{n} | ↑ ↑{d_{1}, …, d_{n}} ⊇ K}. To see that ξ_{D} is Scottcontinuous, let S be a directed family in K_{D}D. Obviously, ⋁_{K∈S}^{↑} ξ_{D}(K) ≤ ξ_{D}(⋂_{↓} S). But if d ≪ ξ_{D}(⋂_{↓} S) then ↑ ↑{d} is an open neighbourhood of ⋂_{↓} S. Sobriety of D allows us to apply Lemma 7.2 (i), so there exists K ∈ S with ↑ ↑{d} ⊇ K. Therefore, d ≤ ξ_{D}(K) ≤ ξ_{D}(⋂_{↓} S). Furthermore, ξ_{D}(K) = ⋀ K for all non-empty compact and saturated K ⊆ D: It is obvious that ξ_{D}(K) is a lower bound for K. If d ∈ D is any lower bound for K then ↑ ↑{d′} is a neighbourhood of K for all d′ ≪ d. So we can find d_{1}, …, d_{n} ∈ D with ↑ ↑{d_{1}, …, d_{n}} ⊇ K and ↑ ↑{d′} ⊇ ↑{d_{1}, …, d_{n}}. Hence d′ ≤ d_{1} ∧ ⋯ ∧ d_{n} ≤ ξ_{D}(K). By Lemma 4.4, ξ_{D} is the unique structure map for D which completes the proof.

Lemma 7.15 Let f: D −^{}E be a morphism of the deflationary semilattices D and E in Dom. Then f preserves meets of non-empty compact sets.

Proof: By the previous lemma we know that meets of non-empty compact sets do exist in D and E. Let K ∈ K_{D}D. Obviously, f(⋀ K) ≤ ⋀ f(K). If e ≪ ⋀ f(K) then f^{−1}(↑ ↑{e}) is an open neighbourhood of K. Hence, by Lemmata 7.2 (i) and 7.3, there are d_{1}, …, d_{n} ∈ D with ↑ ↑{d_{1}, …, d_{n}} ⊇ K and f^{−1}(↑ ↑{e}) ⊇ ↑{d_{1}, …, d_{n}}. Therefore, e ≤ f(d_{1}) ∧ ⋯ ∧ f(d_{n}) = f(d_{1} ∧ ⋯ ∧ d_{n}) ≤ f(⋀ K).

Theorem 7.16 The category of K_{D}-algebras in Dom is the category of deflationary semilattices in Dom.

Proof: Because of Lemma 7.14 and by Proposition 7.11 it remains to prove that any morphism of deflationary semilattices is a morphism of the corresponding K_{D}- algebras. This follows from Lemma 4.4 (ii) and from the last lemma.

These results for F_{D} restricted to Dom have already been published in [Sch93a].

7.2.6 Another Smyth power construction

The above observations call for a smaller Smyth power domain. Following R. Heckmann [Hec92b], we introduce the following notion. Definition. A filter φ in Ω_{ne} X, where X is a T_{0}-space, is called strong if for every U ∈ φ there is a finite subset F of U such that the filter of open neighbourhoods of F is a subset of φ. By F D^{s}D we denote the set of strong filters in Ω_{ne} D ordered by inclusion. Note that the filter of open neighbourhoods of {d_{1}, …, d_{n}} ⊆ D is N_{D}(d_{1}) ∩ ⋯ ∩ N_{D}(d_{n}). Obviously, any strong filter is Scott-open in Ω_{ne} X. The reverse, however, is not true. For an example, see Section 7.3.3. It is not hard to show that the set of strong filters in ΩX is closed with respect to binary intersection and directed union. Hence F D^{s}D is a substructure of the ∧-continuous ∧-semilattice F_{D}D. If X is a sober space then strong filters in Ω_{ne} X correspond to certain compact sets. A subset K of a sober space X is strongly compact iff every neighbourhood of K contains a finitely generated upper set that includes K. The use of strongly compact sets for defining a Smyth power construction is due to Smyth [Smy78]. It is an easy consequence of the Hofmann-Mislove Theorem 7.4 that, for any sober space X, the dcpo of strong filters in Ω_{ne} X is isomorphic to the dcpo of strongly compact subsets of X ordered by reverse inclusion. There are dcpo’s which have compact sets that are not strongly compact: Consider once more the example from Section 7.2.4. A subset K of the dcpo D from this example is strongly compact iff ⋃{A ⊆ [0, 1] | [0, 1] \ A ∈ K} is countable. Hence, if K is a strongly compact subset of D then its meet exists and is the intersection of all open sets in K. Therefore the compact set we used to show that D is not an F_{D}-algebra is not strongly compact. Another example for a compact subset of a dcpo that is not strongly compact is given in Chapter 7 of [Hec92b].

Proposition 7.17 In a domain, every Scott-open filter is strong.

Proof: Let U be an element of the Scott-open filter φ in Ω_{ne} D where D is a domain.

Since U = ⋃_{↑}{↑ ↑{d_{1}, …, d_{n}} | d_{1}, …, d_{n} ∈ U} there are d_{1}, …, d_{n} ∈ U such that

↑ ↑{d_{1}, …, d_{n}} ∈ φ as φ is Scott-open. Obviously, the filter of open neighbourhoods of ↑{d_{1}, …, d_{n}} is a subset of φ.

The preceding proposition tells us that for a domain D we have F_{D}D = F D^{s}D, so by considering the construction F D^{s} instead of F_{D} we do not lose the nice properties we had for F_{D} restricted to Dom. In order to make F D^{s} a functor, we show that the restriction to F D^{s}D and co-restriction to F D^{s}E of F_{D}f is well-defined for any f: D −^{}E in DCPO. We have to prove that {V ∈ Ω_{ne} E | f^{−1}(V) ∈ φ} is a strong filter in Ω_{ne} E for any φ ∈ F D^{s}D. If f^{−1}(V) ∈ φ for some open subset V of E then there is a finite set F ⊆ f^{−1}(V) such that the filter of open neighbourhoods of F is a subset of φ. Consequently, f(F) is a finite subset of V which satisfies

{V ∈ Ω_{ne} E | f(F) ⊆ V } ⊆ {V ∈ Ω_{ne} E | f^{−1}(V) ∈ φ}.

Hence we can make F D^{s} an endofunctor on DCPO if we let F D^{s}f be the restriction and co-restriction of F_{D}f as just described. Furthermore, the co-restriction of N_{D}: D −^{}F_{D}D to F D^{s}D is also well-defined; we will denote this function by N_{D} as well. Therefore, N is a natural transformation from the identity on DCPO to F D^{s}. In order to obtain a monad it remains to give a natural transformation (F D^{s})^{2}−^{}F D^{s}. For that we define □^{s} U := {φ ∈ F D^{s}D | U ∈ φ} = □U ∩ F D^{s}D for any open subset U of some dcpo D. If Φ is an element of F D^{s}D we set μ_{D}(Φ) := {U ∈ Ω_{ne} D | □^{s} U ∈ Φ}. The proof that this defines a filter in Ω_{ne} D is the same as the corresponding proof for the monad (F_{D}, N , μ). To see that μ_{D}(Φ) is indeed a strong filter in Ω_{ne} D let U ∈ μ_{D}(Φ). This implies □^{s} U ∈ Φ. Hence, by strongness of Φ, there are φ_{1}, …, φ_{n} ∈ □^{s} U with N_{F_{D}^{s} D}(φ_{1}) ∩ ⋯ ∩ N_{F_{D}^{s} D}(φ_{n}) ⊆ Φ. As φ_{1}, …, φ_{n} ∈ □^{s} U we have U ∈ φ_{1}, …, φ_{n} by definition of □^{s} U. By strongness of the φ_{i}, there are finite subsets F_{1}, …, F_{n} of U such that the filter of open neighbourhoods of some F_{i} is contained in φ_{i} for all i ∈ {1, …, n}. Consequently, F := F_{1} ∪ ⋯ ∪ F_{n} is a finite subset of U such that the filter of open neighbourhoods of F is a subset of φ_{1} ∩ ⋯∩φ_{n}. If U′ is an open subset of D with U′ ⊇ F then U′ ∈ φ_{1}, …, φ_{n} or φ_{1}, …, φ_{n} ∈ □^{s} U′. By Scott openness of □^{s} U′ we may conclude □^{s} U′ ∈ N_{F_{D}^{s} D}(φ_{1}) ∩ ⋯ ∩ N_{F_{D}^{s} D}(φ_{n}) which yields □^{s} U′ ∈ Φ and therefore U′ ∈ μ_{D}(Φ). Hence the filter of open neighbourhoods of F is a subset of μ_{D}(Φ) which proves strongness of μ_{D}(Φ). The proof that μ_{D} is a Scott-continuous function from (F D^{s})^{2}D to F D^{s}D and that this definition makes μ a natural transformation (F D^{s})^{2}−^{}F D^{s} is the same as for the monad (F_{D}, N , μ). We have shown:

Proposition 7.18 The triple (F D^{s}, N , μ) is a monad on DCPO.

We next show that every F_{D}-algebra is an F D^{s}-algebra. Let (D, ξ) be an F_{D}-algebra. Obviously, the restriction of ξ to F D^{s}D, say ζ, is a left inverse for N_{D}: D −^{}F D^{s}D. Furthermore,

ζ( F D^{s}ζ(Φ)) = ζ({U ∈ Ω_{ne} D | ζ^{−1}(U) ∈ Φ})
= ξ({U ∈ Ω_{ne} D | (ξ^{−1}(U) ∩ F D^{s}D) ∈ Φ})
= ξ({U ∈ Ω_{ne} D | (□U ∩ F D^{s}D) ∈ Φ})  (ξ ∘ F_{D}ξ = ξ ∘ μ_{D})
= ζ({U ∈ Ω_{ne} D | □^{s} U ∈ Φ})
= ζ( μ_{D}(Φ))

for all Φ ∈ (F D^{s})^{2}D. Consequently, every morphism of F_{D}-algebras is a morphism of F D^{s}-algebras. Hence, from Proposition 7.11, we obtain the following result.

Proposition 7.19 The category of F D^{s}-algebras is a subcategory of the category of deflationary semilattices in DCPO.

However, KT^{s} does not give the free deflationary semilattice for DCPO either as is shown on page 18 in [Hec92a].

7.2.7 Unital semilattices

If we want every Smyth power dcpo to have a greatest element then we have to consider Scott-open (or strong) filters in ΩD instead of Ω_{ne} D. The greatest element for both of these changed power constructions, say ˙ F_{D} and Ḟ D^{s}, is the whole set ΩD. Therefore ˙ F_{D}D and Ḟ D^{s}D both are unital deflationary semilattices in DCPO. Note that ΩD is inaccessible by arbitrary joins in both, ˙ F_{D}D and Ḟ D^{s}D, since it is the only filter in ΩD containing the empty set. Hence we get ˙ F_{D}D from ˙ F_{D}D by adjoining a greatest element and the corresponding statement is true for the second Smyth power construction. For every f: D −^{}E in DCPO we define ˙ F_{D}f to be the unique extension of F_{D}f that maps the greatest element in ˙ F_{D}D to the greatest element in ˙ F_{D}E. Therefore any ˙ F_{D}f is a morphism of unital deflationary semilattices in DCPO. In the same manner as Proposition 7.11 we can show that the category of ˙ F_{D}-algebras is a subcategory of the category of unital deflationary semilattices in DCPO. The same is true for the functor Ḟ D^{s} which is defined in the obvious way.

7.2.8 Remarks on the Smyth power dcpo constructions

Unlike the Hoare power construction it is not obvious how a Smyth power dcpo should be defined for non-continuous dcpo’s. Furthermore, we cannot characterize the algebras for either of the two constructions for any subcategory of DCPO larger than Dom. In a way the reason for this is that we cannot say much about the Scott topology on the Smyth power dcpo’s. This seems to be a phenomenon that appears for non-continuous domains in general. For example, there is no order-theoretic characterization of sobriety for dcpo’s. The difficulties in solving the open problems of Section 4.6 in [Hec92b] also seem to originate from this fact. In the next section we will generalize both constructions to arbitrary T_{0}-spaces. The topologies we choose for the different power spaces make it possible to prove much stronger results than what we could show for dcpo’s.

Let us briefly consider the naïve approach to obtaining the free deflationary semilattice for DCPO: Take all finitely generated upper subsets of D and close the resulting set under filtered intersections. If D is continuous then this process is finished after adding filtered intersections of sets of the form ↑{d_{1}, …, d_{n}} where d_{1}, …, d_{n} ∈ D. By that, we obtain K_{D}D = KD^{s}D. If D is not continuous, however, then we might need to add filtered intersections of filtered intersections of … of finitely generated upper subsets of D. This process has to end sometime and we can prove properties of the resulting poset, say KD^{f}D, by transfinite induction. If D is any dcpo then there is at most one structure map for D with respect to KD^{f} by Lemma 4.4. This structure map has to map every element of KD^{f}D to its meet. In order to prove that this function is Scott-continuous we need an analogue to Lemma 7.2: Let S be a filtered family in KD^{f}D. If ⋂_{↓} S ⊆ U for some Scott-open set U ⊆ D then there is K ∈ S with K ⊆ U. So we can only hope to succeed for dcpo’s that satisfy this property. However, KD^{f}D need not have this property even if D does. Hence we cannot make KD^{f} an endofunctor on a reasonable subcategory of DCPO that is larger than Dom.

7.3 Topological spaces

The above Smyth-type power constructions for dcpo’s suggest several possible definitions for a Smyth power space. We considered the domain of Scott-open filters of non-empty open subsets rather than that of non-empty compact saturated sets because the latter does not necessarily have directed joins. For topological spaces, we can drop this requirement. As our studies of power constructions for topological spaces are to help us examining power locale constructions we only consider constructions that preserve sobriety.

7.3.1 A naïve approach

Let us first examine the obvious candidate to give the free deflationary semilattice for the category of T_{0}-spaces. We define KT^{f}X to be the set

{↑{x_{1}, …, x_{n}} | x_{1}, …, x_{n} ∈ X}

with the topology generated by the sets of the form {↑{x_{1}, …, x_{n}} | x_{1}, …, x_{n} ∈ U} where U is an open subset of X. The specialization order for KT^{f}X is reverse inclusion. It is not hard to check that KT^{f}X is a deflationary topological semilattice; the corresponding semilattice operation for KT^{f}X is (binary) union. Any T_{0}-space X is embedded into KT^{f}X by ↑_{X}: x −^{}↑{x}. This function is continuous because

↑_{X}^{−1}({↑{x_{1}, …, x_{n}} | x_{1}, …, x_{n} ∈ U}) = {x ∈ X | ↑_{X}(x) ∈ U} = U

for any open subset U of X. We can make KT^{f} a functor that yields a monad. If

we set KT^{f}f(↑{x_{1}, …, x_{n}}) := ↑{f(x_{1}), …, f(x_{n})} for any f: X −^{}Y in Top and

any x_{1}, …, x_{n} ∈ X then we obtain a continuous function KT^{f}f: KT^{f}X −^{}KT^{f}Y. One easily checks that ↑: id_{Top} −^{}KT^{f} is a natural transformation. To get a natural transformation (KT^{f})^{2} −^{}KT^{f}, let X be a T_{0}-space and let K be an element of (KT^{f})^{2}X. Then _{⋃X}(K) := ⋃ K is a continuous function from (KT^{f})^{2}X to KT^{f}X: As K is of the form ↑{K_{1}, …, K_{n}} for some K_{1}, …, K_{n} ∈ KT^{f}X and since KT^{f}X is closed with respect to binary union, _{⋃X}(K) = K_{1} ∪ ⋯ ∪ K_{n} is an element of KT^{f}X. It is easy to see that this defines a continuous function for any T_{0}-space X and that ⋃ is a natural transformation as desired. Furthermore, (KT^{f}, ↑, ⋃) is a monad. By Lemma 4.4, a T_{0}-space X is an algebra for the monad iff it has meets of sets of the form ↑{x_{1}, …, x_{n}}, that is meets of non-empty finite sets, such that

⋀X: ↑{x_{1}, …, x_{n}} −^{}x_{1} ∧ ⋯ ∧ x_{n}

is a continuous function from KT^{f}X to X. Hence any KT^{f}-algebra is a deflationary semilattice. But if X is a deflationary semilattice then _{⋀X} need not be continuous:

Let U be an open neighbourhood of some x ∈ X and let n be an arbitrary positive integer. Then, by continuity of the n-ary meet function from X × ⋯ × X to X, we can find neighbourhoods U_{1}, …, U_{n} of x such that ∧_{n}(U_{1} × ⋯ × U_{n}) ⊆ U. Hence U′ := U_{1} ∩ ⋯ ∩ U_{n} is an open neighbourhood of x such that x_{1} ∧ ⋯ ∧ x_{n} ∈ U for all x_{1}, …, x_{n} ∈ U′. However, to get continuity of _{⋀X} we need an open neighbourhood U′ of x such that for all n ∈ ℕ and all x_{1}, …, x_{n} ∈ U′ we have x_{1} ∧ ⋯ ∧ x_{n} = ⋀X(↑{x_{1}, …, x_{n}}) ∈ U. This is equivalent to requesting the existence of a semilattice S in U with x ∈ int S. Hence by transferring the Smyth power constructions for dcpo’s to topological spaces, we cannot expect to get the free deflationary semilattice for Top. Rather, the algebras of the corresponding monads all have the following property: Definition. A topological semilattice is said to have small semilattices if each of its points has a neighbourhood basis consisting of semilattices. It is not difficult to show that the KT^{f}-algebras are the deflationary topological semilattices with small semilattices. We are not able to use this functor in order to define a Smyth power locale since it does not give us suitable equations for the open subsets of KT^{f}X. Therefore we will not consider this functor any further. We will rather study the power construction resulting from taking all non-empty compact saturated subsets of a topological space X. After examining this definition for the category of locally compact sober spaces, we proceed to study the power construction corresponding to F D^{s}.

7.3.2 A Smyth power construction

Recall that KX denotes the set of non-empty compact saturated subsets of a T_{0}- space X ordered by ⊃. By K_{T}X we denote the set KX with the topology generated by the sets of the form □U := {K ∈ KX | K ⊆ U} where U is an open subset of X. This definition of a Smyth power space can be traced to M. Smyth [Smy83].

Note that the specialization order for K_{T}X is reverse inclusion and that K_{T}X has binary meets which are given by union. Furthermore, the sets of the form □U, where U is an open subset of X, form a basis for the topology on K_{T}X because

□U ∩ □U′ = □(U ∩ U′) for all U, U′ ∈ ΩX. Also, ⋃_{↑}{□U | U ∈ U} = □ ⋃_{↑} U for any

directed subset U of Ω_{ne} X.

Lemma 7.20 If X is a sober space then so is K_{T}X.

Proof: Let Φ be a completely prime filter of open subsets of K_{T}X. For such a filter, we define K := ⋂_{↓}{U ∈ Ω_{ne} X | □U ∈ Φ}. We will show that K is an element of K_{T}X and that Φ is the filter of open neighbourhoods of K in K_{T}X. To see that K is indeed a non-empty compact saturated subset of X note that {U ∈ Ω_{ne} X | □U ∈ Φ} is a Scott-open filter in Ω_{ne} X. Hence, because of the Hofmann-Mislove Theorem 7.4, K is an element of K_{T}X. Let us show that Φ is the filter of all open neighbourhoods of K in K_{T}X. Assume that U is a subset of ΩX. If ⋃_{U∈U}□U is a neighbourhood of K in K_{T}X then there is U ∈ U with K ∈ □U. Hence U ∈ {U′ ∈ Ω_{ne} X | □U′ ∈ Φ}, again by the Hofmann-Mislove Theorem 7.4, and therefore ⋃_{U∈U}□U ∈ Φ. On the other hand, if ⋃_{U∈U}□U ∈ Φ then there is U ∈ U with □U ∈ Φ by complete primeness of Φ. Then □U and therefore ⋃_{U∈U}□U is an open neighbourhood of K.

Note that the sobrification for K_{T}X is the space of all Scott-open filters on Ω_{ne} X. To make K_{T}an endofunctor on Top, set K_{T}f(K) := ↑ f(K) for any continuous function f: X −^{}Y and any K ∈ K_{T}X. As f(K) is non-empty and compact if K is and since the saturation of a compact set is compact as well, K_{T}f is well-defined. Continuity of K_{T}f follows from the fact that

(K_{T}f)^{−1}(□V) = {K ∈ K_{T}X | ↑ f(K) ⊆ V }
= {K ∈ K_{T}X | f(K) ⊆ V }
= {K ∈ K_{T}X | K ⊆ f^{−1}(V)}
= □f^{−1}(V)

for any open subset V of Y. We can embed X into K_{T}X by setting ↑_{X}(x) := ↑{x} for all x ∈ X. Because ↑_{X}^{−1}(□U) = {x ∈ X | ↑{x} ∈ U} = U for any open subset U of X, the function ↑_{X} is continuous. Thus we obtain a natural transformation ↑: id_{Top} −^{}K_{T} since for all f: X −^{}Y in Top and all x ∈ X we have

K_{T}f(↑_{X}(x)) = ↑ f(↑{x}) = ↑{f(x)} = ↑_{Y} (f(x)).

Furthermore, we can define a natural transformation ⋃: (K_{T})^{2} −^{}K_{T}. For any T_{0}- space X set _{⋃X}(K) := ⋃ K for all K ∈ (K_{T})^{2} X. To show that _{⋃X} is well-defined, we have to prove that ⋃ K is non-empty, compact, and saturated if K is. Obviously, ⋃ K is non-empty if K is. Furthermore, if x ∈ ↑ ⋃ K then there is K ∈ K with x ∈ ↑ K = K ⊆ ⋃ K. Therefore it remains to prove that ⋃ K is compact. Let U be an open cover of ⋃ K. For all K ∈ K there are finitely many elements of U covering K. Hence {□(U_{1} ∪ ⋯ ∪ U_{n}) | U_{1}, …, U_{n} ∈ U} is an open cover of K. By compactness of K, there are U_{1}^{1}, …, U_{n(1)}^{1}, U_{1}^{2}, …, U_{n(m)}^{m} ∈ U with

K ⊆ □(U_{1}^{1} ∪ ⋯ ∪ U_{n(1)}^{1}) ∪ ⋯ ∪ □(U_{1}^{m} ∪ ⋯ ∪ U_{n(m)}^{m}).

Consequently, ⋃ K ⊆ U_{1}^{1} ∪ ⋯ ∪ U_{n(m)}^{m}. The function _{⋃X}: (K_{T})^{2} X −^{}K_{T}X is continuous for any T_{0}-space X as

⋃_{X}^{−1}(□U) = {K ∈ (K_{T})^{2} X | ⋃ K ∈ □U}
= {K ∈ (K_{T})^{2} X | ⋃ K ⊆ U}
= {K ∈ (K_{T})^{2} X | K ⊆ □U}
= □□U

for any open subset U of X. If f: X −^{}Y is a continuous function in Top then, for all K ∈ (K_{T})^{2} X, we have

K_{T}f(_{⋃X}(K)) = ↑ f(⋃ K) = ↑ ⋃_{K∈K} f(K)
= ⋃_{K∈K} ↑ f(K) = ⋃_{K∈K} K_{T}f(K)
= ⋃ ↑{K_{T}f(K) | K ∈ K}
= _{⋃Y} ( ↑ K_{T}f(K)) = _{⋃Y} ( (K_{T})^{2} f(K))

which proves that ⋃: (K_{T})^{2} −^{}K_{T} is a natural transformation.

Proposition 7.21 The triple (K_{T}, ↑, ⋃) is a monad on Top.

Proof: It remains to prove the commutativity of the diagrams defining a monad. Let K be an element of K_{T}X for some T_{0}-space X. We calculate

⋃X ( ↑_{K_{T}X}(K)) = _{⋃X}{K′ ∈ K_{T}X | K′ ⊆ K}
= K
= _{⋃X} ( ↑_{K_{T}X}{↑{x} | x ∈ K})
= _{⋃X} ( K_{T}↑_{X}(K))

which proves the commutativity of the two triangular diagrams. Since we also have

⋃X(_{⋃_{K_{T}}X}(𝔎)) = ⋃(⋃ 𝔎) = ⋃(↑{⋃ K | K ∈ 𝔎}) = _{⋃X}(K_{T}_{⋃X}(𝔎)),

for all 𝔎 ∈ (K_{T})^{3} X, the rectangular diagram also commutes.

Lemma 7.22 Every K_{T}-algebra is a deflationary semilattice in Top.

Proof: First note that any K_{T}X is a deflationary semilattice in Top since binary meet ∪: ⟨K, K′⟩ −^{}K ∪ K′ is a continuous function from K_{T}X × K_{T}X to K_{T}X. To see this, let U be an open subset of X. Then

∪^{−1}(□U) = {⟨K, K′⟩ ∈ K_{T}X × K_{T}X | K ∪ K′ ⊆ U}
= {⟨K, K′⟩ ∈ K_{T}X × K_{T}X | K ⊆ U and K′ ⊆ U}
= □U × □U

which is open in K_{T}X × K_{T}X. Hence any K_{T}-algebra is a deflationary semilattice in Top by Proposition 5.6.

However, the property of being a deflationary topological semilattice is not enough to characterize the category of K_{T}-algebras.

Proposition 7.23 A T_{0}-space X is a K_{T}-algebra iff it has meets of non-empty compact sets and if _{⋀X}: K −^{}⋀ K: K_{T}X −^{}X is a continuous function. A continuous function between K_{T}-algebras is a morphism of these algebras iff it preserves meets of non-empty compact sets.

Proof: This proposition follows from Lemmata 4.4 and 4.6.

There does not seem to be a more satisfactory characterization of the category of K_{T}-algebras. It is particularly disturbing that K_{T}X still appears in the above description. Nevertheless, besides being a deflationary topological semilattice, we can show another property every K_{T}-algebra must have: It must have small semilattices. We have already encountered this property at the beginning of this section. Such unital semilattices with compact Hausdorff topologies give a characterization of continuous lattices (see Theorem VI.3.4 in [GHK^{+}80]). Note that every continuous ∧-semilattice, that is every deflationary semilattice in Dom, has small semilattices.

Proposition 7.24 Every K_{T}-algebra is a deflationary topological semilattice with small semilattices.

Proof: Let (X, ξ_{X}) be a K_{T}-algebra. By Lemma 7.22, X is a deflationary topological semilattice. Therefore it remains to show that X has small semilattices. Let U be an open subset of X and let x be one of the elements of U. Then ↑{x} ∈ ξ_{X}^{−1}(U) and, by continuity of ξ_{X}, there is an open neighbourhood U′ of x such that ξ_{X}(□U′) ⊆ U. We claim that there is a sub-semilattice of X that contains U′ and is contained in U. A sub-semilattice of X that includes U′ is obviously given by S := {x_{1} ∧ ⋯ ∧ x_{n} | x_{1}, …, x_{n} ∈ U′}. To see that S is contained in U, let x_{1}, …, x_{n} be elements of U′. Then ↑{x_{1}, …, x_{n}} ∈ □U′ ⊆ ξ_{X}^{−1}(U) and thus x_{1}∧⋯∧x_{n} = ξ_{X}(↑{x_{1}, …, x_{n}}) ∈ U.

7.3.3 Why K_{T}does not give the free deflationary semilattice with small semilattices for Top

We shall prove that K_{T} does not give the free deflationary semilattice with small semilattices by giving an example of a deflationary topological semilattice with small semilattices which is not a K_{T}-algebra. Since the Scott topology on the product of two dcpo’s need not be the product topology, we cannot re-use the example from Section 7.2.4. Let us consider the unit interval with the usual (Euclidean) topology. Consider KT^{f}[0, 1] as defined at the beginning of this section. In order to show that this gives the desired example we have to show that KT^{f}[0, 1] is a deflationary topological semilattice with small semilattices that is not a K_{T}-algebra. Note that KT^{f}[0, 1] is sober, so even for the category of sober spaces the reverse of Proposition 7.24 is not true. To show sobriety, let A be a non-empty irreducible closed subset of KT^{f}[0, 1]. Since KT^{f}[0, 1] is a subspace of K_{T}[0, 1], the closure of A in K_{T}[0, 1] must be irreducible by Lemma 1.8. By sobriety of K_{T}[0, 1] there is some K ∈ K_{T}[0, 1] such that cl_{K_{T}[0,1]} A is the closure of {K} in K_{T}[0, 1]. But this closure is {K′ ∈ K_{T}[0, 1] | K′ ⊇ K}. So if K ∉ KT^{f}[0, 1], that is if K is not finite, then all of its supersets are infinite, implying A = KT^{f}[0, 1] ∩ \overline{\{K\}} = ∅ which contradicts A being non-empty. Hence K is in KT^{f}[0, 1] and A is the closure of {K} in KT^{f}[0, 1]. The binary meet on KT^{f}[0, 1] is given by ∪: KT^{f}[0, 1] × KT^{f}[0, 1] −^{}KT^{f}[0, 1]. This operation is continuous as it is the restriction and co-restriction of the binary meet on K_{T}[0, 1] whose continuity we showed above. Note further that KT^{f}[0, 1] does have small semilattices. However, KT^{f}[0, 1] is not a K_{T}-algebra since it does not have all compact meets. The subspace {{x} | x ∈ [0, 1]} of KT^{f}[0, 1] consisting of all singleton subsets of [0, 1] is isomorphic to [0, 1]. Hence for every compact subset K of [0, 1] the set {{k} | k ∈ K} is compact in KT^{f}[0, 1]. Its meet in KT^{f}[0, 1], however, would have to be the set K which of course need not be an element of KT^{f}[0, 1]. A similar argument works for every T_{1}-space X (instead of the unit interval) that has at least one non-finite compact subset. So what goes wrong with our construction? Well, it is too big to give the free deflationary topological semilattice with small semilattices as we have just seen. We shall be able to give an alternative definition of a Smyth power space that does exclude examples of the above type.

7.3.4 Locally compact spaces

Before following that path, however, we give a subcategory of Top for which K_{T}does gives the free deflationary semilattice with small semilattices. Recall that LCompS denotes the full subcategory of Top whose objects are the locally compact sober spaces. The following proposition follows from the Hofmann-Mislove Theorem 7.4 and the Lawson duality [Law79].

Proposition 7.25 If X is a locally compact sober space then K_{T}X is a domain with respect to the specialization order. The way below relation in K_{T}X is K′ ≪ K iff int K′ ⊇ K.

Proof: Since K_{T}preserves sobriety by Lemma 7.20, K_{T}X is a dcpo with respect to the specialization order. If int K′ ⊇ K and K ⊆ ⋂_{↓} S for a directed subset S of K_{T}X then, because of Lemma 7.2 (i), there is K′′ ∈ S with K′′ ⊆ int K′ ⊆ K′. Therefore, K′ ≪ K. Let, on the other hand, K′ ≪ K. As K = ⋂_{↓}{K′′ ∈ K_{T}X | int K′′ ⊇ K} there must be K′′ ∈ K_{T}X with int K′′ ⊇ K and K′ ⊇ K′′. So K′ ⊇ K′′ ⊇ int K′′ ⊇ K which proves int K′ ⊇ K. Consequently, ↓ ↓{K} = {K′ ∈ K_{T}X | int K′ ⊇ K}. By local compactness of K_{T}X we know that ↓ ↓{K} is directed. For the same reason,

⋁^{↑} ↓ ↓{K} = ⋂_{↓}{K′ ∈ K_{T}X | int K′ ⊇ K} = K for all K ∈ K_{T}X.

Remark. Note that for the last lemma we do not really need sobriety. In order to make K_{T}X a continuous domain, a locally compact space X only needs the following property: If S is a filtered collection of non-empty compact subsets of X and if U is an open subset of X with ⋂_{↓} S ⊆ U then there is K ∈ S with K ⊆ U.

Lemma 7.26 If X is a locally compact sober space then the Scott topology and the intrinsic topology on K_{T}X coincide.

Proof: As K_{T}X is sober, its topology is coarser than the Scott topology. On the other hand, if U ⊆ K_{T}X is Scott-open and K is an element of U then, by continuity of K_{T}X, there is K′ ≪ K with K′ ∈ U. By the above characterization of the way below relation in K_{T}X we know that int K′ ⊇ K. Hence K ∈ □(int K′) ⊆ U. This proves that the Scott topology for K_{T}X is coarser then the topology K_{T}X carries by definition. Thus the two coincide.

If D is a continuous domain, then it is sober and locally compact with respect to the Scott topology. Therefore we get the following result.

Proposition 7.27 For the category of domains, T_{σ}∘ K_{D}= K_{T}∘ T_{σ}.

Since we cannot show that T_{σ}∘ F_{D} ≅ K_{T}∘ T_{σ} for a larger subcategory of DCPO, we cannot transfer the results we will get for K_{T}to F_{D}. The following characterization of the category of K_{T}-algebras has already been published in [Sch93a].

Proposition 7.28 The category of algebras for the monad (K_{T}, ↑, ⋃) restricted to the category LCompS of locally compact sober spaces, is the category Dom^{∧} of continuous ∧-semilattices.

Proof: By Proposition 7.25 we know that if X is a locally compact sober space then K_{T}X is a domain. Furthermore, we know that K_{T}X is a deflationary topological semilattice, that is a ∧-semilattice with respect to the order of specialization. Because binary meet is continuous for K_{T}X and because K_{T}X and its product both are sober if X is, the binary meet function for K_{T}X is Scott-continuous. Therefore any algebra for the restricted monad (K_{T}, ↑, ⋃) is a continuous ∧-semilattice by Proposition 5.6. Furthermore, the topology on X is the Scott topology: Since X is sober, its topology is coarser than the Scott topology. On the other hand, if U is a Scott-open subset of X then ξ_{X}^{−1}(U) is a Scott-open subset of K_{T}X as ξ_{X} is Scott-continuous. Since the Scott topology for K_{T}X coincides with the intrinsic topology for K_{T}X we get that U = η_{X}^{−1}(ξ_{X}^{−1}(U)) is open in X. Hence any K_{T}-algebra is a continuous ∧-semilattice with its Scott topology. Furthermore, any morphism of K_{T}-algebras is a continuous function between sober spaces and therefore Scott-continuous. Also, according to Proposition 7.23, such a morphism in particular has to preserve finite meets. On the other hand, every continuous ∧-semilattice D (with its Scott topology) is a K_{T}-algebra: By Lemma 7.14, D has a structure map making it an F_{D}-algebra. By Proposition 7.27 the same structure map makes D a K_{T}-algebra. Furthermore, any morphism of continuous ∧-semilattices is a morphism of the corresponding F_{D}- algebras by Lemma 7.15 and therefore also for the corresponding K_{T}-algebras.

Proposition 7.29 Let X be a sober and locally compact space. If X is a deflationary topological semilattice with small semilattices then X is a domain with its Scott topology.

Proof: As X is sober, it is a dcpo. Let x be an element of some open U ⊆ X. By local compactness of X, there is a compact neighbourhood K of x in U. Since X has small semilattices, there is a semilattice S in the interior of K such that x ∈ int S. Consider {↓{s} ∩ K | s ∈ S}. This is a family of closed subsets of K (with respect to the topology K inherits from X) that has the finite intersection property: If s_{1}, …, s_{n} ∈ S then s_{1} ∧ … ∧ s_{n} ∈ (↓{s_{1}} ∩ K) ∩ ⋯ ∩ (↓{s_{n}} ∩ K). By compactness of K, there is some x′ ∈ ⋂{↓{s} ∩ K | s ∈ S}. Obviously, x ∈ int S ⊆ ↑{x′} which proves x ∈ int ↑{x′}. As, by sobriety of X, the topology of X is coarser than the Scott topology, x ≪ x′ holds for all x′ ∈ X with x ∈ int ↑{x′}. On the other hand, for any open neighbourhood U of X we have just shown the existence of some x′ ∈ U with x ∈ int ↑{x′} and thus with x′ ≪ x. We next show that {x′ ∈ X | x ∈ int ↑{x′}} is directed with join x. If x ∈ int ↑{y} and x ∈ int ↑{y′} then int ↑{y} ∩ int ↑{y′} is an open neighbourhood of x and consequently we can find y′′ ∈ int ↑{y} ∩ int ↑{y′} with x ∈ int ↑{y′′}. Since any open neighbourhood of x contains some x′ with x ∈ int ↑{x′}, the join of {x′ ∈ X | x ∈ int ↑{x′}} has to be x. Therefore, {x′ ∈ X | x ∈ int ↑{x′}} is a directed subset of ↓ ↓{x} with join x. This proves continuity of X. It remains to show that the Scott topology for X is coarser than the intrinsic topology for X. If x′ ≪ x then, since x = ⋁^{↑}{x′′ ∈ X | x ∈ int ↑{x′′}}, there is x′′ ∈ X with x ∈ int ↑{x′′} and x′ ≤ x′′. Hence any set of the form ↑ ↑{x′′} is open in X. As a consequence, the Scott topology is coarser than the intrinsic topology for X and thus the two must coincide.

Any domain is sober and locally compact. Furthermore, any continuous ∧-semilattice is a deflationary topological semilattice with small semilattices. Hence Dom^{∧} is a subcategory of the category of deflationary semilattices with small semilattices in LCompS. Together with the preceding proposition this implies the following result.

Theorem 7.30 The category of deflationary semilattices with small semilattices in LCompS coincides with the category of continuous ∧-semilattices.

Theorem 7.31 The functor K_{T} gives the free deflationary semilattice with small semilattices for the category of locally compact sober spaces.

Proof: By Proposition 7.28 the category of K_{T}-algebras is Dom^{∧} which, by the last theorem, is the category of deflationary semilattices with small semilattices for LCompS.

Since there are deflationary semilattices in LCompS that do not have small semilattices, K_{T}does not even give the free deflationary semilattice for the category of locally compact sober spaces: In [Law70], Lawson introduces a compact Hausdorff topological semilattice X that does not have small semilattices. With the help of results from [GHK^{+}80] we can show that this example serves our needs. Interpreting this semilattice structure as meet yields an order on X. The open upper sets then give a topology on X with respect to which X is a sober topological lattice by Proposition VII.1.7 and Corollary VII.1.14. Furthermore, this new topology is just the Scott topology for X. By Exercise VII.1.18, X is locally compact with respect to the Scott topology and the intrinsic topology for X is generated by the Scott topology and the complements of Scott-compact upper sets. If we can show that X does not have small semilattices with respect to the Scott topology then we are done. Assume that this is wrong. By choice of X there is an open neighbourhood O of some x ∈ X that does not contain a semilattice which at the same time is a neighbourhood for x. By the above characterization of the topology on X, there is a Scott-open subset U of X and a Scott-compact upper set K ⊆ X with x ∈ U ∩ (X \ K) ⊆ O. Therefore, U is a Scott neighbourhood of x and, by assumption, contains a semilattice S that also is a Scott neighbourhood for x. Hence there is a Scott-open set U′ in S that contains x. But S ∩ (X \ K) is a semilattice since X \ K is lower closed. Further, S ∩ (X \ K) contains the open set U′ ∩ (X \ K). This contradicts the choice of U and x. Hence, with respect to the Scott topology, X is a deflationary topological semilattice with sober and locally compact topology that does not have small semilattices. As the example from Section 7.3.3 shows, local compactness is necessary in the last theorem.

7.3.5 Another Smyth power construction

The second Smyth power construction for topological spaces suggested by the considerations from Section 7.2 is the following. For a T_{0}-space X let KT^{s}X be the set of non-empty strongly compact and saturated subsets of X with the topology generated by the sets

□^{s} U := {K ∈ KT^{s}X | K ⊆ U} = □U ∩ KT^{s}X

where U is an open subet of X. This definition makes KT^{s}X a subspace of K_{T}X for any T_{0}-space X.

Lemma 7.32 If X is a sober space then so is KT^{s}X. Proof: We will prove the assertion by using that K_{T}X is sober if X is. So let X be a sober space. If A is a non-empty irreducible closed subset of KT^{s}X then the closure of A in K_{T}X is irreducible in K_{T}X. By sobriety of K_{T}X there is K ∈ K_{T}X with cl_{K_{T}X} A = cl_{K_{T}X}{K}. It remains to show that K is in KT^{s}X, that is that K is strongly compact in X. Assume that this is not the case. Then we can find an open neighbourhood U of K such that K is not in ↑ F for any finite F in U. This implies K ∈ cl_{K_{T}X} A ∩ □U ≠ ∅. On the other hand, A ∩ □^{s} U = ∅: If K′ ∈ A ∩ □^{s} U then there is a finite set F in U with K ⊆ K′ ⊆ ↑ F which contradicts K ∈ K_{T}X \ KT^{s}X. Hence K_{T}X \ □^{s} U is a closed superset of A in K_{T}X that does not contain cl_{K_{T}X} A which is a contradiction. Therefore K ∈ KT^{s}X and thus A = cl_{K}_{T}^{s}{K}.

The following result yields a connection between KT^{s} and KT^{f}.

Proposition 7.33 If X is a sober space then KT^{s}X is isomorphic to the sobrification

of KT^{f}X.

Proof: If K is a stronly compact subset of a sober space X then

{↑{x_{1}, …, x_{n}} | x_{1}, …, x_{n} ∈ U | K ⊆ U}

is a completely prime filter in Ω KT^{f}X. If, on the other hand, Φ is a completely prime filter in Ω KT^{f}X then

{□^{s} U | {↑{x_{1}, …, x_{n}} ∈ KT^{f}X | x_{1}, …, x_{n} ∈ U} ∈ Φ}

is a strong filter in Ω KT^{s}X. By sobriety of X and the Hofmann-Mislove Theorem 7.4 this is sufficient to prove the assertion.

Note that images of strongly compact subsets under continuous functions are also strongly compact and that saturation preserves strong compactness. Hence we can make KT^{s} a functor by defining KT^{s}f: KT^{s}X −^{}KT^{s}Y to be the restriction and corestriction of K_{T}f for any continuous f: X −^{}Y. The co-restriction of the function ↑_{X}: X −^{}K_{T}X to KT^{s}X is also well-defined. Thus we obtain a natural transformation

id_{Top} −^{}KT^{s}.

Moreover, we can define _{⋃X}: (KT^{s})^{2}X −^{}K_{T}X by setting _{⋃X}(K) := ⋃ K for every K ∈ (KT^{s})^{2}X. To prove that _{⋃X} is well-defined we have to show that ⋃ K is strongly compact in X for all K ∈ (KT^{s})^{2}X. If U is an open neighbourhood of ⋃ K in X then K is a subset of □^{s} U. Since K is strongly compact there are K_{1}…. , K_{n} ∈ KT^{s}X with K ⊆ ↑{K_{1}, …, K_{n}} ⊆ □^{s} U. Hence K_{1} ∪ ⋯ ∪ K_{n} is a strongly compact subset of U. As a consequence, we can find a finite subset F of U with ⋃ K ⊆ K_{1} ∪⋯∪K_{n} ⊆ ↑ F. The proof that _{⋃X} is continuous and that ⋃ is a natural transformation is the same as for the monad (K_{T}, ↑, ⋃). Also, the proof of the following proposition is the same as for Proposition 7.21.

Proposition 7.34 The triple (KT^{s}, ↑, ⋃) is a monad on Top.

The following proposition can be shown in the same way as Proposition 7.23.

Proposition 7.35 A T_{0}-space X is a KT^{s}-algebra iff the function _{⋀X}: KT^{s}X −^{}X defined by _{⋀X}(K) := ⋀ K is well-defined and continuous. A continuous function f: X −^{}Y between KT^{s}-algebras is a morphism of these algebras iff it preserves meets of non-empty strongly compact sets.

Note that the T_{0}-space from the example considered in Section 7.3.3 is indeed a KT^{s}-algebra. We can show that KT^{s} gives the free deflationary semilattice with small semilattices on a larger category than K_{T} does. Since the sets □^{s} U, where U is an open subset of X, are closed with respect to finite meets, every topological space of the form KT^{s}X has a basis of open semilattices.

Proposition 7.36 Every KT^{s}-algebra is a deflationary semilattice with small semilattices.

Proof: As for Proposition 7.24.

Theorem 7.37 The algebras for the monad (KT^{s}, ↑, ⋃) restricted to the category of sober spaces are all deflationary topological semilattices with small semilattices. The morphisms of this category are morphisms of deflationary topological semilattices.

Proof: Because of the preceding proposition we know that every KT^{s}-algebra is a deflationary semilattice with small semilattices in Sob. Now let X be a such a topological semilattice. We have to show that_{⋀X}: K −^{}⋀ K is a well-defined continuous function from KT^{s}X to X. For any K ∈ KT^{s}X consider the set

φ_{K} := {U ∈ ΩX | K ⊆ int S ⊆ S ⊆ U for some sub-semilattice S of X}.

We claim that φ_{K} is a completely prime filter in ΩX. Obviously, it is an upper set in ΩX. Furthermore, let U and U′ be elements of φ_{K}. Then there are sub-semilattices S and S′ of X in U and U′, respectively, with K ⊆ int S, int S′. Therefore S ∩ S′ is a semilattice in U ∩ U′ with K ⊆ int S ∩ int S′ ⊆ int(S ∩ S′). Moreover, if ⋃ U ∈ φ_{K} for some subset U of ΩX then there is a sub-semilattice S of X in ⋃ U with K ⊆ int S. By strong compactness of K, there are x_{1}, …, x_{n} ∈ int S with K ⊆ ↑{x_{1}, …, x_{n}}. Hence x_{1} ∧ … ∧ x_{n} ∈ S ⊆ ⋃ U. So there must be U ∈ U with x_{1} ∧ ⋯ ∧ x_{n} ∈ U. As X has small semilattices, there is a semilattice S′ in U with x_{1} ∧ ⋯ ∧ x_{n} ∈ int S′. Therefore K ⊆ ↑{x_{1}, …, x_{n}} ⊆ int S′ which proves U ∈ φ_{K}. Since X is sober, there is a unique x ∈ X such that φ_{K} is the filter of open neighbourhoods of x. Because any open neighbourhood of x contains K we know that x is a lower bound for K. If x′ is any lower bound for K in X then every open neighbourhood U of x′ contains a semilattice with x′ ∈ int S as X has small semilattices. Therefore, K ⊆ ↑{x′} ⊆ int S ⊆ S ⊆ U and consequently U is a neighbourhood of x which proves x′ ≤ x. Hence x is the greatest lower bound for K and the function _{⋀X} is well-defined. To see that _{⋀X} is also continuous, let K ∈ KT^{s}X and let U be an open neighbourhood of ⋀ K. Then there is a semilattice S in U with ⋀ K ∈ int S. We claim that □^{s} (int S) is an open neighbourhood of K such that _{⋀X}(□^{s} (int S)) ⊆ U. Obviously, □^{s} (int S) contains K. If K′ is an element of □^{s} (int S) then U is an element of φ_{K′} and therefore an open neighbourhood of ⋀ K′. Hence, by Proposition 7.35, X is a KT^{s}-algebra. It remains to prove that every morphism f: X −^{}Y of deflationary topological semilattices, where X an Y have small semilattices, is a morphism of the corresponding KT^{s}-algebras. Because of Proposition 7.35 it is enough to show that f preserves meets of non-empty strongly compact sets. Since obviously f(⋀ K) ≤ ⋀ f(K), it is sufficient to prove that f(⋀ K) ≥ ⋀ f(K) for all K ∈ KT^{s}X. Let V be an open neighbourhood of ⋀ f(K). There is a semilattice T in V with ⋀ f(K) ∈ int T. As f preserves binary meets, f^{−1}(T) is a semilattice. Furthermore, K ⊆ f^{−1}(int T) ⊆ int f^{−1}(T). By strong compactness of K there are x_{1}, …, x_{n} ∈ int f^{−1}(T) with K ⊆ ↑{x_{1}, …, x_{n}}. Therefore, f(⋀ K) ≥ f(x_{1} ∧ ⋯ ∧ x_{n}) = f(x_{1}) ∧ ⋯ ∧ f(x_{n}) ∈ T ⊆ V.

Hence KT^{s}gives the free deflationary semilattice with small semilattices for the category of sober spaces.

7.3.6 Unital semilattices

If one wants the semilattices resulting from the two Smyth power constructions to be unital semilattices then it is enough to add the empty set to the respective power spaces. As the specialization order for these is reverse inclusion, we thus indeed get a greatest element, that is a unit for the binary meet function. Since any algebra of one of the monads is a retract of its power space, it also must have a greatest element. Furthermore, by Lemma 4.4 (ii), any morphism of algebras has to be a morphism of unital semilattices. The above results therefore remain true for the modified Smyth power constructions if ‘semilattice’ is replaced by ‘unital semilattice’.

7.4 Locales

We have seen before how locales are used to give axiomatic semantics for programming languages. We have also examined how non-deterministic languages can be handled if non-determinism is considered to be angelic. The approach here is to view non-determinism as being demonic just as we did when we introduced the Smyth power construction for algebraic domains at the beginning of this chapter. So a program satisfies a given property iff each of its branches does. Once more we assume that we already know how to handle these branches using a locale A. Then any property a ∈ A gives us a new property □a that a non-deterministic program might (or might not) have. We say that a non-deterministic program p satisfies the property □a iff every of its branches satisfies a. We will denote this by by p⊧ □a. From this approach we can develop some equations that we want to hold for the new properties, that is we want the resulting equations to hold in the locale P_{L}^{♯}A generated by the □a where a ∈ A. For example, p⊧ □a and p⊧ □b hold iff every branch of p satisfies both, a and b iff p⊧ □(a ∧ b). Hence we want the equation □a ∧ □b = □(a ∧ b) to hold for all a, b in A. Furthermore, p ⊧ ⋁_{s∈S}^{↑} □s for a directed subset S of A iff there is some s ∈ S with p ⊧ □s iff all branches of p satisfy s. On the other hand, p ⊧ □ ⋁^{↑}S iff all branches of p satisfy ⋁^{↑}S. We claim that in this case there is some s ∈ S that is satisfied by all branches of p. Assume that this is not the case. All branches of p satisfy ⋁^{↑}S, therefore this is a property we can observe after finite time. Since programs are supposed to be finitely branching, there can only be finitely many branches of p at that time. As S is directed, there must be a property s in S that all branches yet developed (and therefore all branches of p) must satisfy. Hence we want the equation ⋁_{s∈S}^{↑} □s = □ ⋁^{↑}S to hold in P_{L}^{♯}A for any directed subset S of A. Because the greatest element 1_{A} is supposed to be satisfied by each branch of every p, we want 1_{P{♯ \atop L}A} = □1_{A}. Furthermore, we also want 0_{P{♯ \atop L}A} = □0_{A} for the following reason: As 0_{A} is a property that no branch of any p can ever satisfy, □0_{A} will never be satisfied by any p. As the same property holds for 0_{P{♯ \atop L}A}, we want the two to be equal. Note that in order to formulate this motivation we had to get much more involved in the kind of properties we consider and also in the kind of observations we allow than before. These ideas lead to the following definition. Definition. For a locale A the Smyth power locale P_{L}^{♯}A is given by the free frame

F r⟨ □a : a ∈ A | □0_{A} = 0_{P{♯ \atop L}A} , □1_{A} = 1_{P{♯ \atop L}A} , □a ∧ □b = □(a ∧ b) (a, b ∈ A)

⋁_{s∈S}^{↑} □s = □ ⋁^{↑}S (S ⊆ A directed)⟩.

We can write every element of P_{L}^{♯}A in the form ⋁_{s∈S} □s for some subset S of A since every element of P_{L}^{♯}A is the join of finite meets of generators and □a∧□b = □(a∧b) for all a, b ∈ A. The above definition of the Smyth power locale also appears in [Rob88]; the modal operator □ is mentioned in [Win85]. As for the Hoare power construction, there is also another way to introduce nondeterminism. Once more we add the command \underline{or} to a deterministic programming language. Again, every property a of the locale A used to interpret that deterministic language is to give us a new property of the programs we can write using the new command \underline{or}. Because of the view of non-determinism described above we want p \underline{or} q ⊧ □a to hold iff both, p ⊧ □a and q ⊧ □a hold. This is the case iff ⟨p, q⟩ ⊧ □a⊗ □a in P_{L}^{♯}A ⊗ P_{L}^{♯}A where ⟨p, q⟩ is the point of P_{L}^{♯}A ⊗ P_{L}^{♯}A given by ⟨p, q⟩′(□a ⊗ □b) = 1_{2} iff p ⊧ □a and q ⊧ □b. Thus we obtain a connection between P_{L}^{♯}A and its product P_{L}^{♯}A⊗P_{L}^{♯}A. From our considerations of the Hoare power construction we would expect □a −^{}□a ⊗ □a to define a frame homomorphism from P_{L}^{♯}A to P_{L}^{♯}A ⊗ P_{L}^{♯}A. We shall see that this is indeed the case. There is also a motivation for the definition of the Smyth power locale from our studies of the Smyth power constructions for topological spaces. Above we have seen that the equation □U ∩ □U′ = □(U ∩ U′) holds for all U, U′ ∈ ΩX where X is an arbitrary T_{0}-space. Furthermore, □∅ = ∅ and □X = K_{T}X. For any directed subset U of ΩX we have seen that □ ⋃_{↑} U = ⋃_{↑}{□U | U ∈ U}. Note that the corresponding equations also hold for □^{s} and KT^{s}X. As these are the only equations that can be shown to hold for all T_{0}-spaces we get the same definition for a Smyth power locale based on the first of the two Smyth power constructions from the last section. The reader should note that although KT^{s}X has less points than K_{T}X we cannot prove more equations of the above type for the open set lattice of KT^{s}X. Therefore there is only one Smyth power locale construction. We will see later why this construction corresponds to K_{T}rather than KT^{s}.

7.4.1 The monad

As is the case for the power constructions considered so far, we can easily extend P_{L}^{♯} to an endofunctor on Loc. For any morphism f: A −^{}B in Loc we define P_{L}^{♯}f: P_{L}^{♯}A −^{}P_{L}^{♯}B by setting (P_{L}^{♯}f)′(□b) := □f′(b). Since f′ preserves directed joins and finite meets, this does define a continuous function between the locales P_{L}^{♯}A and P_{L}^{♯}B. Also, P_{L}^{♯}is a functor: Obviously, P_{L}^{♯}preserves identities. If f: A −^{}B and g: B −^{}C are continuous functions between locales then for all c ∈ C we have

(P_{L}^{♯}f)′((P_{L}^{♯}g)′(□c)) = (P_{L}^{♯}f)′(□g′(c)) = □f′(g′(c)) = (P_{L}^{♯}(g ∘ f))′(□c).

Hence (P_{L}^{♯}(f ∘ g))′ and (P_{L}^{♯}f ∘ P_{L}^{♯}g)′ are two frame homomorphisms agreeing on the generators of P_{L}^{♯}C and therefore must coincide. In order to give a natural transformation η: id_{Loc} −^{}P_{L}^{♯}, for any locale A we set η′_{A}(□a) := a which obviously defines a morphism in Loc. In order to see that we have thus given a natural transformation as claimed, note that for every f: A −^{}B in Loc we get η′_{A}((P_{L}^{♯}f)′(□b)) = η′_{A}(□f′(b)) = f′(b) = f′(η′_{B}(□b)) for all b ∈ B.

We still have to find a natural transformation μ: (P_{L}^{♯})^{2} −^{}P_{L}^{♯}to obtain a monad. By setting μ′_{A}(□a) := □ □a we define a continuous function from (P_{L}^{♯})^{2}A to P_{L}^{♯}A as, for example,

μ′_{A}(□(a ∧ b)) = □ □(a ∧ b) = □(□a ∧ □b) = □ □a ∧ □ □b.

The other equations needed can be shown analogously. This defines a natural transformation as desired since for all f: A −^{}B in Loc and all b ∈ B we have

((P_{L}^{♯})^{2}f)′(μ′_{B}(□b)) = ((P_{L}^{♯})^{2}f)′(□ □b)
= □(P_{L}^{♯}f)′(□b)
= □ □f′(b)
= μ′_{A}(□f′(b))
= μ′_{A}((P_{L}^{♯}f)′(□b)).

The remaining diagrams defining a monad can also easily be shown to commute. We shall only give one more example for this. For a ∈ A we have η′_{P_{L}^{♯}A}(μ′_{A}(□a)) = η′_{P_{L}^{♯}A}(□ □a) = □a proving μ_{A} ∘ η_{P{♯ \atop L}A} = id_{P{♯ \atop L}A}.

Proposition 7.38 The triple (P_{L}^{♯}, η, μ) is a monad on Loc.

7.4.2 Power space versus power locale

For the Hoare power construction we could show that SpP_{L}^{♭}ΩX ≅ P_{T}^{♭}X for any T_{0}- space X. We will now see that the above Smyth power construction for locales corresponds to the first of the two Smyth power constructions we introduced for topological spaces. What are the points of P_{L}^{♯}A? They are given by frame homomorphisms from P_{L}^{♯}A to 2, that is functions from A to 2 that preserve directed joins, finite meets (including the empty meet) and 1. Such a function is uniquely determined by the inverse image of 1_{2} which has to be a Scott-open filter not containing 0_{A}. Hence the points of P_{L}^{♯}A are the Scott-open filters of A \ {0_{A}}. Thus, if we start with a sober space X then, by the Hofmann-Mislove Theorem 7.4, the points of P_{L}^{♯}ΩX correspond to the elements of K_{T}X. Furthermore, this yields a homeomorphism κ_{X}: K_{T}X −^{}SpP_{L}^{♯}ΩX. The set □U := {K ∈ K_{T}X | K ⊆ U}, for some open subset U of X, corresponds to the set

{p: 2 −^{}P_{L}^{♯}ΩX | ∃K ⊆ U . p′(□U′) = 1_{2} iff K ⊆ U′}
= {p ∈ SpP_{L}^{♯}ΩX | p′(□U) = 1_{2}}
= {p ∈ SpP_{L}^{♯}ΩX | p ⊧ □U}.

Moreover, if f: X −^{}Y is a continuous function then for each K ∈ K_{T}X we have (κ_{Y} (K_{T}f(K)))′(□V) = (κ_{Y} (↑ f(K)))′(□V) = 1_{2} iff V is a neighbourhood of f(K). On the other hand,

( SpP_{L}^{♯}Ωf(κ_{X}(K)))′(□V) = (κ_{X}(K))′((P_{L}^{♯}Ωf)′(□V))
= (κ_{X}(K))′( □Ωf(V))
= (κ_{X}(K))′(□f^{−1}(V))
= 1_{2}

iff f^{−1}(V) is a neighbourhood of K. Therefore SpP_{L}^{♯}Ωf ∘ κ_{X} = κ_{Y} ∘ K_{T}f. These considerations yield the following result.

Proposition 7.39 There is a natural isomorphism K_{T}−^{}SpP_{L}^{♯}Ω for the functors K_{T}, SpP_{L}^{♯}Ω: Sob −^{}Sob.

So, if X is a sober space such that P_{L}^{♯}ΩX is spatial then P_{L}^{♯}ΩX ≅ Ω SpP_{L}^{♯}ΩX ≅ Ω K_{T}X. On the other hand, if A is a spatial locale, then K_{T}SpA ≅ SpP_{L}^{♯}Ω SpA ≅ SpP_{L}^{♯}A.

7.4.3 The algebras for the monad

In this section we will study the category of algebras for the monad (P_{L}^{♯}, η, μ). First of all note that if ξ: P_{L}^{♯}A −^{}A is a structure map for a locale A then ξ must be the upper adjoint for η_{A}. Recall that this is the case if ξ′ is the lower adjoint for η′_{A} in the usual sense. As η′_{A} ∘ ξ′ = id′_{A} it remains to show that ξ′ ∘ η′_{A} ≤ id′_{P_{L}^{♯}A}. Let a be an arbitrary element of A and let ⋁_{i∈I} □a_{i} := ξ′(a). Then ξ′(η′_{A}(□a)) = ⋁_{i∈I} □a_{i}. Since a = η′_{A}(ξ′(η′_{A}(□a))) = η′_{A}( ⋁_{i∈I} □a_{i}) = ⋁_{i∈I} a_{i} we may conclude that a_{i} ≤ a for all i ∈ I. Therefore, □a_{i} ≤ □a for all i ∈ I implying ⋁_{i∈I} □a_{i} ≤ □a. Consequently, ξ′ ∘ η′_{A} ≤ id′_{P_{L}^{♯}A} holds. Hence there is at most one structure map for any locale A which we refer to as ξ_{A} from now on.

Lemma 7.40 Every locale of the form P_{L}^{♯}A is a deflationary localic semilattice. The semilattice structure v: P_{L}^{♯}A ⊗ P_{L}^{♯}A −^{}P_{L}^{♯}A is defined by v′(□a) := □a ⊗ □a.

Proof: Obviously, v′(1_{P{♯ \atop L}A}) = v′(□1_{A}) = □1_{A} ⊗ □1_{A} = 1_{P{♯ \atop L}A} ⊗ 1_{P{♯ \atop L}A} = 1_{P{♯ \atop L}A⊗P_{L}^{♯}A} and v′(0_{P{♯ \atop L}A}) = v′(□0_{A}) = □0_{A} ⊗ □0_{A} = 0_{P{♯ \atop L}A⊗P_{L}^{♯}A}. For a, a′ ∈ A we have

v′(□(a ∧ a′)) = □(a ∧ a′) ⊗ □(a ∧ a′)
= (□a ∧ □a′) ⊗ (□a ∧ □a′)
= (□a ⊗ □a) ∧ (□a′ ⊗ □a′)
= v′(□a) ∧ v′(□a′).

Let S be a directed subset of A. Then

v′(□ ⋁^{↑}S) = (□(⋁^{↑}S)) ⊗ (□(⋁^{↑}S))
= ( ⋁_{s∈S}^{↑} □s) ⊗ ( ⋁_{s∈S}^{↑} □s)
= ⋁_{s∈S}^{↑} ⋁_{s′∈S}^{↑}(□s ⊗ □s′)
= ⋁_{s∈S}^{↑}(□s ⊗ □s)
= ⋁_{s∈S}^{↑}v′(□s).

Hence v′ preserves the relations defining P_{L}^{♯}A and thus defines a continuous function v: P_{L}^{♯}A ⊗ P_{L}^{♯}A −^{}P_{L}^{♯}A. This function is idempotent since

Δ′_{P_{L}^{♯}A}(v′(a)) = Δ′_{P_{L}^{♯}A}(□a ⊗ □a) = □a ∧ □a = □a

for all a ∈ A. Similarly, we can show associativity and symmetry of v. Therefore it remains to prove Δ_{P{♯ \atop L}A} ∘ v ≥ id_{P{♯ \atop L}A⊗P_{L}^{♯}A}, that is v′ ∘ Δ′_{P_{L}^{♯}A} ≤ id′_{P_{L}^{♯}A⊗P_{L}^{♯}A} for the corresponding frame homomorphisms. The calculation

v′(Δ′_{P_{L}^{♯}A}(□a ⊗ □b)) = v′(□a ∧ □b) = v′(□(a ∧ b)) = □(a ∧ b) ⊗ □(a ⊗ b) ≤ □a ⊗ □b

for all a and b in A suffices to prove the desired result.

The semilattice structure from the last lemma is closely related to the semilattice structure P. Johnstone defines for the Vietoris power locale [Joh82a].

Proposition 7.41 The category of P_{L}^{♯}-algebras is a subcategory of Loc^{∧}, the category of deflationary semilattices in Loc.

Proof: Because of the last lemma as well as because of Lemma 5.5 we may conclude that every P_{L}^{♯}-algebra is a deflationary localic semilattice. Now for morphisms. Let f: A −^{}B be a morphism of the P_{L}^{♯}-algebras (A, ξ_{A}) and (B, ξ_{B}). First of all we show that P_{L}^{♯}f is a morphism of the deflationary semilattices P_{L}^{♯}A and P_{L}^{♯}B. Let the semilattice structures from Lemma 7.40 for P_{L}^{♯}A and P_{L}^{♯}B be denoted by v and w, respectively. For all b ∈ B we have

v′((P_{L}^{♯}f)′(□b)) = v′(□f′(b))
= □f′(b) ⊗ □f′(b)
= (P_{L}^{♯}f)′(□b) ⊗ (P_{L}^{♯}f)′(□b)
= (P_{L}^{♯}f ⊗ P_{L}^{♯}f)′(□b ⊗ □b)
= (P_{L}^{♯}f ⊗ P_{L}^{♯}f)′(w′(□b)).

By Lemma 5.5 we know that the semilattice structures for A and B are given by ξ_{A} ∘ v ∘ (η_{A} ⊗ η_{A}) and ξ_{B} ∘ w ∘ (η_{B} ⊗ η_{B}), respectively. Hence

f ∘ (ξ_{A} ∘ v ∘ (η_{A} ⊗ η_{A}))
= ξ_{B} ∘ P_{L}^{♯}f ∘ v ∘ (η_{A} ⊗ η_{A})  (f morphism of P_{L}^{♯}-algebras)
= ξ_{B} ∘ w ∘ (P_{L}^{♯}f ⊗ P_{L}^{♯}f) ∘ (η_{A} ⊗ η_{A}) (P_{L}^{♯}f morphism of P_{L}^{♯}-algebras)
= ξ_{B} ∘ w ∘ ((P_{L}^{♯}f ∘ η_{A}) ⊗ (P_{L}^{♯}f ∘ η_{A}))
= ξ_{B} ∘ w ∘ ((η_{B} ∘ f) ⊗ (η_{B} ∘ f))  (Naturality of η)
= ξ_{B} ∘ w ∘ (η_{B} ⊗ η_{B}) ∘ (f ⊗ f).

For T_{0}-spaces we have seen that every algebra for one of the two Smyth power functors has an additional property besides being a deflationary semilattice. Such an algebra has small semilattices, that is any open set is the union of the interiors of the semilattices it contains. We were able to deduce this from the fact that any space of the form K_{T}X or KT^{s}X has a basis of open semilattices. A subset S of a ∧-semilattice is a sub-semilattice iff ∧_{n}(S × ⋯ × S) ⊆ S or S × ⋯ × S ⊆ ∧_{n}^{−1}(S) for all n ∈ ℕ. If A is a locale then for any element a of A we get that □a is a semilattice (with respect to the semilattice structure of Lemma 7.40) in the above sense: For all n ∈ ℕ we have v′_{n}(□a) = □a ⊗ ⋯ ⊗ □a. Put differently, this means that we have

a = ⋁{b ∈ P_{L}^{♯}A | ∀n ∈ ℕ .b ⊗ ⋯ ⊗ b ≤ v′_{n}(a)}

for all a ∈ P_{L}^{♯}A. Definition. We say that a deflationary localic semilattice (A, v) has small semilattices if, for all a ∈ A,

a = ⋁{b ∈ A | ∀n ∈ ℕ .b ⊗ ⋯ ⊗ b ≤ v′_{n}(a)}.

The following lemma tells us that this definition indeed captures our intentions.

Proposition 7.42 Let (A, v) be a deflationary localic semilattice with small semilattices. Then the deflationary topological semilattice SpA has small semilattices.

Proof: Because of Lemma 5.2, SpA is a deflationary topological semilattice. The semilattice structure is induced by Spv: Sp(A ⊗ A) −^{}SpA. By Proposition 2.4, Sp(A ⊗ A) and SpA × SpA are isomorphic. Abusing our notation slightly, let Spv denote the semilattice structure on SpA. Recall that the open sets in SpA are the sets of the form U_{a} := {p ∈ SpA | p ⊧ a} where a ∈ A. So let p ∈ SpA be an element of some U_{a}. We have to show that there is a semilattice in U_{a} that is a neighbourhood of p. Since a = ⋁{b ∈ A | ∀n ∈ ℕ . b ⊗ ⋯ ⊗ b ≤ v′_{n}(a)} there is some b ∈ A with b ⊗ ⋯ ⊗ b ≤ v′_{n}(a) for all n ∈ ℕ such that p ⊧ b. Let

S := ⋃_{n∈ℕ}{p_{1} ∧ ⋯ ∧ p_{n} | p_{1}, …, p_{n} ∈ U_{a}} = ⋃_{n∈ℕ}(Spv_{n})(U_{a} × ⋯ × U_{a}).

Obviously, S is a semilattice that contains U_{b} and therefore is a neighbourhood of p. It remains to prove that S is a subset of U_{a}. Since b ⊗ ⋯ ⊗ b ≤ v′_{n}(a) we have U_{b} × ⋯ × U_{b} ⊆ (Spv_{n})^{−1}(U_{a}) which proves Spv_{n}(U_{b} × ⋯ × U_{b}) ⊆ U_{a} for all n ∈ ℕ.

Lemma 7.43 Every P_{L}^{♯}-algebra has small semilattices.

Proof: Let (A, ξ_{A}) be a -algebra. By Lemma 7.40 and Proposition 7.41 both, P_{L}^{♯}A and A, are deflationary semilattices with semilattice structures, say w and v, respectively. Furthermore, ξ_{A} is a morphism of the P_{L}^{♯}-algebras (P_{L}^{♯}A, μ_{A}) and (A, ξ_{A}) and therefore is a semilattice morphism from (P_{L}^{♯}A, w) to (A, v), also by Proposition 7.41. For all a ∈ A we have

a = η′_{A}(ξ′_{A}(a))
= η′_{A}(⋁{a ∈ P_{L}^{♯}A | ∀n ∈ ℕ . a ⊗ ⋯ ⊗ a ≤ w′_{n}(ξ′_{A}(a))})
= ⋁{η′_{A}(a) | ∀n ∈ ℕ . a ⊗ ⋯ ⊗ a ≤ (ξ_{A} ⊗ ⋯ ⊗ ξ_{A})′(v′_{n}(a))}
≤ ⋁{η′_{A}(a) | ∀n ∈ ℕ . (η_{A} ⊗ ⋯ ⊗ η_{A})′(a ⊗ ⋯ ⊗ a)
≤ (η_{A} ⊗ ⋯ ⊗ η_{A})′(ξ_{A} ⊗ ⋯ ⊗ ξ_{A})′(v′_{n}(a)) = v′_{n}(a)}
≤ ⋁{b ∈ A | ∀n ∈ N . b ⊗ ⋯ ⊗ b ≤ v′_{n}(a)}.

If b ⊗ ⋯ ⊗ b ≤ v′_{n}(a) for all n ∈ ℕ then b ≤ a. Therefore, we are done.

We cannot show in general, however, that every deflationary localic semilattice with small semilattices is an algebra for the monad. Neither can we give a counterexample. Since frames that are presented via generators and relations are not easy to handle, the search for a counterexample might be quite a tedious task. Problem. Is there a deflationary semilattice with small semilattices in Loc that is not a P_{L}^{♯}-algebra? Note that we cannot re-use our example for topological spaces (see Section 7.3.3) since a topological semilattice in general does not give rise to a localic semilattice. Hence we only have the following result which follows from Lemma 4.6.

Proposition 7.44 A deflationary localic semilattice with small semilattices A is a P_{L}^{♯}-algebra iff η_{A}: A −^{}P_{L}^{♯}A has an upper adjoint ξ_{A} in Loc with ξ_{A} ∘ η_{A} = id_{A}.

7.4.4 Locally compact locales

From our studies of locally compact spaces it follows that, restricted to the category LCompL of locally compact locales, P_{L}^{♯}does give the free deflationary semilattice with small semilattices.

Proposition 7.45 If A is a deflationary semilattice with small semilattices in the category of locally compact locales then it is isomorphic to the open set lattice of a domain.

Proof: By Proposition 7.42, SpA is a deflationary topological semilattice with small semilattices. Furthermore, it is a locally compact sober space. Hence, by Theorem 7.30, SpA is a domain with its Scott topology. Since A is isomorphic to Ω SpA, we are done.

Proposition 7.46 If A is a locally compact locale so is P_{L}^{♯}A.

Proof: We will show that □a ≪ □b in P_{L}^{♯}A if a ≪ b in A. Suppose that S is a directed subset of P_{L}^{♯}A with □b ≤ ⋁^{↑}S. Let S′ := {□c ∈ P_{L}^{♯}A | ∀c′ ≪ c.∃s ∈ S . □c′ ≤ s}. Then S′ is a lower set of generators for P_{L}^{♯}A that is closed with respect to directed joins: If T is a directed subset of S′ then for all c′ ≪ ⋁^{↑}{c ∈ A | □c ∈ T } there is □c ∈ T with c′ ≤ c. Therefore □c′ ≤ s for some s ∈ S which proves that □ ⋁^{↑}{c ∈ A | □c ∈ T } ∈ S′. Furthermore, the join of S′ is the join of S. Consequently, S′ is a C-ideal in the sense of Theorem 1.11. Hence □b ∈ S′ which proves □a ≤ s for some s ∈ S. Thus {□a ∈ P_{L}^{♯}A | a ≪ b} is a directed subset of ↓ ↓{□b} with join □b. Therefore, ↓ ↓{□b} is directed with join □b. Since a = ⋁{□b | □b ≤ a} and ↓ ↓{a} ⊇ {↓{□b} | □b ≤ a} for every a ∈ P_{L}^{♯}A, we are done.

For a locally compact locale A we know that P_{L}^{♯}A ≅ Ω SpP_{L}^{♯}Ω SpA ≅ Ω K_{T}SpA by Proposition 7.39. Furthermore, this isomorphism is natural P_{L}^{♯}−^{}Ω K_{T}Sp; we will denote it by ι. Note that the components of ι are defined by ι′_{A}(□U_{a}) = □a where U_{a} := {p ∈ SpA | p ⊧ a} denotes the open subset of Ω SpA that is generated by a. In order to characterize the category of algebras for the monad (P_{L}^{♯}, η, μ) restricted to LCompL we try to find an equivalent monad which is built from (K_{T}, ↑, ⋃). We already have a natural isomorphism ι: P_{L}^{♯}−^{}Ω K_{T}Sp. In addition, Ω ↑ Sp: Ω Sp −^{}Ω K_{T}Sp is a natural transformation. There is a natural isomorphism α from the identity of the full subcategory of spatial locales to Ω Sp (see Chapter 2). Recall that the components of α are defined by α_{A}(U_{a}) := a for all a ∈ A. Thus we get a natural transformation \overline{η} := (Ω ↑ Sp) • α: id_{LCompL} −^{}Ω K_{T}Sp. We claim that ι • η = Ω ↑ Sp•α. Let A be a locally compact locale. For every a ∈ A we calculate

α′_{A}((Ω ↑_{Sp A})′(□U_{a})) = α′_{A}(↑_{Sp A}^{−1}(□U_{a}))
= α′_{A}(U_{a})
= a
= η′_{A}(□a)
= η′_{A}(ι′_{A}(□U_{a})).

We still have to find a natural transformation (Ω K_{T}Sp)^{2} −^{}Ω K_{T}Sp in order to obtain a monad for the functor Ω K_{T}Sp. Let γ: SpΩ −^{}id_{Sob} be the natural isomorphism we introduced in Chapter 2. Since Ω K_{T}γ K_{T}Sp: Ω K_{T}SpΩ K_{T}Sp −^{}Ω (K_{T})^{2} Sp and Ω ⋃ Sp: Ω (K_{T})^{2} Sp −^{}Ω K_{T}Sp are natural, their composite \overline{μ} := Ω(⋃ •(K_{T}γ K_{T})) Sp is a natural transformation as needed. We want to show that

ι ∘ μ = (Ω ∘ (⋃ •(K_{T}∘γ ∘ K_{T})) ∘ Sp) • (ι ∘ ι)

since, by Proposition 4.1, this implies that (Ω K_{T}Sp, \overline{η}, \overline{μ}) is a monad that is equivalent to (P_{L}^{♯}, η, μ). We achieve this by verifying that for any locale A we have

ι_{A} ∘ μ_{A} = Ω ⋃_{Sp A} ∘Ω K_{T}γ_{K_{T}Sp A} ∘ ι_{Ω K_{T}Sp A} ∘ (P_{L}^{♯}ι_{A}).

This computation is a bit messy since we have to keep track of all the natural transformations involved but it is not difficult. Recall that γ_{X}^{−1}(V) = U_{V} for any open subset V of some topological space X. Let a be an arbitrary element of some locale A. Then (ι_{A} ∘ μ_{A})′(□U_{a}) = μ′_{A}(ι′_{A}(□U_{a})) = μ′_{A}(□a) = □ □a. On the other hand we have

(P_{L}^{♯}ι_{A})′(ι′_{Ω K_{T}Sp A}((Ω K_{T}γ_{K_{T}Sp A})′((Ω ⋃_{Sp A})′(□U_{a}))))
= (P_{L}^{♯}ι_{A})′(ι′_{Ω K_{T}Sp A}((Ω K_{T}γ_{K_{T}Sp A})′ ⋃_{Sp A}^{−1}(□U_{a})))
= (P_{L}^{♯}ι_{A})′(ι′_{Ω K_{T}Sp A}((K_{T}γ_{K_{T}Sp A})^{−1}(□□U_{a})))
= (P_{L}^{♯}ι_{A})′(ι′_{Ω K_{T}Sp A}(□γ_{K_{T}Sp A}^{−1}(□U_{a})))
= (P_{L}^{♯}ι_{A})′(ι′_{Ω K_{T}Sp A}(□U_{□U_{a}}))
= (P_{L}^{♯}ι_{A})′(□ □U_{a})
= □ι′_{A}(□U_{a})
= □ □a.

Hence we have shown the following proposition.

Proposition 7.47 The monad (P_{L}^{♯}, η, μ) is equivalent to the monad (Ω K_{T}Sp, \overline{η}, \overline{μ}) on LCompL.

Theorem 7.48 The category of algebras for the monad (P_{L}^{♯}, η, μ) on the category LCompL of locally compact locales is the category of deflationary semilattices that have small semilattices in LCompL.

Proof: Let A be a deflationary semilattice with small semilattices in LCompL. By Proposition 7.42, SpA is a deflationary semilattice with small semilattices in the category of locally compact sober spaces. Hence, by Theorem 7.31, there is a structure map ⋀_{Sp A}: K_{T}SpA −^{}SpA making SpA a K_{T}-algebra. Then (Ω SpA, Ω ⋀_{Sp A}) is an algebra for the monad (Ω K_{T}Sp, \overline{η}, \overline{μ}). By Proposition 4.2 this implies that (Ω SpA, Ω ⋀_{Sp A} ∘ι_{A}) is a P_{L}^{♯}-algebra. Since Ω SpA and A are (naturally) isomorphic, we can indeed find a structure map for A. Analogously we can show that every morphism of deflationary semilattices with small semilattice is a morphism of the corresponding P_{L}^{♯}-algebras.

The proof we gave for this characterization of P_{L}^{♯}-algebras in the category of locally compact locales is not ‘localic’ in nature. Hence it does not tell us why we could not show an analogous characterization for arbitrary locales. However, the only other proof we were able to come up with is to consider open set lattices of locally compact sober spaces and then more or less repeat the proof for topological spaces. We consider this proof even less expressive than the one we gave. That we have not been able to find a truly ‘pointless’ proof is quite unsatisfactory.

7.4.5 Unital semilattices

If we want to make every Smyth power locale a unital semilattice in Loc then we drop the equation □0_{A} = 0_{P{♯ \atop L}A} from the above definition. We thus get a slightly different Smyth power construction, say Ṗ_{L}^{♯}. Obviously, we also get a monad for Ṗ_{L}^{♯}. To see that every locale of the form Ṗ_{L}^{♯}A is a unital deflationary semilattice, note that Ṗ_{L}^{♯}A carries a semilattice structure similar to the one we have for P_{L}^{♯}A. We obtain this semilattice structure v: Ṗ_{L}^{♯}A ⊗ Ṗ_{L}^{♯}A −^{}Ṗ_{L}^{♯}A by setting v′(□a) := □a ⊗ □a for all a ∈ A. To show that this definition makes Ṗ_{L}^{♯}A a deflationary semilattice we can proceed in the same way as we did for P_{L}^{♯}A. By Proposition 5.8 it is sufficient to show that o′(□a) := 1_{2} defines a continuous function from 2 to Ṗ_{L}^{♯}A in order to prove that v has a unit. Since the element 0_{Ṗ_{L}^{♯}A} is freely adjoint to P_{L}^{♯}A, the above definition makes o the unit for the semilattice structure for Ṗ_{L}^{♯}A. For locally compact locales we can also prove that every unital deflationary semilattice with small semilattices is a Ṗ_{L}^{♯}-algebra.

Appendix.

Name Objects
C^{SLat} semilattices in C
C^{∨}  inflationary semilattices in C
C^{∧}  deflationary semilattices in C

Table 1: Subcategories of an order-enriched category C; in each case the morphisms are all semilattice homomorphisms.

Name  Objects  Morphisms: Scott-continuous

functions that additionally preserve

DCPO directed complete partial orders
DCPO^{∨} dcpo’s that have binary joins  binary joins

= up-complete ∨-semilattices

DCPO^{∧} dcpo’s that have binary meets binary meets

= ∧-continuous ∧-semilattices

Dom  continuous dcpo’s = domains
Dom^{∨} continuous ∨-semilattices  binary joins
Dom^{∧} continuous ∧-semilattices  binary meets
AlgD  algebraic dcpo’s
AlgD^{∨} algebraic ∨-semilattices  binary joins
AlgD^{∧} algebraic ∧-semilattices  binary meets

Table 2: Subcategories of DCPO

Name  Objects
Top  T_{0}-spaces
Sob  sober spaces
LCompT locally compact T_{0}-spaces
LCompS locally compact sober spaces

Table 3: Full Subcategories of Top

Name  Objects
Loc  locales
LCompL locally compact locales

Table 4: Full subcategories of Loc

Name Operation
Ω_{ne} X non-empty open subsets of a topological space X
CD  compact elements of an algebraic dcpo D
KX  non-empty compact saturated subsets of a T_{0}-space X

ordered by reverse inclusion

K_{D}D non-empty compact upper subsets of a dcpo D

ordered by reverse inclusion

KD^{s}D non-empty strongly compact upper subsets of a dcpo D

ordered by reverse inclusion

PS  power set of a set S
P_{fin}S  finite subsets of a set S
P_{fne}S finite non-empty subsets of a set S
P_{L}P  lower subsets of a poset P ordered by inclusion
P_{U}P upper subsets of a poset P ordered by reverse inclusion

Table 5: Operators

Name From to  Operator
Idl Pos −^{}AlgD ideal completion

Ω

Top −^{}Loc

X −^{}the open set lattice of X

Sp

Loc −^{}Top

A −^{}the spatialization of A

Pt

Loc −^{}DCPO

A −^{}the set of points for A

with the specialization order

A  Top −^{}DCPO^{∨}  X −^{}({A ⊆ X | ∅ ≠ A closed}, ⊆)
A^{D}

DCPO −^{}DCPO

D −^{}({A ⊆ D | ∅ ≠ A Scott-closed}, ⊆)

T_{σ}

DCPO −^{}Top

D −^{}the set D with the Scott topology

T_{ω}

DCPO^{∨} −^{}Top

L −^{}the set L with the weak topology Table 6: Functors

Name Category Operator
P_{A}^{♭}≅ A^{D}

AlgD

D −^{}Idl(P_{fne}CD, ≤^{♭})
P_{D}^{♭}  DCPO P_{D}^{♭}= A_{D}
P_{T}^{♭}  Top  P_{T}^{♭}= T_{ω}A
P_{L}^{♭}  Loc  A −^{}F r⟨⋄a : a ∈ A | ⋄1_{A} = 1_{P_{L}^{♭}A}

⋁_{s∈S} ⋄s = ⋄ ⋁ S (S ⊆ A)⟩

Table 7: Hoare power constructions

Name Category Algebras
P_{A}^{♭}≅ A^{D} AlgD  AlgD^{∨}
P_{D}^{♭} DCPO DCPO^{∨}
Dom  Dom^{∨}
P_{T}^{♭} Top Top^{∨} ∩ Sob
LCompT Top^{∨} ∩ LCompS
P_{L}^{♭} Loc Loc^{∨}
LCompL Loc^{∨} ∩ LCompL

Table 8: Algebras for Hoare power constructions

Name  Category Operator

P_{A}^{♯} AlgD

D −^{}Idl(P_{fne}CD, ≤^{♯})

K_{D}(≅ F_{D} Dom

D −^{}({K ⊆ D | ∅ ≠ K saturated and
≅ F D^{s})  Scott-compact}, ⊇)

F_{D} DCPO

D −^{}{φ ⊆ Ω_{ne} D | φ Scott-open filter}

F D^{s} DCPO

D −^{}{φ ⊆ Ω_{ne} D | φ strong filter}

K^{T} Top

X −^{}({K ⊆ X | ∅ ≠ K saturated and

compact}, □-topology)

KT^{s} Top

X −^{}({K ⊆ X | ∅ ≠ K saturated and

strongly compact}, □^{s} -topology)

KT^{f} Top X −^{}{↑{x_{1}, …, x_{n}} | x_{1}, …, x_{n} ∈ X}

as a subspace of K_{T}X

P_{L}^{♯} Loc A −^{}F r⟨□a : a ∈ A | □0_{A} = 0_{P{♯ \atop L}A} , □1_{A} = 1_{P{♯ \atop L}A},

□a ∧ □b = □(a ∧ b) (a, b ∈ A),

⋁_{s∈S}^{↑} □s = □ ⋁^{↑}S (S ⊆ A directed)⟩

Table 9: Smyth power constructions

Name  Category Algebras
P_{A}^{♯} AlgD AlgD^{∧}
K_{D}(≅ F_{D} ≅ F D^{s}) Dom  Dom^{∧}
F_{D}  DCPO  ⊂ DCPO^{∧}
F D^{s}  DCPO  ⊂ DCPO^{∧}
K^{T}  Top  ⊂ Top_{ssl}^{∧}
LCompS Top_{ssl}^{∧} ∩ LCompS ≅Dom^{∧}
KT^{s}  Sob  Top_{ssl}^{∧} ∩ Sob
KT^{f}  Top  Top_{ssl}^{∧}
P_{L}^{♭}  Loc  ⊆ Loc_{ssl}^{∧}
LCompL Loc_{ssl}^{∧} ∩ LCompL

Table 10: Algebras for Smyth power constructions; C_{ssl}^{∧} is the full subcategory of semilattices with small semilattices of C^{∧} 160

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