EXTREME THEORY
On Quotients and Completions of Partially Ordered Monoids - Samy Abbes
Thursday - 15.11.2007
In any monoid (= semigroup with unit element), left divisibility induces a pre-order and thus a partial order in the collapse monoid. Hence, we say that u is less than v if v=ux for some x in the monoid. That's the "on the one hand" part.
On the other hand, monoids are often defined as quotients of other monoids with respect to congruences--typically, monoids presented by generators and relations.
Now on the third hand, partial orders can be completed in order to give a sense to l.u.b. of suitable subsets (countable chains and directed subsets are typical examples).
So when mixing these three ingredients, natural questions arise :
- Does the quotient of a completion coincide with the completion of the quotient? The answer is "no" in general for partial orders, but "yes" for partial orders arising from monoids.
- When completing the free semigroup for lub of countable chains, we find that the result is chain-complete ("countable" has been droped). Can we extend this result to quotients? The answer is "yes", and the method I use involves topological tools.
The talk will summarize these results, and try to picture the situation as clearly as possible. Proofs will be given on particular examples, and emphasis will be put on the topological aspects.