We first consider the existence of finite equational axiomatisations. In order to express an interesting class of equational axioms we embed the processes into a simply typed lambda calculus, allowing equation schemes with metasubstitutions to be expressed by pure equations. Two equivalences over the lambda terms are defined, an extensional equality and a higher order bisimulation. Under a restriction to first order variables these are shown to coincide and an examination of the coincidence shows that no finite equational axiomatisation of strong bisimulation can exist. We then encode the processes of Basic Process Algebra with iteration and zero (BPA_\delta^*) into this lambda calculus and show that it too is not finitely equationally axiomatisable, in sharp contrast to the extant positive result for the fragment without zero.
For the implicational theory, we show the existence of finite computable complete sets of unifiers for finite sets of equations between processes (with zero order variables). It follows that the soundness of sequents over these is decidable.
Some applications to the theories of higher order process calculi and non-well-founded sets are made explicit.
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