Open maps as a bridge between algebraic observational equivalence and bisimilarity

There are two widely accepted notions of behavioural equivalence, formalizing the idea of observational indistinguishability: observational equivalence for algebras (which are models for sequential systems) and bisimulation equivalence (bisimilarity) for concurrent processes. In this paper we show that the observational equivalences for standard, partial and regular algebras are bisimulation equivalences. This is done in the setting of open maps, proposed in JNW93] as an abstract approach to behavioural equivalences of processes. The main advantage of the results is capturing the models for sequential and concurrent systems in a uniform framework. In such an abstract setting we formulate the property of determinism, shared by all the algebras considered in this paper, and identify some interesting facts about bisimilarity in the deterministic case. All the results for standard, regular and partial algebras are obtained by the applications of a general machinery developed in the paper, which we expect to be applicable also in other contexts.