Concurrent semantics for fusions: Weak prime domains and connected event structures

Stable event structures, and their duality with prime algebraic domains, represent a landmark of concurrency theory, since they provide a neat characterisation of causality in computations. As such, they have been used for defining the concurrent semantics of many formalisms, from Petri nets to (linear) graph rewriting systems. Stability however is restrictive for formalisms with “fusion”, where a computational step may merge parts of the state. This happens e.g. for graph rewriting systems with non-linear rules, which are used to cover some relevant applications (such as the graphical encoding of calculi with name passing). Guided by the need of giving semantics to such formalisms, we leave aside stability and characterise a class of domains, referred to as weak prime domains, naturally generalising prime algebraic domains. We then identify a corresponding class of event structures, that we call connected event structures, via a duality result formalised as an equivalence of categories.