# Domains and Event Structures for Fusions

## TL;DR

This paper extends the theory of event structures to include 'fusion' phenomena by introducing weak prime domains and connected event structures, enabling semantics for non-linear graph rewriting systems with merging steps.

## Contribution

It introduces weak prime domains and connected event structures, generalizing stable event structures to handle fusion in concurrent computations.

## Key findings

- Connected event structures correspond to semantics of non-linear graph rewriting.
- Weak prime domains generalize prime algebraic domains for fusion scenarios.
- Category of unstable event structures coreflects into weak prime domains.

## Abstract

Stable event structures, and their duality with prime algebraic domains arising as partial orders of configurations, are a landmark of concurrency theory, providing a clear characterisation of causality in computations. They have been used for defining a concurrent semantics of several formalisms, from Petri nets to (linear) graph rewriting systems, which in turn lay at the basis of many visual modelling frameworks. Stability however is restrictive when dealing with formalisms with "fusion", i.e., where a computational step can not only consume and produce but also merge parts of the state. This happens, e.g., for graph rewriting systems with non-linear rules, which are needed to cover some relevant applications (such as the graphical encoding of calculi with name passing). Guided by the need of capturing the semantics of formalisms with fusion we leave aside stability and we characterise, as a natural generalisation of prime algebraic domains, a class of domains, referred to as weak prime 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. We show that connected event structures are exactly the class of event structures that arise as the semantics of non-linear graph rewriting systems. Interestingly, the category of general unstable event structures coreflects into our category of weak prime domains, so that our result provides a characterisation of the partial orders of configurations of such event structures.