Petri nets are monoids: a new algebraic foundation for net theory

The composition and extraction mechanisms of Petri nets are at present inadequate. This problem is solved by viewing place/transition Petri nets as ordinary, directed graphs equipped with two algebraic operations corresponding to parallel and sequential composition of transitions. A distributive law between the two operations captures a basic fact about concurrency. Novel morphisms are defined, mapping single, atomic transitions into whole computations, thus relating system descriptions at different levels of abstraction. Categories equipped with products and coproducts (corresponding to parallel and nondeterministic compositions) are introduced for Petri nets with and without initial markings. It is briefly indicated how the approach yields function spaces and novel interpretations of duality and invariants. The results provide a formal basis for expressing the semantics of concurrent languages in terms of Petri nets and an understanding of concurrency in terms of algebraic structures over graphs and categories that should apply to other models and contribute to the conceptual unification of concurrency.<>