Process Semantics of Petri Nets over Partial Algebra

“Petri nets are monoids” is the title and the central idea of the paper [7]. It provides an algebraic approach to define both nets and their processes as terms. A crucial assumption for this concept is that arbitrary concurrent composition of processes is defined, which holds true for place/transition Petri nets where places can hold arbitrarily many tokens. This paper defines a similar concept for elementary Petri nets, which are elementary net systems with arbitrary initial marking. Since markings of elementary nets cannot be added arbitrarily, some operators are only defined partially; hence we employ concepts of partial algebra. The main result of the paper states that the semantics based on process terms agrees with the classical partial-order process semantics for elementary net systems. More precisely, we provide a syntactic equivalence notion for process terms and a bijection from according equivalence classes of process terms to isomorphism classes of partially ordered processes.