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. A decade earlier, [10] presented a similar concept for elementary Petri nets, i.e. nets where no place can ever carry more than one token. Since markings of elementary Petri nets cannot be added arbitrarily, concurrent composition is defined as a partial operation. The present papers provides a general approach to process term semantics. Terms are equipped with the minimal necessary information to determine if two process terms can be composed concurrently. Applying the approach to elementary nets yields a concept very similar to the one in [10]. The second result of this 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. This result slightly generalizes a similar observation given in [11].