Petri nets with generalized algebra: a comparison

Abstract In the last decade we can see substantial effort to develop an abstract and uniform constructions for Petri nets. Most of such abstractions are based on algebraic characterizations of Petri nets. They work mostly over commutative monoids and their various subclasses, namely cancellative commutative monoids or cones of Abelian groups. In the paper we study relationships between Petri nets with generalized underlying algebra. More precisely, we study Petri nets over commutative monoids, cancellative commutative monoids, cones of Abelian groups, and fully ordered cones of Abelian groups. As the main result, we show that classes of reachability graphs of Petri nets over cancellative commutative monoids and cones of Abelian groups coincide (up to isomorphism). In other words, partial order on used cancellative commutative monoid plays no role in expressive power of Petri nets. However, as shows the fact that the class of reachability graphs of nets over fully ordered cones is a proper subclass of the class of reachability graphs of nets over cancellative commutative monoids, the total order on used monoids plays an important role in expressive power of Petri nets.