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## Behavior and Realization Construction for Petri Nets Based on Free Monoid and Power Set Graphs.

Padberg, Julia;
Ehrig, Hartmut;
Rozenberg, Grzegorz
In:
H. Ehrig, G. Juhás, J. Padberg, G. Rozenberg (Eds.): *LNCS 2128: Unifying Petri Nets - Advances in Petri Nets*, pages 230-pp.
Springer Verlag,
December 2001.

Abstract:
Starting from the algebraic view of Petri nets as monoids (as advocated by
Meseguer and Montanari in [MM90]) we present the marking graphs of place
transition nets as free monoid graphs and the marking graphs of specific
elementary nets as powerset graphs. These are two important special cases
of a general categorical version of Petri nets based on a functor M,
called M-nets. These nets have a compositional marking graph semantics in
terms of F-graphs, a generalization of free monoid and powerset graphs.
Moreover we are able to characterize those F-graphs, called reflexive
Fgraphs, which are realizable by corresponding M-nets. The main result
shows that the behavior and realization constructions are adjoint functors
leading to an equivalence of the categories MNet of M-nets and RFGraph of
reflexive F-graphs. This implies that the behavior construction preserves
colimits so that the marking graph construction using F-graphs is
compositional. In addition to place transition nets and elementary nets we
provide other interesting applications of M-nets and F-graphs. Moreover we
discuss the relation to classical elementary net systems. The behavior and
realization constructions we have introduced are compatible with
corresponding constructions for elementary net systems (with initial
state) and elementary transition systems in the sense of [NRT92].

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