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## Petri Algebras.

Badouel, Eric;
Chenou, Jules;
Guillou, Goulven
In:
*Lecture Notes in Computer Science, Vol. 3580*, pages 742-754.
2005.

Abstract:
The firing rule of Petri nets relies on a residuation operation for the
commutative monoid of natural numbers. We identify a class of residuated
commutative monoids, called Petri algebras, for which one can mimic the
token game of Petri nets to define the behaviour of generalized Petri net
whose flow relation and place contents are valued in such algebraic
structures. We show that Petri algebras coincide with the positive cones
of lattice-ordered commutative groups and constitute the subvariety of the
(duals of) residuated lattices generated by the commutative monoid of
natural numbers. We introduce a class of nets, termed lexicographic Petri
nets, that are associated with the positive cones of the lexicographic
powers of the additive group of real numbers. This class of nets is
universal in the sense that any net associated with some Petri algebras
can be simulated by a lexicographic Petri net. All the classical decidable
properties of Petri nets however are undecidable on the class of
lexicographic Petri nets. Finally we turn our attention to bounded nets
associated with Petri algebras and show that their dynamics can be
reformulated in term of MV-algebras.

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