Notions of Duality and Symmetry in Petri Nets.

After having recalled some basic definitions and properties of the Petri nets, we synthesize some algebraic results, based upon a theorem of compatibility of linear inequalities systems (near of the Farkas lemma). There are found, as corollaries of equivalence theorems between algebraic propositions and Petri nets properties, two theorems synthesizing new results and scattered ones in the literature. Moreover, there are rediscovered a necessary condition for a Petri net to be live only based upon the structure of the graph, and also a symmetrical necessary condition for a Petri net to be bounded and live.