A characterization of concurrency-like relations

In the paper algebraic properties of symmetric and irreflexive relations (called sir-relations) are discussed. Such relations are of importance in the Petri nets theory (Best[1], Petri[11,12], Mazurkiewicz[7]). It was proved that for every sir-relation C≤ X×X there is a family of functions (called representations of C) of the form r:X ↣ 2U, where U is a set, such that (a,b)εC ⇔ r(a) ⌢ r(b) = ø. The properties of that family and the relationship between the theory of covers and the theory of sir-relations are discussed. The notion of K-density for sir-relations is introduced, and some of its properties are proved.