Homology groups of asynchronous systems, Petri nets, and trace languages

The paper is devoted to the homology groups of mathematical models for concurrent systems. It is proved that the homology groups of a set with a trace monoid action is isomorphic to the homology groups of a corresponding semi-cubical set. Homology groups of Petri nets and Mazurkiewicz trace languages are introduced. It is shown that in dimensions $n\geqslant2$, the homology groups of Petri nets and Mazurkiewicz languages can be arbitrary, up to direct summands which are equal to the homology groups of generalized tori. Examples of the computing the homology groups of state spaces and Petri nets are considered. The integral homology groups of some partially ordered sets of traces are investigated.