Homology groups of cubical sets

The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of Abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in the diagram of Abelian groups are isomorphic to the homology groups of normalized complex of the cubical Abelian group corresponding to this diagram. The main result shows that the homology groups of a cubical set with coefficients in a contravariant system of Abelian groups are isomorphic to the values of left derived functors of the colimit functor on this contravariant system. This is used to obtain the isomorphism criterion for homology groups of cubical sets with coefficients in contravariant systems, and also to construct spectral sequences for the covering of a cubical set and for a morphism between cubical sets.Comment: 24 page

Homology and Bisimulation of Asynchronous Transition Systems and Petri Nets

Homology groups of labelled asynchronous transition systems and Petri nets are introduced. Examples of computing the homology groups are given. It is proved that if labelled asynchronous transition systems are bisimulation equivalent, then they have isomorphic homology groups. A method of constructing a Petri net with given homology groups is found.Comment: 21 page

Generalized Asynchronous Systems

The paper is devoted to a mathematical model of concurrency the special case of which is asynchronous system. Distributed asynchronous automata are introduced here. It is proved that the Petri nets and transition systems with independence can be considered like the distributed asynchronous automata. Time distributed asynchronous automata are defined in standard way by the map which assigns time intervals to events. It is proved that the time distributed asynchronous automata are generalized the time Petri nets and asynchronous systems.Comment: 8 page