Nets, sequential components and concurrency relations

A lattice of unmarked nets is introduced and studied. It is proved that unmarked nets representing the static structure of sequential systems are atoms of that lattice. Marking classes defined by the decomposition of nets into sequential components are introduced and properties (safeness, fireability, etc.) of nets with those marking classes are investigated. The notion of concurrency relation on the system level is defined and discussed. Two different definitions of that relation are given. The first one starts with a given in advance decomposition of a net into sequential components; the second one is constructed on the basis of a given in advance marking class. Both definitions follow from a general concept of the symmetric and irreflexive relation define by a set covering. Petri's postulate about a common element for every global system state and every sequential component is carried up the system level and its strength is discussed. It turns out that if a net is safe and each transition has a possibility to be fired then that postulate implies that the net is decomposable into finite state machines.