The Structure of Facts in Occurrence Nets

A fact in a C/E-system Σ: = (B, E; F, C) is a conceivable “dead” pure transition t which could be added to the system Σ without changing its behaviour. The transition t is completely described by the corresponding pair (·t, t·); and therefore facts are essentially pairs (I, J) with I, J ⊆ B. Facts with minimal (I, J) are called basic facts; and it is immediately clear that a basic fact (I₀, J₀) implies all facts (I, J) with I₀ ⊆ I ^ J₀ ⊆ J.Processes of Σ can be conceived as mappings from appropriate occurrence nets into Σ. Then every fact of Σ can be derived from the facts in the corresponding occurrence nets. Therefore, knowledge of the general structural properties of facts in occurrence nets may be helpful to understand how facts can be realized by C/E-systems. The aim of this paper is to show that facts in occurrence nets have a rather simple structure provided some finiteness conditions hold true. Obviously, to this end it is suffucient to investigate the basic facts of occurrence nets; it will turn out that for all basic facts (I₀, J₀) of an occurrence net we have | I₀ | ≤ 2; and that the set of all of these basic facts can be partitioned into six classes each of which contains facts of a special structure.