Cycles of partial orders
The approach of characterising certain properties of concurrent structures by traces, i.e. partial orders of event occurrences has already been used by several authors in particular by S. Patil, A.W. Holt, P.E. Lauer. The approach also has a close connection to causal-nets introduced by C.A. Petri.
As to the formal linguistic approaches to Petri nets, the aspects of (string-) languages and commutative grammars have already been carefully investigated, e.g. by M. Hack and S. Crespi Reghizzi respectively. The interconnections between Petri nets and traces have been studied by F. Furtek, and later in particular by A. Mazurkiewicz who proved that the behaviour of 1-safe nets can always be characterised by regular languages of traces.
This paper is inspired by an interest in whether the converse statement is true, that is, whether regular trace languages can always be represented by 1-safe nets. An affirmative answere is given for a simple special class which, however, has a number of practical applications.
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