On Three Alternative Characterizations of Combined Traces

Combined traces (i.e., comtraces), a generalization of Mazurkiewicz traces, were introduced by Janicki and Koutny in 1995 as congruence classes of step sequences, where the congruence relation is induced by two relations simultaneity and serializability on events. They also showed that comtraces corresponds to some class of labeled stratified order structures, but the question "what class of labeled stratified orders represent comtraces?" was left open. In this work, we proposed a class of labeled stratified order structures that captures exactly the comtrace notion. Our main technical contributions are representation theorems showing that comtrace quotient monoid, \emph{combined dependency graph} (Kleijn and Koutny 2008) and our labeled stratified order structure characterization are three different and yet equivalent ways to represent comtraces. This paper is a revised and expanded version of D. T. M. Le (Proc. of PETRI NETS 2010, LNCS 6128, pp. 104-124).