On congruences and partial orders

Abstract

Mazurkiewicz trace theory is not powerful enough to describe concurrency paradigms as, for instance, the “Producer / Consumer”. We propose in this paper a generalization of Mazurkiewicz trace monoids which allows to model such problems. We consider quotients of the free monoids by congruences which preserve the commutative images of words. An equivalence class in the quotient monoid consists of all the sequential observations of a distributed computation. In order to characterize congruences which do model concurrency, we study the relationship of this approach and the classical representation of distributed computations with partial orders. We show that the only congruences for which the classes can be represented by partial orders and the concatenation transfers modularly to partial orders are congruences generated by commutations, that is trace congruences. We prove necessary conditions and sufficient conditions on congruences so that their classes can be represented by partial orders. In particular, an important sufficient condition covers both trace congruences and the “Producer / Consumer” congruence.