The poset of infinitary traces

Partially commutative monoids, also called trace monoids, are among the most-studied formalisms to describe the behaviour of distributed systems. In order to model systems which never stop, we have to consider an extension of traces, namely infinite traces. Finite-trace monoids are strongly related to partial-order sets (PoSets), domains and event structures, which are other models to describe the behaviour of distributed systems. The aim of this paper is to establish similar connexions between infinite-trace monoids, PoSets and event structures. We prove that the set of finite and infinite traces with the prefix order is a Scott domain and a coherently complete prime algebraic PoSet. Moreover, we establish a representation theorem between the class of finite- and infinite-trace PoSets and a subclass of labelled prime event structures.