Topological Aspects of Traces

Abstract

This paper is a little mathematical study of some models of concurrency. The most elementary one is the concept of an independence structure, which is nothing but a set L with a binary, irreflexive and symmetric relation on it, the independence relation. This leads to the notion of a trace: a string of elements of L, modulo the equivalence generated by swapping adjacent, independent elements of the string.
There are two aspects of finite traces: they form an order, hence a topology; on the other hand they form a monoid, a quotient of the free monoid on L. Unfortunately, these two points of view are hard to bring together, since the monoid structure can never be continuous or even order-preserving. It is therefore not surprising that many papers on trace theory consist of two, disjoint, parts. In this paper I concentrate on the order-theoretic and topological aspects.