Trace monoid

Abstract: It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized finite automata.

Abstract: The {\em height} of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier-Foata decomposition. Let $h(t)$ and $|t|$ stand respectively for the height and the length of a trace $t$. Roughly speaking, $|t|$ is the `sequential' execution time and $h(t)$ is the `parallel' execution time. We prove that the bivariate commutative series $\sum_t x^{h(t)}y^{|t|}$ is rational, and we give a finite representation of it. We use the rationality to obtain precise information on the asymptotics of the number of traces of a given height or length. Then, we study the average height of a trace for various probability distributions on traces. For the uniform probability distribution on traces of the same length (resp. of the same height), the asymptotic average height (resp. length) exists and is an algebraic number. To illustrate our results and methods, we consider a couple of examples: the free commutative monoid and the trace monoid whose independence graph is the ladder graph.

Abstract: Trace monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and unicity of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including an immanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function from an abelianization procedure.