The Generalized Rank of Trace Languages

Abstract

Given a partially commutative alphabet and a set of words [Formula: see text], the rank of [Formula: see text] expresses the amount of shuffling required to produce a word belonging to [Formula: see text] from two words whose concatenation belongs to the closure of [Formula: see text] with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.