Lyndon heaps: An analogue of Lyndon words in free partially commutative monoids

Abstract

Given a free partially commutative monoid, we interpret equivalence classes of words (also called ‘traces’) in terms of Viennot's heaps of pieces. We study the notion of conjugacy applied to this context. Each heap has a unique representative in the set of ‘standard words’ which are lexicographically ordered. Then, by definition, Lyndon heaps are the minimal elements of the conjugacy classes of heaps. Most combinatorial properties of Lyndon words have a counterpart in the theory of Lyndon heaps. For instance, a ‘standard factorization’ exists, the set of Lyndon heaps provides a complete factorization of the monoid, etc.