Generalised Nyldon words

Abstract

One of the most studied famous classes of words is the class of Lyndon words. Their studies are mainly motivated by the property that they factorise the free monoid as shown in the famous Chen-Fox-Lyndon Theorem. Several generalisations of Lyndon words as anti-Lyndon words, Nyldon words or inverse Lyndon words were made over time. In 2014, Grinberg introduced Nyldon words as a new perspective on the factorisation of the free monoid of words. In particular, for Nyldon words the famous Chen-Fox-Lyndon Theorem is considered w.r.t. a reversed lexicographical order, i.e., a lexicographically non-decreasing factorisation where each factor is smaller or equal than its successor. Further, a generalised lexicographical order is defined by equipping each position i in a word in ${\Sigma }^{⁎}$ with a total order ${◃}_i$ on Σ. For combining the concept of a generalised order as for generalised Lyndon words and the class of Nyldon words, we investigate a non-decreasing factorisation of the free monoid w.r.t. this generalised ordering and introduce generalised Nyldon words. We show that those words even force a unique non-decreasing factorisation, form a right Hall set, and coincide with the anti-Lyndon words.