Properties of Pseudo-Primitive Words and their Applications

TL;DR

This paper explores the properties of pseudo-primitive words related to antimorphic involutions and applies these findings to improve solutions for the extended Lyndon-Schützenberger equation, revealing structural conditions for the involved words.

Contribution

It introduces new properties of pseudo-primitive words and demonstrates their application in solving a generalized Lyndon-Schützenberger equation with specific word length conditions.

Findings

Pseudo-primitive words are linked to primitive words, palindromes, and commutativity.

For l ≥ 4, n, m ≥ 3, the equation implies u, v, w are related through a common word t and its image.

Several cases with l = 3 are also analyzed.

Abstract

A pseudo-primitive word with respect to an antimorphic involution \theta is a word which cannot be written as a catenation of occurrences of a strictly shorter word t and \theta(t). Properties of pseudo-primitive words are investigated in this paper. These properties link pseudo-primitive words with essential notions in combinatorics on words such as primitive words, (pseudo)-palindromes, and (pseudo)-commutativity. Their applications include an improved solution to the extended Lyndon-Sch\"utzenberger equation u_1 u_2 ... u_l = v_1 ... v_n w_1 ... w_m, where u_1, ..., u_l \in {u, \theta(u)}, v_1, ..., v_n \in {v, \theta(v)}, and w_1, ..., w_m \in {w, \theata(w)} for some words u, v, w, integers l, n, m \ge 2, and an antimorphic involution \theta. We prove that for l \ge 4, n,m \ge 3, this equation implies that u, v, w can be expressed in terms of a common word t and its image \theta(t). Moreover, several cases of this equation where l = 3 are examined.