Properties of Pseudo-Primitive Words and their Applications
Abstract
A pseudo-primitive word with respect to an antimorphic involution θ is a word which cannot be written as a catenation of occurrences of a strictly shorter word t and θ(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 u1 u2 ... ul = v1 ... vn w1 ... wm, where u1, ..., ul ∈ u, θ(u), v1, ..., vn ∈ v, θ(v), and w1, ..., wm ∈ w, (w) for some words u, v, w, integers l, n, m 2, and an antimorphic involution θ. We prove that for l 4, n,m 3, this equation implies that u, v, w can be expressed in terms of a common word t and its image θ(t). Moreover, several cases of this equation where l = 3 are examined.