Properties of Pseudo-Primitive Words and their Applications

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-Schützenberger equation u1u2 · · ·u` = v1 · · · vnw1 · · ·wm, where u1, . . . , u` ∈ {u, θ(u)}, v1, . . . , vn ∈ {v, θ(v)}, and w1, . . . , wm ∈ {w, θ(w)} for some words u, v, w, integers `, n,m ≥ 2, and an antimorphic involution θ. We prove that for ` ≥ 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 ` = 3 are examined.