On a Test of Square-Free Morphisms

Hanoi Institute of Mathematics 18 Hoang Quoc Viet Road, Cau Giay, 10307 Hanoi, Vietnam

^* Corresponding author: nhlam@math.ac.vn

Received: 22 July 2022
Accepted: 7 July 2023

Abstract

A square-free word is one which does not contain two consecutive occurrences of the same factor; a square-free morphism is one which produces a square-free word whenever applied to a square-free word. For testing square-freeness of a morphism, the previous researches (Berstel, Ehrenfeucht-Rozenberg, Crochemore, Hsiao et al., …) dealt with the compactness claim: they proved that there is a bound (sometimes tight, depending on the morphism) such that a morphism is square-free if it is so on the words on the source alphabet of length up to this bound. In particular, when the morphism is ternary (the source alphabet of three letters) this bound is universally 5 and 5 is sharp on the target alphabet of 5 letters. In this paper we undertake a different approach: we do not search for any compactness bound or verify square-freeness excerpt for a few prerequisites; instead we define the relator on the source alphabet, the existence of which is relatively easy to verify by matching words for a common factor. As applications, we easily deduce all the previous bounds and we manifest the simplicity of performance by giving a short proof of a Lallement’s result. More essentially, using it we show the optimum bound 5 for the ternary morphism on the alphabet of 4 letters and, by a more elaborate construction, on the alphabet of 3 letters. That gives the current problem the finishing touches.