# Finite test sets for morphisms which are square-free on some of Thue's square-free ternary words

## TL;DR

This paper characterizes when a morphism preserves square-freeness on infinite words over a ternary alphabet, showing it suffices to check factors up to length 8 for certain sets of factors.

## Contribution

It introduces finite test sets for morphisms to determine square-freeness on specific classes of infinite words, simplifying verification processes.

## Key findings

- Square-freeness of morphisms can be tested on finite factors of length 8 or less.
- The result applies to words avoiding certain factor sets $S$.
- Provides a practical criterion for morphism analysis in combinatorics on words.

## Abstract

Let $S$ be one of $\{aba,bcb\}$ and $\{aba, aca\}$, and let $w$ be an infinite square-free word over $\Sigma=\{a,b,c\}$ with no factor in $S$. Suppose that $f:\Sigma\rightarrow T^*$ is a non-erasing morphism. Word $f(w)$ is square-free if and only if $f$ is square-free on factors of $w$ of length 8 or less.