On the product of square-free words (Q1060021)

Square-free words in a free monoid \(A^*\) are of great interest in different fields such as algebra (Burnside problem for semigroups and groups), ergodic theory, games, etc. The reader interested in this topic should consult the book by \textit{M. Lothaire} [Combinatorics on words (1983; Zbl 0514.20045)]. In this paper we consider the problem of characterizing when the product of two or more square-free words is itself square-free. In the case of three words we show that: If u, v and w the three words of \(A^+\) such that uv, vw are square-free, then the product uvw is square-free if and only if there does not exist a nontrivial prefix p of w and a nontrivial suffix s of u such that ps Cof v, where Cof is a relation in \(A^*\) defined as u Cof v if and only if there exists a word x such that \(u=xvx\) or \(v=xux\). Some further results are derived in counting the squares which can be produced in the product of square-free words. An interesting corollary is that the number of squares in the product of two words u and v is upperlimited by the minimum value of the lengths of u and v. In the last section we consider the product of more than three words. In particular the following result is shown: Let \(\phi\) be a morphism \(\phi:B^+\to A^+\) (A and B finite) such that \(\phi\) (B) satisfies the following conditions: i. no word is a factor of another word. ii. there are no overlaps between two different words. iii. no word w is the product of a nontrivial prefix and a nontrivial suffix of two other words different from w. One has that if w is a square-free word over B and all the factors of length 3 of \(\phi\) (w) are square-free then \(\phi\) (w) is square-free. Some corollaries are derived.