On a combinatorial property of Sturmian words. (English) Zbl 0872.68144

Summary: Recall that a semigroup has the property \(P^{\ast}_{n}\) if for any sequence of \(n\) of its elements, two differently permuted products of these \(n\) elements are equal. Let \(s\) be an infinite Sturmian word (on a 2-letter alphabet \(A\)). We prove that the Rees quotient of \(A^{\ast}\) by the set of the non-factors of \(s\) has \(P^{\ast}_{4}\) and that this result is the best possible. We prove also that if \(St\) is the set of all finite Sturmian words, then the Rees quotient \(A^{\ast}/(A^{\ast}-St)\) has \(P^{\ast}_{8}.\)


68R15 Combinatorics on words
20M05 Free semigroups, generators and relations, word problems
Full Text: DOI


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