## 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 $$P^{\ast}_{8}.$$

### MSC:

 68R15 Combinatorics on words 20M05 Free semigroups, generators and relations, word problems

### Keywords:

Sturmian words; infinite words; finite words; Rees quotient
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### References:

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