On a combinatorial property of Sturmian words

Recall that a semigroup has the property P-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* by the set of the non-factors of s has P-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*/(A* - St) has P-8*.