Permutation properties and the Fibonacci semigroup.(English)Zbl 0663.20063

A semigroup S is said to have the weak permutation property is there exists $$n\geq 2$$ such that for any elements $$x_ 1,...x_ n\in S$$ there exist two permutations $$\sigma$$, $$\tau$$ of the set $$\{$$ 1,...,n$$\}$$ such that $$\sigma\neq \tau$$ and $$x_{\sigma (1)}...x_{\sigma (n)}=x_{\tau (1)}...x_{\tau (n)}$$. Groups of this type were described by R. D. Blyth [J. Algebra 116, 506-521 (1988; Zbl 0647.20033)], who also showed that for groups the above property is equivalent to the permutation property: there exists $$n\geq 2$$ such that for any $$x_ 1,...,x_ n\in S$$ there is a permutation $$\sigma\neq 1$$ such that $$x_{\sigma (1)}...x_{\sigma (n)}=x_ 1...x_ n$$. An example of a semigroup S with the weak permutation property which is finitely generated, periodic but infinite is given. (It is the Rees factor of the free semigroup $$<a,b>$$ by the ideal consisting of all non-factors of the infinite Fibonacci word in a,b.) This is in contrast to the result of A. Restivo and C. Reutenauer [J. Algebra 89, 102-104 (1984; Zbl 0545.20051)] asserting that every periodic semigroup with the permutation property must be locally finite.
Reviewer: J.Okniński

MSC:

 20M05 Free semigroups, generators and relations, word problems 20M10 General structure theory for semigroups

Citations:

Zbl 0647.20033; Zbl 0545.20051
Full Text:

References:

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