Permutation properties and the Fibonacci semigroup
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 \textit{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 \textit{A. Restivo} and \textit{C. Reutenauer} [J. Algebra 89, 102-104 (1984; Zbl 0545.20051)] asserting that every periodic semigroup with the permutation property must be locally finite.
scientific article; zbMATH DE number 4083944