Permutation properties and the Fibonacci semigroup
type: Item
instance of: scholarly article
title: Permutation properties and the Fibonacci semigroup
zbMATH Open document ID: 0663.20063
DOI: 10.1007/BF02573241
author: Antonio Restivo
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134964
review text: 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.
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 4083944
zbMATH Keywords: weak permutation property
zbMATH Keywords: Rees factor
zbMATH Keywords: free semigroup
zbMATH Keywords: Fibonacci word
zbMATH Keywords: periodic semigroup
reviewed by: Jan Okniński
MaRDI profile type: Publication
cites work: Q3310057
cites work: A permutational property of groups
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OpenAlex ID: W2334575004
description: scientific article; zbMATH DE number 4083944