Comments on the permutation property for semigroups
type: Item
instance of: scholarly article
title: Comments on the permutation property for semigroups
zbMATH Open document ID: 0665.20028
DOI: 10.1007/BF02573290
author: J. Justin
author: Giuseppe Pirillo
published in: Semigroup Forum
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134980
review text: Let S be a semigroup and n an integer (n\(\geq 2)\). We say that S has the property \(P_ n\) if for any \(x_ 1,x_ 2,...,x_ n\) in S there exists a permutation \(\sigma\) of \(\{\) 1,2,...,n\(\}\), \(\sigma\neq id\), such that \(x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}\). We say that S has the property P (the ``permutation property'') if S satisfies \(P_ n\) for some n. Similarly, we say that S has the property \(P^*_ n\) if for any \(x_ 1,x_ 2,...,x_ n\) in S, there exist two distinct permutations \(\sigma\), \(\tau\) such that \[ x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}=x_{\tau (1)}x_{\tau (2)}...x_{\tau (n)}, \] and we say that S has the property \(P^*\) if it satisfies \(P^*_ n\) for some n. In this paper we prove the following propositions: Proposition 1. The bicyclic semigroup B has the property \(P^*_ 5\) but it does not have the property P. Also, any direct product of a finite number of copies of the bicyclic semigroup has the property \(P^*\). - Proposition 2. Let S and D be semigroups, with D finite, and let \(\phi\) be a morphism from S into D. If for every idempotent e of \(\phi\) (S), \(\phi^{-1}(e)\) has the property P (resp. \(P^*)\), then S has the property P (resp. \(P^*)\).
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 4087924
zbMATH Keywords: property \(P_ n\)
zbMATH Keywords: permutation property
zbMATH Keywords: bicyclic semigroup
zbMATH Keywords: property \(P^*\)
zbMATH Keywords: idempotent
MaRDI profile type: Publication
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OpenAlex ID: W2062706610
description: scientific article; zbMATH DE number 4087924