Interassociates of monogenic semigroups.
type: Item
instance of: scholarly article
title: Interassociates of monogenic semigroups.
zbMATH Open document ID: 1059.20056
author: Karen A. Linton
author: Amy Wildsmith Nelson
author: Matthew I. Gould
published in: Semigroup Forum
publication date: 2004-07-01T00:00:00Z
review text: A semigroup \((S,*)\) is said to be an `interassociate' of the semigroup \((S,\cdot)\) if \(x\cdot(y*z)=(x\cdot y)*z\) and \(x*(y\cdot z)=(x*y)\cdot z\) for all \(x,y,z\) in \(S\). Two problems that arise very naturally in the study of interassociates is their determination and whether they are isomorphic. In this paper these problems are considered in the case of monogenic semigroups. Let \(S\) be generated by \(a\) and define an operation \(*_k\) on \(S\) by \(a^x*_k a^y=a^{x+k+y-2}\), then the interassociates of \(S\) are of the form \((S,*_k)\) for \(k\geq 1\). These are all interassociates one of another, and there are \(|S|\) of them. If \(S\) has infinite order then no two of these are isomorphic. If \(S\) has index \(i\) and period \(p\), then the `nexus', \(n\), of \(S\) is the least residue of \(i\) modulo \(p\), and conditions for isomorphism are given in terms of \(p\), \(i\) and \(n\).
reviewed by: Sheila Oates-Williams
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 2077993
zbMATH Keywords: monogenic semigroups
zbMATH Keywords: cyclic semigroups
zbMATH Keywords: interassociates
MaRDI profile type: Publication
full work available at URL: s00233-002-0028-y
OpenAlex ID: W2010312897
DOI: 10.1007/S00233-002-0028-Y
description: scientific article; zbMATH DE number 2077993