The structure of commutative semigroups with the ideal extension property
type: Item
instance of: scholarly article
title: The structure of commutative semigroups with the ideal extension property
zbMATH Open document ID: 0918.20049
author: Karen D. Aucoin
published in: Semigroup Forum
publication date: 1999-08-16T00:00:00Z
review text: A semigroup \(S\) is said to have the ideal extension property (IEP) if for each subsemigroup \(T\) of \(S\) and each ideal \(I\) of \(T\) there is an ideal \(J\) of \(S\) such that \(J\cap T=I\). In this paper, the author characterizes the structure of commutative semigroups having IEP. 1. Case \(S\) is a commutative Archimedean semigroup. The following are equivalent: (i) \(S\) has IEP. (ii) If \(x,y\in S\) and \(xy\neq 0\) then \(xy=x^2=y^2\). (iii) Every subsemigroup of \(S\) is an ideal of \(S\). 2. Assume that \(S\) is a periodic Archimedean semigroup, that is, \(S\) is an ideal extension of an Abelian group \(G\) by an Archimedean semigroup \(N\) with zero. \(S\) has IEP if and only if \(G\) and \(N\) have IEP. 3. Case \(S\) is a commutative semigroup. Let \(S=\bigcup_{e\in E_S} C(e)\) where \(E_S\) is the set of idempotents of \(S\) and \(C(e)\) the Archimedean component containing \(e\), and \(M(C(e))\) is the Abelian group in \(C(e)\). \(S\) has IEP if and only if (i) \(C(e)\) has IEP for all \(e\in E_S\) and (ii) if \(e,f\in E_S\), \(e\neq f\), \(x\in C(e)\), \(y\in C(f)\) then either \(xy\in M(C(ef))\), \(xy=x\) or \(xy=y\). Finally, the author considers the three properties, CEP, ideal semigroups, and \(\Delta\)-semigroups. A semigroup \(S\) is said to have the congruence extension property (CEP) if for each subsemigroup \(T\) of \(S\) and each congruence \(\sigma\) on \(T\), there is a congruence \(\overline\sigma\) on \(S\) such that \(\overline\sigma\cap(T\times T)=\sigma\). A semigroup \(S\) is called a \(\Delta\)-semigroup if its congruences form a chain under inclusion. By an ideal semigroup \(S\) we mean a semigroup \(S\) in which every congruence is a Rees congruence, \((I\times I)\cup S\) for some ideal \(I\) of \(S\). In this section, the author characterizes each of the following in terms of the properties of multiplication: (i) commutative ideal semigroups, (ii) commutative ideal semigroups having IEP, (iii) commutative ideal semigroups havi…
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 1251281
zbMATH Keywords: ideal extension property
zbMATH Keywords: commutative semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: ideal semigroups
zbMATH Keywords: \(\Delta\)-semigroups
zbMATH Keywords: congruence extension property
zbMATH Keywords: Rees congruences
reviewed by: Takayuki Tamura
MaRDI profile type: Publication
full work available at URL: s002339900013
OpenAlex ID: W4245498213
DOI: 10.1007/S002339900013
description: scientific article; zbMATH DE number 1251281