Extending ideals in regular topological semigroups
type: Item
instance of: scholarly article
title: Extending ideals in regular topological semigroups
zbMATH Open document ID: 0934.22012
author: Kirk Benningfield
author: Karen D. Aucoin
published in: Semigroup Forum
publication date: 1999-11-16T00:00:00Z
review text: The authors define three concepts of ideal extension properties of a topological semigroup \(S:S\) has the ideal extension property (IEP) if, for each closed subsemigroup \(T\) of \(S\) and each closed ideal \(I\) of \(T\), there is a closed ideal \(J\) of \(S\) such that \(J\cap T=I\). \(S\) has the weak ideal extension property (WIEP) if there is an ideal \(J\) (not necessarily closed) of \(S\) extending each closed ideal of each closed subsemigroup. \(S\) has the feeble ideal extension property (FIEP) if, for each closed subsemigroup \(T\) of \(S\) and each closed ideal \(I\) of \(T\), there is a congruence \(\sigma\) (not necessarily closed) extending the Rees congruence \(\Delta_T\cup(I\times I)\) on \(T\) (i.e., \(\sigma\cap(T\times T)=\Delta_T\cup(I\times I))\). Let \(S\) be a topological semigroup and let \(a\in S\). Let \(\theta(a)=\{a^n:n\in Z^+\}\). Then \(\Gamma(a)=\overline{\theta(a)}\) is called the monothetic subsemigroup of \(S\) generated by \(a\). If \(S=\Gamma(a)\) for some \(a\in S\) then \(S\) is called a monothetic semigroup. A topological semigroup \(S\) is called \(\Gamma\)-compact if each of its monothetic subsemigroups is compact. The main theorem of this paper is the following: If \(S\) is a \(\Gamma\)-compact regular semigroup then the following four conditions are equivalent: (i) \(S\) has monothetic index 1. (ii) \(S\) is completely regular. (iii) \(S\) satisfies WIEP. (iv) \(S\) satisfies FIEP.
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 1330706
zbMATH Keywords: topological semigroup
zbMATH Keywords: ideal extension property
zbMATH Keywords: weak ideal extension property
zbMATH Keywords: feeble ideal extension property
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 1330706