Generalized transformation semigroups whose bi-ideals and quasi-ideals coincide.
type: Item
instance of: scholarly article
title: Generalized transformation semigroups whose bi-ideals and quasi-ideals coincide.
zbMATH Open document ID: 1060.20053
author: Y. Kemprasit
author: Chaiwat Namnak
published in: Southeast Asian Bulletin of Mathematics
publication date: 2004-09-22T00:00:00Z
review text: A subsemigroup \(J\) of a semigroup \(S\) is a `quasi-ideal' of \(S\) if \(SJ\cap JS\subseteq J\) and it is a `bi-ideal' of \(S\) if \(JSJ\subseteq J\). Let \(X\) and \(Y\) be nonempty sets and let \(P(X,Y)\), \(T(X,Y)\), \(I(X,Y)\), \(M(X,Y)\), and \(E(X,Y)\) denote, respectively, the collection of all partial transformations of \(X\) into \(Y\), the collection of all transformations of \(X\) into \(Y\), the collection of all 1-1 partial transformations of \(X\) into \(Y\), the collection of all 1-1 transformations of \(X\) into \(Y\), and the collection of all surjections of \(X\) onto \(Y\). If \(S(X,Y)\) is any of the previous collections and \(\theta\in S(Y,X)\) then \((S(X,Y),\theta)\) denotes the semigroup where the product \(\alpha*\beta\) of any two elements \(\alpha,\beta\in(S(X,Y),\theta)\) is given by \(\alpha*\beta=\alpha\theta\beta\). Denote by \(\mathbf{BQ}\) the class of all semigroups whose collections of all bi-ideals and quasi-ideals coincide. The authors prove that \((S(X,Y),\theta)\in\mathbf{BQ}\) if \(S(X,Y)\) is any one of \(P(X,Y)\), \(T(X,Y)\), or \(I(X,Y)\). They go on to prove that if \(S(X,Y)\) is either \(M(X,Y)\) or \(E(X,Y)\), then \((S(X,Y),\beta)\in\mathbf{BQ}\) if and only if \(|X|=|Y|<\infty\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 2101755
zbMATH Keywords: transformation semigroups
zbMATH Keywords: quasi-ideals
zbMATH Keywords: bi-ideals
zbMATH Keywords: sandwich semigroups
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2101755