Weakly \(E\)-unitary locally inverse semigroups.
type: Item
instance of: scholarly article
title: Weakly \(E\)-unitary locally inverse semigroups.
zbMATH Open document ID: 1033.20075
DOI: 10.1016/S0021-8693(03)00395-8
author: Bernd Billhardt
author: Mária B. Szendrei
published in: Journal of Algebra
publication date: 2003-09-25T00:00:00Z
review text: If a completely simple semigroup \(T\) acts on a regular semigroup \(K\) by endomorphisms on the left, then \(\{(a,t)\mid{^{t^0}a}=a\}\) endowed with the multiplication \((a,t)(b,u)=(a\cdot{^tb},tu)\) is a regular semigroup which is called a restricted semidirect product of \(K\) by \(T\). If \(T\) has the Rees matrix representation \(T={\mathcal M}(G;I,\Lambda;P)\) and the group \(G\) acts on the regular semigroup \(K\) by automorphisms on the left, then \(K\times T\) endowed with the multiplication \((a,(i,g,\lambda))(b,(j,h,\mu))=(a\cdot{^{^{gp}\lambda j}b},(i,gp_{\lambda j}h,\mu))\) is a semigroup called a Pastijn product of \(K\) by \(T\). It is proved that the notion of a Pastijn product is self-dual. The authors show that if \(K\) is a regular semigroup and \(T\) a completely simple semigroup, then every restricted semidirect product of \(K\) by \(T\) is isomorphic to a Pastijn product of a regular subsemigroup of \(K\) by \(T\), and, each Pastijn product of \(K\) by \(T\) is embeddable into a restricted semidirect product of a power of \(K\) by \(T\). A locally inverse semigroup \(S\) is weakly \(E\)-unitary if and only if the least completely simple congruence on \(S\) is idempotent pure. It is shown that each weakly \(E\)-unitary locally inverse semigroup can be embedded into a restricted semidirect product [Pastijn product] of a normal band \(B\) and a completely simple semigroup. If in particular the maximal subsemilattices of \(S\) are pairwise disjoint, then \(B\) can be chosen to be a semilattice.
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 1985874
zbMATH Keywords: locally inverse semigroups
zbMATH Keywords: weakly \(E\)-unitary semigroups
zbMATH Keywords: restricted semidirect products
zbMATH Keywords: Pastijn products
zbMATH Keywords: completely simple semigroups
zbMATH Keywords: regular semigroups
zbMATH Keywords: semilattices
reviewed by: Francis Pastijn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1985874