Rees matrix covers for regular semigroups
type: Item
instance of: scholarly article
title: Rees matrix covers for regular semigroups
zbMATH Open document ID: 0543.20041
DOI: 10.1016/0021-8693(84)90217-5
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: A regular semigroup S is called locally P (for some property P) if each local submonoid eSe has P (where \(e\in E(S))\). A homomorphism from a regular semigroup is called a local isomorphism if it is one-to-one on all local submonoids. In [Trans. Am. Math. Soc. 277, 727-738 (1983; Zbl 0516.20039)] the author showed that S is locally inverse if and only if S is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup. In this paper he proves analogs of this theorem for locally E-solid semigroups (regular semigroups whose idempotent generated part is completely regular), for locally orthodox semigroups and for locally L-unipotent semigroups. He proves that S is locally E-solid [locally orthodox, locally L-unipotent] if and only if S is a locally isomorphic image of a regular Rees matrix semigroup over an E-solid [orthodox, L-unipotent] semigroup. The proof utilises a construction which begins with a Rees matrix semigroup \({\mathcal M}(S;E,E;Q)\), where \(E=E(S)\) and Q is any \(E\times E\) matrix such that (i) \(q_{f,e}\in fSe\) and (ii) \(q_{e,e}=e\), for all e,\(f\in E\). One then considers the regular subsemigroup \(W=\{(e,x,f):\quad x\in eSf\}.\) The key result is that if \(\psi\) :\(W\to T\) is a local isomorphism onto a regular semigroup T, then S is a locally isomorphic image of a regular Rees matrix semigroup over T. This is coupled with results of \textit{T. E. Hall} [Semigroup Forum 25, 35-48 (1982; Zbl 0497.20049)] to obtain these covering theorems. Locally isomorphic images of regular Rees matrix semigroups over completely regular semigroups are characterized as those regular semigroups which are locally complete regular and whose principal ideals form a semilattice under intersection. Finally it is shown that if a locally E-solid semigroup S is the multiplicative semigroup of a ring, then S is in fact a semilattice of groups.
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 3863480
zbMATH Keywords: local isomorphism
zbMATH Keywords: local submonoids
zbMATH Keywords: locally isomorphic image
zbMATH Keywords: regular Rees matrix semigroup
zbMATH Keywords: inverse semigroup
zbMATH Keywords: locally E-solid semigroups
zbMATH Keywords: locally orthodox semigroups
zbMATH Keywords: locally L-unipotent semigroups
zbMATH Keywords: covering theorems
zbMATH Keywords: multiplicative semigroup of a ring
zbMATH Keywords: semilattice of groups
author: Donald B. McAlister
reviewed by: Peter M. Higgins
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90217-5
OpenAlex ID: W2081181165
cites work: On the compatibility of the natural order on a regular semigroup
cites work: Q5343445
cites work: On regular semigroups whose idempotents form a subsemigroup
cites work: Orthodox semigroups
cites work: Some properties of local subsemigroups inherited by larger subsemigroups
cites work: Demi-groupes réguliers
cites work: Q3929783
cites work: Rees Matrix Covers for Locally Inverse Semigroups
cites work: The natural partial order on a regular semigroup
cites work: Q3892397
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description: scientific article; zbMATH DE number 3863480