Clifford semigroups and seminear-rings of endomorphisms.
type: Item
instance of: scholarly article
title: Clifford semigroups and seminear-rings of endomorphisms.
zbMATH Open document ID: 1184.16052
author: M. Samman
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2009-07-06T00:00:00Z
review text: A `seminear-ring' is a triple \((U,+,\cdot)\), where \(U\) is a nonempty set such that both \((U,+)\) and \((U,\cdot)\) are semigroups, and the left distributive law is satisfied, i.e., \(a\cdot(b+c)=a\cdot b+a\cdot c\) for all \(a,b,c\in U\). An element \(d\) of \(U\) is a `distributive' element if \((a+b)\cdot d=a\cdot d+b\cdot d\) for all \(a,b\in U\). A seminear-ring is `distributively generated' (dg, for short) if there exists a (multiplicative) subsemigroup \((T,\cdot)\) of distributive elements in \(U\) that generates \(U\) as a seminear-ring. The main purpose of the paper is to prove the theorem: Let \(\Lambda=\{0,1,\dots,n\}\) be a semilattice with \(n\geq\cdots\geq 1\geq 0\). Let \(\{G_i\}_{i\in\Lambda}\) be a family of isomorphic groups and consider \(S=\bigcup_{i\in\Lambda}G_i\), a Clifford semigroup defined by isomorphisms. Then the dg seminear-ring \(E(S)\) generated by \(\text{End}(S)\), the set of all endomorphisms of \(S\), has a Clifford semigroup structure that forms its additive semigroup \((E(S),+)\). Moreover, \(E(S)=\bigcup_{i\in\Lambda}(E(G_i),+)\), where each \(E(G_i)\) is a dg near-ring generated by a set of endomorphisms of \(G_i\) of a specific type.
reviewed by: Johan Meyer
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 5575489
zbMATH Keywords: seminear-rings
zbMATH Keywords: Clifford semigroups
zbMATH Keywords: endomorphisms
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5575489