Left orders in completely \(0\)-simple semigroups
type: Item
instance of: scholarly article
title: Left orders in completely \(0\)-simple semigroups
zbMATH Open document ID: 0855.20048
DOI: 10.1007/BF02574126
published in: Semigroup Forum
publication date: 1996-08-28T00:00:00Z
full work available at URL: https://eudml.org/doc/135475
review text: An element \(a\) of a semigroup \(S\) is said to be square-cancellable if, for all \(x,y\in S^1\), \(a^2x=a^2y\) implies \(ax=ay\) and \(xa^2=ya^2\) implies \(xa=ya\). Let \(S\) be a subsemigroup of \(Q\). Then \(Q\) is a semigroup of left quotients of \(S\) and \(S\) is a left order in \(Q\) if every square-cancellable element of \(S\) lies in a subgroup of \(Q\) and every \(q\in Q\) can be written as \(q=a^\# b\) where \(a\) is square-cancellable, \(b\in S\), and \(a^\#\) is the inverse of \(a\) in a subgroup of \(S\) which contains \(a\) [see \textit{J. Fountain, M. Petrich}, J. Algebra 101, 365-402 (1986; Zbl 0589.20041)]. A semigroup \(S\) with zero is called prime if for any \(a\neq 0\neq b\) in \(S\) there is an \(x\in S\) with \(axb\neq 0\). Let \(\lambda\) and \(\rho\) be binary relations on \(S\) such that \(a\lambda b\) iff \(Sa\cap Sb\neq 0\), and \(a\rho b\) iff \(aS\cap bS\neq 0\). Denote by \(\lambda^t\) and \(\rho^t\) the transitive closure of \(\lambda\) and \(\rho\), respectively. The main result says that a semigroup \(S\) with zero is a left order in a completely 0-simple semigroup iff (i) \(S\) is prime and \(S^1\) categorical at zero, (ii) there exists an \(a\in S\) such that \(aSa\) is a left order in a group with zero, (iii) if \(x(\lambda^t\cap\rho^t)y\) in \(S\) then, for any \(z\in S\), \(zx=zy\neq 0\) implies \(x=y\) and \(xz=yz\neq 0\) implies \(x=y\). This theorem is used to obtain several alternative characterizations of left orders in completely 0-simple semigroups.
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 895855
zbMATH Keywords: semigroups of left quotients
zbMATH Keywords: square-cancellable elements
zbMATH Keywords: completely \(0\)-simple semigroups
author: Ánh Pham Ngoc
author: John Fountain
author: L. Márki
MaRDI profile type: Publication
cites work: COMPLETELY 0-SIMPLE SEMIGROUPS OF LEFT QUOTIENTS OF A SEMIGROUP
cites work: Completely 0-simple semigroups of quotients
cites work: Completely 0-simple semigroups of quotients III
cites work: Left orders in regular ℋ-Semigroups II
OpenAlex ID: W1977855089
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description: scientific article; zbMATH DE number 895855