Preorderings on semigroups and semirings of right quotients
type: Item
instance of: scholarly article
title: Preorderings on semigroups and semirings of right quotients
zbMATH Open document ID: 0989.20046
author: Jakob Cimprič
published in: Semigroup Forum
publication date: 2002-04-22T00:00:00Z
review text: Let \((S,\cdot)\) be a semigroup and \(\Pi_n(S)\) the set of all finite products \(z\) in \(S\) such that every factor \(s\in S\) of \(z\) has a multiplicity which is divisible by \(n\). A subsemigroup \(T\) of \(S\) is called a preordering (of exponent \(n\)) if \(\Pi_n(S)\subseteq T\) for some integer \(n\). Now, let \(Q=Q_r(S,\Sigma)\) be the semigroup of right quotients of \(S\) with respect to a subsemigroup \(\Sigma\) of \(S\). A preordering \(T\) of \(S\) is called \(\Sigma\)-closed if \(\xi a\eta\in T\) implies \(a\in T\) for every \(a\in S\) and \(\xi,\eta\in\Sigma\cap T\). It is shown that every preordering \(T\) of \(S\) that has an extension to a preordering \(T'\) of \(Q\) is necessarily \(\Sigma\)-closed. Moreover, if such a preordering \(T'\) exists then it equals \(T(T\cap\Sigma)^{-1}\). A sufficient condition for \(T'\) to be a preordering of \(Q\) is (1) \(a\xi c\in T\) implies \(ac\in T\) for every \(a,c\in S\) and \(\xi\in T\cap\Sigma\). The necessary and the sufficient condition coincide if \(T\) has the permutation property, i.e., \(s_1\cdots s_k\in T\) iff \(s_{\pi(1)}\cdots s_{\pi(k)}\in T\) for every permutation \(\pi\). Corresponding results are proved for semirings \((S,+,\cdot)\) and semirings \(Q_r(S,\Sigma)\) of right quotients.
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 1445558
zbMATH Keywords: preorderings
zbMATH Keywords: semigroups of right quotients
zbMATH Keywords: semirings of right quotients
zbMATH Keywords: permutation properties
reviewed by: Udo Hebisch
MaRDI profile type: Publication
full work available at URL: s002339910032
OpenAlex ID: W1987265956
DOI: 10.1007/S002339910032
description: scientific article; zbMATH DE number 1445558