Divisible and exponent actions of semigroups
type: Item
instance of: scholarly article
title: Divisible and exponent actions of semigroups
zbMATH Open document ID: 0774.20045
DOI: 10.1007/BF02573560
published in: Semigroup Forum
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/135225
review text: Suppose that \(S\) is a monoid (i.e., semigroup with identity) and \(X\) a set. An \textit{action of \(S\) on} \(X\) is a function \((s,x)\mapsto sx\colon S\times X\to X\) such that \(1x=x\) and \((st)x=s(tx)\) for all \(s,t\in S\), \(x\in X\). For each monoid and each set we always have a \textit{trivial action} given by \(sx=x\). If \(X^ X\) denotes the monoid of all self-maps of \(X\) under composition, then there is a bijection between the set of all actions of \(S\) on \(X\) and all monoid morphisms (\textit{representations}) \(S\to X^ X\) sending \((s,x)\mapsto sx\) to \(s\mapsto (x\mapsto sx)\). (The reviewer writes actions on the left, the author on the right.) If \(X\) is itself a monoid then for an action of \(S\) on \(X\) every map \(x\mapsto sx\) is a monoid endomorphism of \(X\) iff the associated representation maps \(S\) into \(\Hom(X,X)\subseteq X^ X\). The author calls these actions \textit{exponent actions}; the reviewer's preference would be \textit{homomorphic actions}; if \(X\) is a vector space and \(\Hom(X,X)\) denotes the monoid of all linear self-maps, then one would speak of \textit{linear actions} and \textit{linear representations} anyhow. We set \(\Hom_{\text{Sur}}(X,X)=\text{Sur}(X)\cap\Hom(X,X)\). The author calls an action \textit{divisible}, respectively, \textit{uniquely divisible} if for all \(s\in S\) and \(a\in X\) the equation \(sx=a\) has a solution, respectively, a unique solution. We note that an action is divisible in this sense if the associated representation maps \(S\) into the subsemigroup \(\text{Sur}(X)\) of surjective self-maps of \(X\). [This subsemigroup is Green's \(\mathcal R\)-class \({\mathcal R}(\text{id}_ X)\) of the identity (see e.g.\ \textit{A. H. Clifford} and \textit{G. B. Preston}, The algebraic theory of semigroups, Vol. I (1961; Zbl 0111.03403), pp. 52, 53, Theorem 2.9(iv)). We also note that for a vector space \(X\) the monoid is again the \(\mathcal R\)-class of \(\text{id}_ X\) in \(\Hom(X,X)\) (see…
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M30
zbMATH DE Number: 167671
zbMATH Keywords: monoid
zbMATH Keywords: semigroup with identity
zbMATH Keywords: monoid of self-maps
zbMATH Keywords: actions
zbMATH Keywords: monoid morphisms
zbMATH Keywords: monoid endomorphism
zbMATH Keywords: exponent actions
zbMATH Keywords: homomorphic actions
zbMATH Keywords: linear actions
zbMATH Keywords: linear representations
zbMATH Keywords: surjective self-maps
zbMATH Keywords: idempotents
zbMATH Keywords: completely simple
zbMATH Keywords: divisible action
author: John A. Hildebrant
reviewed by: Karl H. Hofmann
MaRDI profile type: Publication
cites work: Q3661823
cites work: Q3848243
cites work: Semigroups acting on continua
cites work: Clan acts and codimension
cites work: Swelling actions of \(\Gamma\)-compact semigroups
cites work: Q5545552
OpenAlex ID: W2051787948
description: scientific article; zbMATH DE number 167671