Multiplicative closed maps on the set of \(S\)-subsets of an \(S\)-set
type: Item
instance of: scholarly article
title: Multiplicative closed maps on the set of \(S\)-subsets of an \(S\)-set
zbMATH Open document ID: 0849.20048
DOI: 10.1007/BF02574094
author: Zensiro Goseki
published in: Semigroup Forum
publication date: 1996-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/135446
review text: Let \(S\) be a monoid with zero 0, \(M\) be a right \(S\)-act with zero \(0_M\) (\(0_M=0_M\cdot s=m0\), \(\forall m\in M\), \(s\in S\)) and \(G(M)\) is the set of subsets of \(M\). A map \(\rho:G(M)\to G(M)\) is said to be multiplicative closed, if (1) \(\rho(A)=\rho(\rho(A))\), \(A\subseteq\rho(A)\), \(\forall A\in G(M)\), (2) \(A\subseteq B\Rightarrow\rho(A)\subseteq\rho(B)\), (3) \(\rho(A\cap B)=\rho(A)\cap\rho(B)\), \(\forall A,B\in G(M)\). In the paper multiplicative closed maps are studied. It is shown that the construction of such a map is closely connected with a radical system on \(G(M)\). The special case, when the radical system is a family of \(\rho\)-irreducible \(S\)-subsets of \(M\), where \(\rho\) is a multiplicative closed map, is considered.
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M11
Mathematics Subject Classification ID: 06A15
zbMATH DE Number: 858472
zbMATH Keywords: monoids
zbMATH Keywords: right acts
zbMATH Keywords: sets of subsets
zbMATH Keywords: multiplicative closed maps
zbMATH Keywords: radical systems
zbMATH Keywords: irreducible \(S\)-subsets
reviewed by: Vladimir Fleischer
MaRDI profile type: Publication
cites work: Theory of \(x\)-ideals
cites work: On \(\rho\)-irreducible \(S\)-subsets of an \(S\)-set
cites work: Closure Operations on Ideals and Submodules
cites work: Commutative semi-primary $x$-semigroups
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description: scientific article; zbMATH DE number 858472