Inverse semigroups with isomorphic bundles of \(i\)-correspondences
type: Item
instance of: scholarly article
title: Inverse semigroups with isomorphic bundles of \(i\)-correspondences
zbMATH Open document ID: 0860.20049
author: S. M. Goberstein
published in: Journal of Algebra
publication date: 1997-04-13T00:00:00Z
review text: Let \(S\) be a semigroup and \(C(S)\) the set of all correspondences of \(S\) (i.e. subsemigroups of \(S\times S\) including the empty subset). The bundle (it is denoted \(C(S)\), too) of all correspondences of \(S\), i.e. the algebraic system \((C(S),\circ,{}^{-1},\subset)\) is considered. If \(S\) is an inverse semigroup, then the set of all \(i\)-correspondences (inverse subsemigroups of \(S\times S\)) forms a subbundle \(C_i(S)\) of the bundle \(C(S)\). If \(\Phi:C(S)\to C(T)\) (resp. \(C_i(S)\to C_i(T))\) is an isomorphism of bundles, it is called a \(C\) (\(C_i\))-isomorphism \(S\) onto \(T\) and these semigroups are called \(C\)-(\(C_i\))-isomorphic. Some consequences of the fact that the inverse semigroups \(S\) and \(T\) are \(C_i\)-isomorphic are given. These results lead to the main theorem, which states that every \(C_i\)-isomorphism \(\Phi\) of a fundamental inverse semigroup \(S\) onto an inverse semigroup \(T\) is induced by a unique isomorphism of \(S\) onto \(T\). Moreover, that isomorphism and its composition with the natural involution on \(T\) are the only bijections of \(S\) onto \(T\) inducing \(\Phi\). Comparing \(C\)- and \(C_i\)-isomorphisms, the author obtains that if \(\Psi\) is a \(C\)-isomorphism of an inverse semigroup \(S\) with \(\text{ind }S>1\) onto a semigroup \(T\) then \(T\) is an inverse semigroup and \(\Psi|C_i(S)\) is a \(C_i\)-isomorphism of \(S\) onto \(T\). As a corollary, the main result of the author's paper [J. Algebra 125, No. 2, 474-488 (1989; Zbl 0677.20050)] is obtained.
reviewed by: Grigori I. Zhitomirski
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 934247
zbMATH Keywords: stable binary relations
zbMATH Keywords: bundles of correspondences
zbMATH Keywords: inverse subsemigroups
zbMATH Keywords: \(i\)-correspondences
zbMATH Keywords: fundamental inverse semigroups
MaRDI profile type: Publication
full work available at URL: jabr.1996.0254
OpenAlex ID: W2071492017
DOI: 10.1006/JABR.1996.0254
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description: scientific article; zbMATH DE number 934247