Decomposition of regular subsemigroup lattices
type: Item
instance of: scholarly article
title: Decomposition of regular subsemigroup lattices
zbMATH Open document ID: 0806.20054
DOI: 10.1007/BF02573478
published in: Semigroup Forum
publication date: 1994-12-15T00:00:00Z
full work available at URL: https://eudml.org/doc/135341
review text: In an earlier paper [Algebra Univers. 30, 463-468 (1993; Zbl 0797.20046)], the author showed that for any \(E\)-solid or locally inverse regular semigroup \(S\), the set of regular subsemigroups forms a complete lattice \({\mathcal L}{\mathcal R}(S)\). The author considers the decomposition of this lattice induced by the relation \(\theta\) that identifies two regular subsemigroups if and only if they have the same idempotents. Those sets of idempotents that so arise are termed `closed' and have been characterized in various ways. They naturally form a complete lattice \({\mathcal L}{\mathcal C}(E)\). That \(\theta\) is a complete congruence follows from the fact that the map which assigns to each regular subsemigroup its set of idempotents is a complete morphism from \({\mathcal L}{\mathcal R}(S)\) onto \({\mathcal L}{\mathcal C}(E)\). Each \(\theta\)-class is isomorphic to the lattice of full regular subsemigroups of its maximum element and it is thus of interest to note that the latter is a complete sublattice of the lattice of all subsemigroups. The author points out a parallel with the `trace' congruence on the lattice of congruences of a regular semigroup.
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 06B15
zbMATH DE Number: 611122
zbMATH Keywords: locally inverse regular semigroup
zbMATH Keywords: regular subsemigroups
zbMATH Keywords: complete lattice
zbMATH Keywords: idempotents
zbMATH Keywords: complete congruence
zbMATH Keywords: complete morphism
zbMATH Keywords: lattice of full regular subsemigroups
zbMATH Keywords: lattice of congruences
author: Katherine G. Johnston
reviewed by: Peter R. Jones
MaRDI profile type: Publication
cites work: Q3934450
cites work: Q3956998
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cites work: THE EXISTENCE OF E-FREE OBJECTS IN E-VARIETIES OF REGULAR SEMIGROUPS
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description: scientific article; zbMATH DE number 611122