A concept of variety for regular biordered sets
type: Item
instance of: scholarly article
title: A concept of variety for regular biordered sets
zbMATH Open document ID: 0814.20041
DOI: 10.1007/BF02573495
author: Raymond Broeksteeg
published in: Semigroup Forum
publication date: 1995-04-25T00:00:00Z
full work available at URL: https://eudml.org/doc/135358
review text: A regular biordered set \((E,\cdot,\omega^ l, \omega^ r)\) is a set \(E\) together with two quasiorders \(\omega^ l\) and \(\omega^ r\) and a partial binary operation \(\cdot\), satisfying a finite (but somehow complicated) system of axioms. Regular biordered sets were introduced by \textit{K. S. S. Nambooripad} [Mem. Am. Math. Soc. 224 (1979; Zbl 0457.20051)] and are an abstract characterization of the partial groupoid \(E(S)\) of idempotents of a regular semigroup \(S\). Nambooripad also introduced suitable concepts of morphic image and substructure of regular biordered sets (called ``regular bimorphic image'' and ``relatively regular biordered subset''). In the paper under review, a concept of variety for regular biordered sets is introduced by putting a class \(\mathcal E\) of regular biordered sets to be a ``bivariety'' if \(\mathcal E\) is closed under taking direct products (which are defined in an obvious way), relatively regular biordered subsets, and regular bimorphic images. In the main theorem it is shown that the mapping \({\mathcal V} \mapsto \{E(S) \mid S \in {\mathcal V}\}\) is a complete surjective morphism from the lattice of all existence varieties of regular semigroups onto the lattice of all bivarieties of regular biordered sets. Another interesting result of the paper states that a bivariety \(\mathcal E\) of regular biordered sets contains all free objects if and only if \(\mathcal E\) consists entirely of local semilattices or solid biordered sets. Precisely in these cases, regular biordered sets can be equipped with a total binary operation (extending the partial biordered set product) so that the bivarieties are precisely the varieties of the resulting algebras.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08A55
zbMATH DE Number: 682638
zbMATH Keywords: partial groupoid of idempotents
zbMATH Keywords: lattice of existence varieties of regular semigroups
zbMATH Keywords: lattice of bivarieties of regular biordered sets
zbMATH Keywords: quasiorders
zbMATH Keywords: partial binary operation
zbMATH Keywords: system of axioms
zbMATH Keywords: regular biordered sets
zbMATH Keywords: regular bimorphic images
zbMATH Keywords: local semilattices
zbMATH Keywords: solid biordered sets
reviewed by: K. Auinger
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 682638