Varieties of semigroups and varieties of completely regular semigroups closed for certain extensions
type: Item
instance of: scholarly article
title: Varieties of semigroups and varieties of completely regular semigroups closed for certain extensions
zbMATH Open document ID: 0811.20052
author: Xiaoying Yan
author: Francis Pastijn
published in: Journal of Algebra
publication date: 1995-04-17T00:00:00Z
review text: Let \(\mathcal V\) be any variety of universal algebras and let \(C_{\mathcal V}\) be the category whose members are the members of \(\mathcal V\) and whose morphisms are the surjective homomorphisms. For a given functor \(F : C_{\mathcal V} \to C_{\mathcal V}\), a subvariety \(\mathcal W\) of \(\mathcal V\) is said to be closed for \(F\) if for every \(A \in {\mathcal W}\) also \(F(A) \in {\mathcal W}\). Let \(L_ F\) be the partially ordered set of all subvarieties of \(\mathcal V\) which are closed for \(F\). The authors are interested in situations when \(L_ F\) is a complete sublattice of \(L({\mathcal V})\), the lattice of all subvarieties of \(\mathcal V\). They find the following -- very transparent -- sufficient condition for this case: for every \(A \in {\mathcal V}\) and for every family (\(\phi_ i, i\in I\)) of morphisms in \(C_{\mathcal V}\) with domain \(A\), \(\bigcap_{i \in I} \phi_ i \phi_ i^{-1} = \iota_ A\) implies \(\bigcap_{i \in I} (F \phi_ i)(F \phi_ i)^{- 1} = \iota_{F(A)}\). An appropriate analogue to this theorem holds for pseudovarieties as well. Though in the above situation \(F\) is an arbitrary functor, one should think of \(F(A)\) being obtained from \(A\) by a certain canonical construction. Two examples of such constructions are elaborated; however, the general theory will have many more applications. The first example is the functor \(B : C_{\mathcal S} \to C_{\mathcal S}\) where \(\mathcal S\) is the variety of all semigroups, and for any semigroup \(S\), \(B(S)\) denotes the well known Bruck extension of \(S\). This leads to a functor of the above kind since any surjective homomorphism \(\phi : S \to T\) may be canonically extended to a surjective homomorphism \(B(\phi) : B(S) \to B(T)\). The above mentioned theorem applies and so the set \(L_ B\) of all semigroup varieties closed for the Bruck extension forms a complete sublattice of the lattice of all semigroup varieties. Several properties of the lattice \(L_ B\) are investigate…
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 20M50
zbMATH DE Number: 554148
zbMATH Keywords: lattice of subvarieties
zbMATH Keywords: category
zbMATH Keywords: surjective homomorphisms
zbMATH Keywords: functor
zbMATH Keywords: pseudovarieties
zbMATH Keywords: Bruck extension
zbMATH Keywords: semigroup variety
zbMATH Keywords: least variety
zbMATH Keywords: completely regular semigroups
zbMATH Keywords: idempotent generated extension
zbMATH Keywords: complete sublattice
zbMATH Keywords: complete congruences
reviewed by: K. Auinger
MaRDI profile type: Publication
full work available at URL: jabr.1994.1043
OpenAlex ID: W2042295080
DOI: 10.1006/JABR.1994.1043
description: scientific article; zbMATH DE number 554148