Dualities in lattices of semigroup varieties
type: Item
instance of: scholarly article
title: Dualities in lattices of semigroup varieties
zbMATH Open document ID: 0687.20055
DOI: 10.1007/BF02573251
published in: Semigroup Forum
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135004
review text: The subvariety lattice of a variety \({\mathfrak V}\) of universal algebras is denoted by L(\({\mathfrak V})\). Varieties \({\mathfrak V}\) and \({\mathfrak V}'\) are said to be dual to one another if L(\({\mathfrak V})\) and L(\({\mathfrak V}')\) are dual lattices; \({\mathfrak V}\) is selfdual if L(\({\mathfrak V})\) is selfdual. Let \({\mathfrak X}\) be a class of semigroup varieties, then a semigroup variety \({\mathfrak V}\) is said to admit duality in \({\mathfrak X}\) if there is \({\mathfrak V}'\in {\mathfrak X}\) which is dual to \({\mathfrak V}\); \({\mathfrak C}\) admits duality if \({\mathfrak X}\) is the class of all varieties of semigroups. If all subvarieties of \({\mathfrak V}\) admit duality in \({\mathfrak X}\) (admit duality) then \({\mathfrak V}\) is said to hereditarily admit duality in \({\mathfrak X}\) (hereditarily admit duality). The abbreviations h.a.d. in \({\mathfrak X}\) and h.a.d. will be used for these properties. Similarly \({\mathfrak V}\) is hereditarily selfdual if all its subvarieties are selfdual. This paper investigates h.a.d. and hereditarily selfdual varieties. Two results on hereditarily selfdual varieties of universal algebras are given, together with three equivalents to semiduality in a variety of semigroups. In the case of h.a.d. varieties, attention is restricted to semigroup varieties. Results are given for h.a.d. varieties in general and also for varities h.a.d. in \({\mathcal K}\), the class of all varieties of semigroups in which every periodic group is locally finite. In the latter case we have the interesting result that, if one of two additional conditions is satisfied, a variety which is h.a.d. in \({\mathcal K}\) is also hereditarily selfdual.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 4125706
zbMATH Keywords: subvariety lattice
zbMATH Keywords: dual lattices
zbMATH Keywords: semigroup varieties
zbMATH Keywords: semiduality
zbMATH Keywords: varieties of semigroups
zbMATH Keywords: hereditarily selfdual
author: Boris Vernikov
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4125706