A problem concerning the lattice varietal product
type: Item
instance of: scholarly article
title: A problem concerning the lattice varietal product
zbMATH Open document ID: 0652.06006
DOI: 10.1007/BF01229960
author: Tom Harrison
published in: Algebra Universalis
publication date: 1988-01-01T00:00:00Z
review text: In general, the product of two quasivarieties of universal algebras is again a quasivariety. Thus although the product of two varieties of lattices is not necessarily a variety, it is a quasivariety and so closed under the formation of sublattices, direct products, and ultraproducts. McKenzie showed (in a forthcoming paper) that for any lattice variety \({\mathcal V}\) both \({\mathcal D}\circ {\mathcal V}\) and \({\mathcal M}\circ {\mathcal V}\) are varieties, where \({\mathcal D}\) refers to the variety of all distributive lattices and \({\mathcal M}\) refers to the variety of all modular lattices. There exist continuum many nonmodular varieties \({\mathcal V}\) such that \({\mathcal V}\circ {\mathcal D}\) is not a variety. McKenzie postulated that if \({\mathcal V}\) is a nonmodular lattice variety then \({\mathcal V}\circ {\mathcal D}\) is a variety only in the trivial case that \({\mathcal V}\) is the variety of all lattices. This paper confirms McKenzie's conjecture, showing that for any nontrivial modular lattice variety \({\mathcal P}\), if \({\mathcal V}\) is nonmodular and \({\mathcal V}\circ {\mathcal P}\) is a variety than \({\mathcal V}\) is the variety of all lattices.
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 4063162
zbMATH Keywords: lattice varietal product
zbMATH Keywords: distributive lattices
zbMATH Keywords: modular lattices
reviewed by: Adalbert Csaba Hatvany
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4063162