Products of lattice varieties
type: Item
instance of: scholarly article
title: Products of lattice varieties
zbMATH Open document ID: 0587.06002
DOI: 10.1007/BF01187554
author: G. Grätzer
author: David Kelly
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: The concept of variety product originated in \textit{H. Neumann}'s work on ''Varieties of groups'' [Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37 (1967; Zbl 0251.20001)] and was generalized to universal algebra by \textit{A. I. Mal'cev} [Sib. Mat. Zh. 8, 346-365 (1967; Zbl 0228.08007); The metamathematics of algebraic systems (1971; Zbl 0231.02002)]. Given two classes of lattices \({\mathcal V}\) and \({\mathcal W}\), their product \({\mathcal V}\circ {\mathcal W}\) consists of all lattices L which admit a congruence \(\Theta\) such that all congruence classes (as lattices with the induced order) belong to \({\mathcal V}\), while L/\(\Theta\) is in \({\mathcal W}\). The product operation preserves closedness under isomorphic copies, sublattices, products and ultraproducts, but unfortunately not closedness under homomorphic images. E.g., if \({\mathcal V}\) is a nondistributive variety generated by a finite lattice and \({\mathcal D}\) consists of all distributive lattices, then \({\mathcal V}\circ {\mathcal D}\) is not a variety (Theorem 1). This contrasts with the situation for groups, where products of varieties are again varieties. The main result of the paper is the embedding theorem (Theorem 2). If the class \({\mathcal V}\) is closed under formation of ideal- and filter-lattices, then every object L of \({\mathcal V}\circ {\mathcal W}\) by virtue of a congruence \(\Theta\) can be embedded into a lattice \(\hat L,\) which belongs to \({\mathcal V}\circ {\mathcal W}\) by virtue of an extension \({\hat \Theta}\) of \(\Theta\), and all \({\hat \Theta}\)-classes of \(\hat L\) are complete lattices. Moreover, L/\(\Theta\) \(\cong \hat L/{\hat \Theta}\), and every congruence of L extends to \(\hat L.\) This leads to a sufficient condition for the product of two lattice varieties to be a variety. The authors then provide a construction scheme, which for varieties \({\mathcal V}\) and \({\mathcal W}\) produces all the lattices \(\hat L\) for the objects L o…
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 3941602
zbMATH Keywords: ultraproducts
zbMATH Keywords: finite lattice
zbMATH Keywords: distributive lattices
zbMATH Keywords: products of varieties
zbMATH Keywords: embedding theorem
zbMATH Keywords: ideal- and filter-lattices
zbMATH Keywords: congruence
zbMATH Keywords: complete lattices
zbMATH Keywords: lattice varieties
zbMATH Keywords: dimension for lattice varieties
zbMATH Keywords: dimension formula for products
reviewed by: J. Koslowski
MaRDI profile type: Publication
cites work: Splitting lattices generate all lattices
cites work: The arithmetical theory of Birkhoff lattices
cites work: Q4097341
cites work: Q3893972
cites work: Subdirectly Irreducible Members of Products of Lattice Varieties
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Varieties of groups
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description: scientific article; zbMATH DE number 3941602