Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II
type: Item
instance of: scholarly article
title: Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II
zbMATH Open document ID: 0848.08004
DOI: 10.1007/BF00739404
author: V. A. Gorbunov
published in: Algebra and Logic
publication date: 1996-10-24T00:00:00Z
full work available at URL: https://eudml.org/doc/187733
review text: The paper under review is the second part of the author's work which will be published in three parts [for the first part see Algebra Logika, 34, No. 2, 142-168 (1995); English translation: Algebra Logic 34, No. 2, 73-86 (1995; Zbl 0841.08005)]. The aim of the whole work is to provide a unified approach to the study of lattices of varieties and lattices of quasivarieties. The work consists of 8 sections. The present second part contains Sections 4-6. In the first part of the work the notions of (quasi-) Birkhoff classes of algebraic systems are introduced. In Section 4 a general characterization of lattices of varieties and lattices of quasivarieties in terms of (quasi-) Birkhoff classes is given. In Sections 5 and 6 a method for constructing homomorphic images of such lattices is presented. As an application, it is proved that the lattice of varieties of modular lattices has a complete homomorphism onto the Boolean lattice of subsets of a countable set. The author gives sufficient conditions for embedding the free lattice with \(\omega\) generators in a given lattice of quasivarieties. As a corollary, it is noted that the variety of commutative rings with 1 and the variety of mono-unary algebras are \(Q\)-universal. Other applications and examples are given.
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 06B25
zbMATH DE Number: 907276
zbMATH Keywords: Birkhoff classes
zbMATH Keywords: quasi-Birkhoff classes
zbMATH Keywords: lattices of varieties
zbMATH Keywords: lattices of quasivarieties
zbMATH Keywords: homomorphic images
zbMATH Keywords: varieties of modular lattices
zbMATH Keywords: free lattice
zbMATH Keywords: variety of commutative rings
zbMATH Keywords: variety of mono-unary algebras
reviewed by: Boris Vernikov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 907276