Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II
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.
scientific article; zbMATH DE number 907276
scientific article