Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. I
The paper under review is the first part of the author's work which will be published in three parts. The aim of the whole work is to provide a unified approach to study lattices of varieties and lattices of quasivarieties. The author says that this approach often helps us to see where and why differences and similarities in structures of these two types arise. The work consists of 8 sections. The present first part contains the Introduction (where results of all three parts are announced) and Sections 1-3. In Sections 1 and 2 the basic definitions are given and some preliminary results are presented. In particular, it is shown that an algebraic system \(A\) is limit-projective in a prevariety \({\mathcal V}\) if and only if \(A\) is a retract of some system with finitely many defining relations in \({\mathcal V}\) (Theorem 2.1). In Section 3 the notions of (quasi-)Birkhoff classes of algebraic systems are introduced and discussed. They are one of the main notions in the whole work. Let \({\mathcal K}\) be an arbitrary class of algebraic systems. A subclass of \({\mathcal K}\) is called a \({\mathcal K}\)-(quasi)variety if it is closed in \({\mathcal K}\) under taking subsystems, subdirect products, and homomorphic images (respectively subsystems, subdirect products, and superdirect limits). This is a natural generalization of pseudo(quasi)varieties which are exactly \({\mathcal F}\)-(quasi)varieties where \({\mathcal F}\) is the class of all finite systems of a given type. The author says that \({\mathcal K}\) is a (quasi-)Birkhoff class iff every \({\mathcal K}\)-(quasi)variety is definable in \({\mathcal K}\) by some set of (quasi)identities. The author introduced also strict and uniform (quasi-)Birkhoff classes. The main results dealing with all these notions will be published in the second part of the work.
scientific article; zbMATH DE number 848948
scientific article