Varieties: Spectra, Skeletons, Categories

Abstract

The aim of the present chapter is to apply the methods, results and constructions considered in the first two chapters to “external” studies of universal algebra varieties. “External” studies of varieties imply consideration and description of not the algebras incorporated into a given variety but of the variety as a whole, i.e., studies of the variety as a single object the elements of which are the algebras of the variety with basic algebraic relations and operations among them such as isomorphisms, epimorphisms, embeddings, Cartesian products, etc.. Studies of the “external” structure of a variety imply, first of all, those of the categories of the algebras belonging to the variety in the case when the morphisms of the category are all homomorphisms between algebras of the given variety. Indeed, the overwhelming part of the notions related to an algebra can be formulated in terms of these categories and, therefore, the varieties with “the same” categories must be “almost the same” themselves, as we will see in the first theorems proved in section 8. Another, rougher “external” characteristic of a variety is its spectrum and its fine spectrum. We have already discussed in section 6 some results for algebras with a minimal spectrum, these impose very rigid limitations and allow only three variants for the varieties generated by such algebras. Below, in section 8, we will present a result describing to the accuracy of “the same category” all the varieties with a minimal fine spectrum of a certain quite definite type, as well as a number of other results on spectra and fine spectra. Well-known descriptions of category transformations also pertain to the results characterizing varieties with a fine spectrum. On the other hand, in the case when the fine spectrum of a variety is big, i.e., when the number of the types of the isomorphisms of the algebras of a given variety is big, it is interesting to study various relations and operations between the types of the isomorphism induced by algebraically important relations and operations between the algebras of the variety themselves. This results in the definition of the notion of the skeleton of a variety, and the greater part of the present chapter is devoted to studying skeletons of congruence-distributive varieties for which the application of Boolean constructions is most efficient. In particular, a number of results on countable skeletons of congruence-distributive varieties make it possible, using the language of “external” description of varieties, to express such facts as the degeneration of a variety with a quasi-primal algebra with no proper subalgebras, finite generation of discriminator varieties, etc.