Theorems of Birkhoff type in pseudovarieties and e-varieties of regular semigroups

Abstract

This thesis is concerned with the problem of being able to use, or generalize, Birkhoff's fundamental theorems for classes of algebras which do not form varieties - particularly in pseudovarieties and e-varieties. After giving an introduction to these areas in Chapter 1, we first look at pseudovarieties, focusing on certain generalized varieties. Let Com, Nil, and N denote the generalized varieties of all commutative, nil, and nilpotent semigroups respectively. For a class W of semigroups let L (W) and G (W) denote respectively the lattices of all varieties and generalized varieties of semigroups contained in W. Almeida has shown that the mapping L (Nil ∩ Com) U {Nil ∩ Com} — G (N ∩ Com) given by W - W ∩ N is an isomorphism, and asked whether the extension of this mapping to L (Nil) U {Nil} is also an isomorphism. In Chapter 2 we consider this question. In Section 2.2 we show that the extension is not surjective. Non-injectivity is then established in Sections 2.4 - 2.6; this involves analysing sequences of words of unbounded lengths derived from the defining identities of certain nil varieties. Results of a more general nature are also given, in Section 2.3, involving the question of when two arbitrary semigroup varieties possess the same set of nilpotent semigroups. In Chapter 3 we turn to the problem of establishing analogues of Birkhoff's theorems for e-varieties. In Section 3.1 Auinger's Birkhoff-style theory for locally inverse e-varieties is expanded, to obtain a unified theory for e-varieties of locally inverse or of E-solid semigroups - that is, for the entire lattice of e-varieties in which nonmonogenic bifree objects exist. In addition an alternative unification, based on the techniques used by Kadourek and Szendrei to describe a Birkhoffstyle theory for E-solid e-varieties, is given in Section 3.2. In Section 3.3 we show that trifree objects on at least three generators exist in an e-variety V of regular semigroups if and only if V is locally E-solid; this extends Kadourek's work on the existence of trifree objects in locally orthodox e-varieties and generalizes Yeh's result on the existence of bifree objects. In conclusion, a theory of n-free objects is outlined in Section 3.4, indicating how analogues of the concept of a free object can be defined for any e-variety. The results presented in Sections 2.4 - 2.6 appear in [12]. The results of Chapter 3 will appear in [13].