Theorems of Birkhoff type in pseudovarieties and e-varieties of regular semi-groups

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 ‚Äöv†¬© Com) U {Nil ‚Äöv†¬© Com} ‚ÄövÑvÆ G (N ‚Äöv†¬© Com) given by W - W ‚Äöv†¬© 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
-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]."