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

Churchill, Genevieve 1998 , 'Theorems of Birkhoff type in pseudovarieties and e-varieties of regular semi-groups', PhD thesis, University of Tasmania.

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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
The results presented in Sections 2.4 - 2.6 appear in [12]. The results of
Chapter 3 will appear in [13].

Item Type: Thesis - PhD
Authors/Creators:Churchill, Genevieve
Keywords: Semigroup algebras, Fundamental theorem of algebra
Copyright Holders: The Author
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Copyright 1998 the Author - The University is continuing to endeavour to trace the copyright
owner(s) and in the meantime this item has been reproduced here in good faith. We
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Additional Information:

Thesis (Ph.D.)--University of Tasmania, 1999. Includes bibliographical references

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