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

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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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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].

Item Type: Thesis (PhD)
Keywords: Semigroup algebras, Fundamental theorem of algebra
Copyright Holders: The Author
Copyright Information:

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
would be pleased to hear from the copyright owner(s).

Additional Information:

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

Date Deposited: 25 Nov 2014 00:49
Last Modified: 06 Jul 2016 01:25
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