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Small semigroup related structures with infinite properties

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Jackson, MG (1999) Small semigroup related structures with infinite properties. PhD thesis, University of Tasmania.

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Abstract

In mathematics, one frequently encounters constructions of a pathological or
critical nature. In this thesis we investigate such structures in semigroup theory
with a particular aim of finding small, finite, examples with certain associated infinite
characteristics.
We begin our investigation with a study of the identities of finite semigroups.
A semigroup (or the variety it generates) whose identities admit a finite basis is
said to be finitely based. We find examples of pairs of finite (aperiodic) finitely
based semigroups whose direct product is not finitely based (answering a question
of M. Sapir) and of pairs of finite (aperiodic) semigroups that are not finitely based
whose direct product is finitely based. These and other semigroups from a large
class (the class of finite Rees quotients of free monoids) are also shown to generate
varieties with a chain of finitely generated supervarieties which alternate between
being finitely based and not finitely based. Furthermore it is shown that in a natural
sense, "almost all" semigroups from this class are not finitely based.
Not finitely based semigroups that are locally finite and have the property that
every locally finite variety containing them is also not finitely based are said to be
inherently not finitely based. We construct all minimal inherently not finitely based
divisors in the class of finite semigroups and establish several results concerning a
fundamental example with this property; the six element Brandt semigroup with
adjoined identity element, B. 1/2. We then find the first examples of finite semigroups admitting a finite basis
of identities but generating a variety with uncountably many subvarieties (indeed
with a chain of subvarieties with the same ordering as the real numbers). For some
well known classes, a complete description of the members with this property are
obtained and related examples and results concerning joins of varieties are also
found. A connection between these results and the construction of varieties with
decidable word problem but undecidable uniform word problem is investigated.
Finally we investigate several embedding problems not directly concerned with
semigroup varieties and show that they are undecidable. The first and second of
these problems concern the fundamental relations of Green; in addition some small
examples are found which exhibit unusual related properties and a problem of M.
Sapir is solved. The third of the embedding problems concerns the potential embeddability
of finite semigroup amalgams. The results are easily extended to the class
of rings.

Item Type: Thesis (PhD)
Keywords: Infinite groups, Semigroups, Finite simple groups
Copyright Holders: The Author
Copyright Information:

Copyright 1999 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: 19 Dec 2014 02:39
Last Modified: 04 Apr 2017 23:17
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