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U.W. Bangor - School of Informatics
Mathematics Preprints 2000
Semigroup and Automata Theory
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00.05 : KHAN, T.A.
The relationship between the local and global structure of semigroups
Summary:
The main aim of this thesis is to generalise McAlister's theory of
locally inverse regular semigroups to the class of semigroups with
local units in which the local submonoids have commuting idempotents.
We prove that if such a semigroup has what we call a McAlister
sandwich function then the semigroup can be covered by means of a
Rees matrix semigroup over a semigroup with commuting idempotents.
Examples of such semigroups are easily constructed.
Indeed, if T is a semigroup with local units having an
idempotent e such that T = TeT,
and eTe has commuting idempotents, then all the local submonoids
of T have commuting idempotents and T is equipped
with a McAlister sandwich function.
We prove that the semigroups with local units having local submonoids
with commuting idempotents S which can be embedded in such a
semigroup T in such a way that S = STS are precisely
the ones having a McAlister sandwich function.
Finally, in a different direction,
we study variants of semigroups concentrating on the relationship
between the local structure of a semigroup and the global structure
of its variants.
Published in:
U.W. Bangor, Ph. D. thesis
(2001).
Download:
gzipped postscript of the thesis:
khan.ps.gz
00.10 : KHAN, T.A. & LAWSON, M.V.
A characterisation of a class of semigroups with locally commuting idempotents
Abstract:
McAlister proved that a necessary and sufficient condition for a regular
semigroup S to be locally inverse is that it can be embedded as a
quasi-ideal in a semigroup T which satisfies the following two
conditions:
(1) T = TeT, for some idempotent e; and
(2) eTe is inverse.
We generalise this result to the class of semigroups with
local units in which all local submonoids have commuting idempotents.
Published in:
Periodica Mathematica Hungarica
40 (2000) 85-107.
00.20 : HINES, P.
A categorical approach to Kleene's theorem
Abstract:
The aim of this paper is to make an analogy between propositions/proofs,
and formal languages/finite state machines. In particular,
we consider similarities between the connectives of linear logic,
as represented in the Geometry of Interaction,
and the closure properties of the languages
recognised by finite state automata.
Although a formal correspondence is not given,
the categorical structures produced by an analysis of Kleene's theorem
provide good supporting evidence for such a correspondence.
An important part of this procedure is to abstract the essential features
required by the monoid of relations in order to give the closure
of the regular languages under the operations described by Kleene's theorem.
This opens the way to the construction of analogues of Kleene's theorem
in other monoids.
Published in:
Download:
gzipped postscript file:
00_20.ps.gz
00.28 : HINES, P.
The categorical theory of self-similarity
Abstract:
We demonstrate how the identity $N\otimes N \cong N$ in a monoidal
category allows us to construct a functor from the full subcategory
generated by N and \otimes to the endomorphism monoid
of the object N.
This provides a categorical foundation for one-object
analogues of the symmetric monoidal categories used by J.-Y. Girard
in his Geometry of Interaction series of papers,
and explicitly described in terms of inverse semigroup theory.
This functor also allows the construction of one-object analogues
of other categorical structures. We give the example of one-object
analogues of the categorical trace, and
compact closedness.
Finally, we demonstrate how the categorical theory of
self-similarity can be related to the algebraic theory,
and Girard's dynamical algebra,
by considering one-object analogues of projections and inclusions.
Published in:
Theory and Applications of Categories
6 (2001) 33-46.