Proceedings of the International Conference on Semigroups : Braga, Portugal, 18-23 June 1999

296 Pages·2000·4.907 MB·English

by  Paula Smith, EmiМЃlia Giraldes, Paula Martins (eds.)

PROCEEDI NGS OF THE
I NTERNATIONAL CONFERENCE ON
SEMIGROUPS
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PROCEEDINGS OF THE
INTERNATIONAL CONFERENCE ON
SEMIGROUPS
Braga, Portugal 18 - 23 June 1999
Editors
Paula Smith
Emilia Giraldes
Paula Martins
(Mat - University of Minho, Portugal
b
l
World Scientific
II
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
SEMIGROUPS
Proceedings of the International Conference
Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means.
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ISBN 981-02-4392-8
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Introduction
This volume contains papers which are based on talks given at the Inter
national Conference on Semigroups which was held at the University of Minho,
in Braga, Portugal, between 18 - 23 June 1999. The Organising Committee
consisted of the editors of this volume. In addition, John Howie, Donald
McAlister, Norman Reilly and Lev Shevrin participated as members of the
Scientific Committee. A total of 90 mathematicians from different countries
(Australia, Austria, Belgium, Canada, Czech Republic, Estonia, France, Ger
many, Hungary, Israel, Japan, Latvia, Poland, Portugal, Russia, Spain, U.K.,
U.S.A. and Yugoslavia) attended the conference. The scientific program in
cluded ten plenary lectures, most of which were surveys, and 38 contributed
talks.
The purpose of the conference was to bring together experts in semigroup
theory and its applications, to stimulate discussion of recent results in this do
main and to allow young scientists to meet leading researchers in an informal
environment. The inclusion of a series of surveys of important sub-domains of
semigroup theory was intended to provide an accessible introduction to certain
areas for anyone interested in acquiring a working knowledge of a new field.
We are confident that the conference attained, and indeed exceeded, the ori
ginal objectives and that these Proceedings will be an important contribution
to the literature on pure and applied semigroup theory.
We acknowledge, with much gratitude, the financial support received from
the following organisations: Research Centre of Mathematics of the Univer
sity of Minho, Pundar;iio para a Ciencia e Tecnologia, NATO, Department
of Mathematics of the University of Minho and the Banca Portugues do
Atlantica. The following local sponsors also contributed in a variety of ways:
Aguas do Luso, Cafes Delta, Cafes Silveira, Camara Municipal de Melgar;o,
Livraria Minho, Pastelaria S. Victor, Quintas de Melgar;o and Sumos Com
pal. For their contribution, which made the conference so much easier to run
and so pleasant to attend, we are indeed very grateful! We also express our
thanks to all who contributed their work to be included in this volume and
respected the publisher'S instructions and editors' deadlines. Finally, a special
acknowledgement is due to the members of the Scientific Committee for their
work in refereeing the papers published in this volume.
Braga, 30th November 1999
Paula Smith
Enulia Giraldes
Paula Martins
v
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Contents
Introduction . . . . . . . . . . . .. . ........................................ v
Computing with semigroups in GAP - a tutorial
1. Araujo and A. Solomon ............................................ 1
The semigroup efficiency of direct powers of groups
H. Ayik, C. M. Campbell*, 1. J. O'Connor and N. Ru§kuc ........... 19
Inverse transversals - a guided tour
T. S. Blyth .......................................................... 26
Semigroups satisfying some variable identities
M. Cirie, T. Petkovie and S. Bogdanovie ............................. 44
Some variations on the notion of locally testable language
J. C. Costa .......................................................... 54
Solid varieties of semirings
K. Denecke* and H. Hounnon ....................................... 69
Deciding some embeddability problems for semigroups of mappings
* ...................................
D. H. Premlin and P. M. Higgins 87
On the semigroup with very good magnifiers
M. Gutan ............................................................ 96
Locally uniformly 7r-regular semigroups
M. Mitrovie, S. Bogdanovie and M. Cirie ........................... 106
Introduction to E-inversive semigroups
H. Mitsch .......................................................... 114
Rings graded by inverse semigroups
W. D. Munn ....................................................... 136
vii
viii
Varieties of bands
M. Petrich ......................................................... 146
Characterization of a semidirect product of groups by its endomorphism semi
group
P. Puusemp ........................................................ 161
Generalized N-semigroups
* .............................
J. C. Rosales and J. 1. Garcia-Garcia 171
PG=BG: Redux
B. Steinberg ........................................................ 181
'Transformation semigroups: Past, present and future
R. P. Sullivan ...................................................... 191
The finite basis problem for finite semigroups: a survey
M. V. Volkov ....................................................... 244
List of Participants ..... ...... ..... ................ ............ ... 291
COMPUTING WITH SEMIGROUPS IN GAP - A TUTORIAL
ISABEL M. ARAUJO AND ANDREW SOLOMON
School of Mathematical and Computational Sciences,
The University of St. Andrews, Scotland
E-mail: {isabel.andrews}@dcs.st-and.ac. uk
With the release of Version 4.1, GAP becomes an integrated environment for per
forming calculations with semigroups and developing algorithms for them. In the
Introduction, we give a brief account of the state of computational semigroup the
ory and outline what we see as the main challenges for the future. The rest of
the paper takes the form of a tutorial and is intended to be a semigroup theorist's
introduction to the use of GAP, showcasing the features directly connected with
semigroups.
Introduction
Computational algebra as a discipline deals with a spectrum of questions,
from the 'theoretical' ones of computability and complexity of algebraic prob
lems, to the 'practical' issues of using computers to perform algebraic calcu
lations. Theoretical computational semigroup theory is very well developed
whereas the practical aspects of computing with semigroups have been some
what neglected, especially compared with group theory and commutative al
gebra where there is a great deal of software for performing calculations, and
a large body of work relying on it.
There have been two main obstacles to the development of practical com
putational semigroup theory. The first is a general pessimism about the com
plexity of problems in arbitrary semi groups compared with groups - recall
Kozen's remark [13] that the problem of deciding membership of a transfor
mation semigroup given by a particular generating set is PSPACE complete.
However, this is not an argument against the development of computational
semigroup theory - there are many 'small' semigroups of interest where the
asymptotic complexity of the algorithms does not become an issue. Fur
thermore, as in any discipline, computational semigroup theory will develop
through an apprehension of what are reasonable questions to address and
semigroups to study. This should not be defined by what is reasonable in
computational group theory or commutative algebra.
The other, more concrete reason that computational semigroup theory
has lagged behind, is the lack of an integrated framework in which to develop
algorithms. For well over a decade, computational group theorists have had
tools like Cayley, Magma [2] and GAP [5] in which to develop and integrate
2
algorithms. The computational group theorist now relies on many diverse
algorithms developed over this period and uses them together with ease. In
contrast, a number of excellent programs have been developed for various dif
ferent types of computation with semigroups but it is virtually impossible to
use them together for calculations. Another problem facing these packages is
that they are developed by one or two mathematicians, and as their circum
stances change, the program ceases to be developed and maintained. These
are the issues we have sought to address in GAP 4.1.
Semigroups and GAP
Due to its ready availability - it is free and is ported to many platforms
- GAP has come to be the most widely used tool for computational group
theory. It is used both as a 'desk calculator' for automating large or tedious
hand calculations, and also as a software development platform in which to
implement mathematical algorithms.
Over the last five years, various third party plug-ins such as Glissando [7)
and Monoid [12) have contributed piecemeal functionality for semigroups in
GAP, but with version 4.1, basic support for semigroups is provided as core
GAP functionality for the first time. This new status of semigroup functional
ity in GAP will allow software developed in this setting to outlive the interest
of any single developer.
In this tutorial we tour all the basic features provided for semigroups in
GAP at the time of the 4.1 release. It is expected that this will form the basis
for further development of semigroup functionality in GAP in response to user
requests and the work of the large and growing GAP developer community.
The tutorial
This tutorial emphasises the use of GAP as a desk calculator for semigroups.
There are five sections, and in each section we use different features to inves
tigate the structure of a well known semigroup. The material in each section
is intended to be independent from the rest, so they can be approached in
any order. Functions and constructions which are not fully explained in this
tutorial may be found in the GAP Reference Manual [6).
1. Endomorphisms of a finite chain introduces the reader to transfor
mation semigroups, congruences and Green's relationsj
2. Orientation preserving mappings uses transformation semigroups as
a starting point for working with congruences and quotient semigroups,
and Rees matrix semigroupsj