Table Of ContentSEMIGROUPS
AND
ApPLICATIONS
Proceedings of the Conference on
SEMIGROUPS
AND
ApPLICATIONS
Sf Andrews, UK 2-9 July 1997
Editors
John M Howie & Nik Ruskuc
University of Sf Andrews, UK
, I»
World Scientific
II
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SEMIGROUPS AND APPLICATIONS
Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd.
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v
CONTENTS
Contents ................................................................... v
Introduction .............................................................. vii
List of participants ........................................................ Lx
On a problem of Brzozowski and Fich
J. Almeida ................................................................ 1
The classification of monoids by flatness properties of acts
S. Bulman-Fleming ....................................................... 18
Monoids of orientation-preserving transformations of a finite chain and their
presentations
P.M. Catarino ............................................................ 39
The lattice of varieties of bands
M. Ciric and S. Bogdanovic ...................... , ........................ 4i
Type II theorem and hyperdecidability of pseudovarieties of groups
M. Delgado ............................................................... 62
A flatness property of acts over monoids
A. Golchin and J. Renshaw ............................................... i2
The converse of Lallement's lemma
P.M. Higgins .............................................................. i8
On special amalgamation bases
P.M. Higgins and K. Shoji ................................................ 8i
Remarks on deletions, scattered deletions and related operations on languages
M. Ito and P.V. Silva ..................................................... 9i
Growth of a class of inverse semigroups
J. Lau ................................................................... 106
Fixed points of order-preserving transformations
T.G. Lavers, R.L. Ollerton and A.G. Shannon ........................... 115
Bass-Serre theory and inverse monoids
J.C. Meakin and A. Yamamura .......................................... 125
Equationally compact acts: coproducts
P. Normak ............................................................... 141
vi
Uniform decision problems for certain restricted classes of finite monoid pre
sentations - a survey on recent undecidability results
F. Otto .................................................................. 152
On a problem of B.I. Plotkin
J.S. Ponizovskil .......................................................... 171
Varieties and pseudovarieties generated by 'V-chains
N.R. Reilly and S. Zhang ................................................ 179
Finite .J-trivial transformation semigroups as inductuvely constructed semi
groups
T. Saito ................................................................. 194
A finite non-embeddable E-unitary regular semigroup
M.B. Szendrei ........................................................... 202
The lattices of pseudovarieties of DA and B
P.C. Trotter ............................................................. 215
The semigroups of order-preserving mappings: quest for quasiidentities
A.S. Vernitskir ........................................................... 229
The finite basis problem for the pseudovariety PO
M.V. Volkov ............................................................. 239
Congruences on bicyclic extensions
R.J. Warne .............................................................. 258
On weakly V-regular semigroups
M.-C. Zhang and H.-B. Kan ............................................ 270
vii
INTRODUCTION
This volume had its origin in the conference on "Semigroups and Applications"
held in St Andrews, Scotland from 2 to 9 July, 1997. Some 90 mathematicians
participated in the conference, and the extensive range of topics covered was
representative of the ever-widening scope of semigroup theory. Most of the
papers featuring here are based on lectures given at the conference.
The notion of a pseudo variety (a class of semigroups closed under the
formation of subsemigroups, homomorphic images and finite direct products)
continues to give rise to deep and interesting work, and this is reflected in
several of the papers in this collection. Moreover, the notion provides, via an
important theorem of Eilenberg, a link between semigroups and theoretical
computer science, and so this work is an important interface between semi
groups and the "outside world" .
The emergence of a theory of pseudovarieties has not lessened the impor
tance of the classical notion of a variety. In the present volume this area also is
represented, in the articles by Ciric and Bogdanovic and by Reilly and Zhang.
The study of presentations, for many years a topic of interest in group the
ory, has recently become a major strand of research in semigroup theory. Here
too, through questions of decidability, there are important links to theoretical
computer science.
Sets of mappings closed under composition have always been the most
natural source of examples of semigroups, and the study of semigroups of
this kind has for many years given rise to interesting and non-trivial ideas,
while providing links, both in results and methodology, with combinatorial
mathematics. Again, this strand of thought is well represented here.
Sets of mappings closed under composition have always been the most
natural source of examples of semigroups, and the study of semigroups of
this kind has for many years given rise to interesting and non-trivial ideas,
while providing links, both in results and methodology, with combinatorial
mathematics. Again, this strand of thought is well represented here.
It is possible to generalize the classical idea of a module over a ring to
a non-additive context. If S is a monoid, then a set X is said to be a (left)
S-act if there is an action x t-+ sx : S x X -t X satisfying the axioms (st)x =
s(tx) (s, t E S, x E X), and Ix = x (x E X). This simple idea has
proved remarkably fruitful, and has been used to make considerable progress
in amalgamation theory. The article by Higgins and Shoji provides an example
of this progress, while the survey article by Sydney Bulman-Fleming is an
excellent introduction to the ideas, techniques and achievements of this aspect
of semigroup theory.
Mathematical journals rarely if ever contain survey articles, partly for rea-
viii
sons of space, partly because mathematicians tend to think of a survey article
as being in some way a lesser achievement than a piece of original research.
This is unfortunate, for it means that it is very hard for mathematicians to
acquire even a basic working knowledge of any field outside their own narrow
specialisms. It is, we believe, one of the chief values of Proceedings volumes
such as this that they tend to contain a number of not too technical accounts
which enable the reader, whether attending the conference or not, to get the
gist of an area without necessarily ploughing through the fine detail.
While hoping that there are no errors of substance, we take full responsi
bility for any that have crept in during the editing process. We are confident
that these Proceedings will be a valuable addition to the literature on pure and
applied semigroup theory.
John M. Howie
Nik Ruskuc
August1998
ix
LIST OF PARTICIPANTS
Jorge Almeida Centro de Matematica, Universidade do Porto, P. Gomes
Teixeira, 4000 Porto, Portugal, e-mail: jalmeidaClfc1.fc.up.pt
Paulo Almeida Quinta da Corga, 3360 Penacova, Portugal, e-mail:
palmeidaCldmat.uevora.pt
Jorge Andre Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama
Pinto, 2, 1699 Lisboa Codex, Portugal, e-mail: jandreClptmat .lmc. fc.
ul.pt
Isabel Araujo Mathematical Institute, University of St Andrews, St An
drews, Fife KY16 9SS, Scotland, U.K., e-mail: isabelClmcs.st-and.
ac.uk
Robert Arthur Mathematical Institute, University of St Andrews, St An
drews, Fife KY16 9SS, Scotland, U.K., e-mail: robertaClmcs.st-and.
ac.uk
Karl Auinger Institut fUr Mathematik, Strudlhofgasse 4, Universitat Wien,
1090 Wien, Austria, e-mail: karlaClbanach.mat.univie.ac.at
Hayrullah Ayik Mathematical Institute, University of St Andrews, St An
drews, Fife KY16 9SS, Scotland, U .K., e-mail: ayiUmcs.st-and.ac .
uk
Assis Azevedo Department of Mathematics, University of Minho, 4700
Braga, Portugal, e-mail: assisClmath.uminho.pt
Bernd Billhardt Fachbereich Mathematik-Informatik, GHS Kassel, 34109
Kassel, Germany, e-mail: billardClhrz.uni-kassel.de
Tom Blyth Mathematical Institute, University of St Andrews, St Andrews,
Fife KY16 9SS, Scotland, U.K., e-mail: tsbClst-and.ac.uk
Stojan Bogdanovic University of Nis, Faculty of Economics, Department of
Mathematics, Trg JNA 11, P.O. Box 121, 18000 Nis, Yugoslavia e-mail:
sbogdanClarchimed.filfak.ni.ac.yu
Mario Branco Centro de Algebra da Universidade de Lisboa, Av. Prof.
Gama Pinto, 2, 1699 Lisboa Codex, Portugal, e-mail: mbrancoClptmat.
lmc.fc.ul.pt
Sydney Bulman-Fleming Wilfrid Laurier University, Waterloo, Ontario,
N2L 3C5, Canada, e-mail: sbulmanClmachl.wlu.ca
Colin Campbell Mathematical Institute, University of St Andrews, St An
drews, Fife KY16 9SS, Scotland, U .K., e-mail: cmcClst-and.ac . uk
Paula Catarino Department of Mathematics, Quinta de Prados, Univer
sity of Tras-os-Montes e Alto Douro, 5000 Vila Real, Portugal, e-mail:
pcatarinClutad.pt
