Semigroups and Their Applications: Proceedings of the International Conference “Algebraic Theory of Semigroups and Its Applications” held at the California State University, Chico, April 10–12, 1986

213 Pages·1987·9.24 MB·English

by  Jorge Almeida

Semigroups and Their Applications
Semigroups
and
Their Applications
Proceedings ofthe International Conference
''Algebraic Theory of Semigroups and Its Applications"
held at the California State University, Chico, April 10-12, 1986
edited by
Simon M. Goberstein
Department of Mathematics, California State University. Chico, U.S.A.
and
Peter M, Higgins
Department of Mathematics, California State University, Chico
and Division of Computing and Mathematics, Deakin University, U.S.A.
D. REIDEL PUBLISHING COMPANY
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
DORDRECHTI BOSTON I LANCASTER I TOKYO
Library of Congress Cataloging in Publication Data
International Conference, "Algebraic Theory of Semigroups and Its Applications"
(1986: California State University, Chico)
Semigroups and their applications.
Includes index.
I. Semigroups-Congresses. I. Gobcrstein, Simon M., 1949-
II. Higgins, Peter M., 1956- .III. Title.
QA171.l59 1986 512'.2 87-4591
ISBN-13 978-94-0 I 0-8209-9 e-ISBN-13 978-94-009-3839-7
DOII0.I007J978-94-009-3839-7
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
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All Rights Reserved
© 1987 by D. Reidel Publishing Company, Dordrecht, Holland
Soft cover reprint of the hardcover Is t edition 1987
No part of the material protected by this copyright notice may be reproduced or
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TABLE OF CONTENTS
Preface vii
ix
List of participants
xiii
Scientific program
J. Almeida and A. Azevedo, Implicit operations on certain
classes of semigroups
C. J. Ash, Finite idempotent-commuting semigroups 13
J.-C. Birget, S. Margolis, and J. Rhodes, Finite semigroups
whose idempotents commute or form a subsemigroup 25
K. Byleen, Inverse semigroups with countable universal semi-
lattices 37
M. P. Drazin, The identity element in inverse semigroup
algebras 43
J. A. Gerhard, Free bands and free *-bands 47
T. E. Hall, Finite inverse semigroups and amalgamation 51
J. M. Howie, Rank properties in semigroups of mappings 57
K. G. Johnston and P. R. Jones, Inverse semigroups whose
lattices of full inverse subsemigroups are modular 61
P. R. Jones, Basis properties, exchange properties and
embeddings in idempotent-free semi groups 69
W. Lex, Lattices of torsion theories for semi-automata 83
K. D. Magill, Jr., The' countability indices of certain trans-
formation semigroups 91
S. W. Margolis, J. C. Meakin, and J. B. Stephen, Some decision
problems for inverse monoid presentations 99
W. D. Munn, A class of inverse semigroup algebras 111
vi TABLE OF CONTENTS
W. R. Nico, Categorical extension theory -- revisited 121
F. Pastijn, A class of uniform chains 125
M. Petrich, Cayley theorems for semigroups 133
J. E. Pin, Power semigroups and related varieties of
finite semigroups 139
N. R. Reilly, On the lattice of varieties of completely
regular semigroups 153
J. Rhodes, New techniques in global semigroup theory 169
H. Straubing and D. Therien, Finite J-trivial monoids and
partially ordered mono ids 183
T. Tamura, On the recent results in the study of power
semigroups 191
B. Tilson, Type II redux 201
Index 207
PREFACE
Most papers published in this volume are based on lectures presented at
the Chico Conference on Semigroups held on the Chico campus of the Cal
ifornia State University on April 10-12, 1986. The conference was spon
sored by the California State University, Chico in cooperation with the
Engineering Computer Sciences Department of the Pacific Gas and Electric
Company. The program included seven 50-minute addresses and seventeen
30-minute lectures. Speakers were invited by the organizing committee
consisting of S. M. Goberstein and P. M. Higgins.
The purpose of the conference was to bring together some of the
leading researchers in the area of semigroup theory for a discussion
of major recent developments in the field. The algebraic theory of
semigroups is growing so rapidly and new important results are being
produced at such a rate that the need for another meeting was well
justified. It was hoped that the conference would help to disseminate
new results more rapidly among those working in semi groups and related
areas and that the exchange of ideas would stimulate research in the
subject even further. These hopes were realized beyond all expectations.
All talks presented at the conference were of high quality; together,
they covered a vast array of topics: semigroup algebras, transformation
semigroups, varieties and pseudovarieties of some classes of semigroups,
global semigroup theory, various topics in the theory of inverse semi
groups, semigroups presented by relations, and connections with languages
and automata. An enlightening lecture presented at the conference by
T. E. Hall about C. Ash's brilliant result that every finite monoid with
commuting idempotents is a morphic image of a subsemigroup of a finite
inverse semigroup stimulated J.-C. Birget, S. Margolis and J. Rhodes
to extend Ash's ideas and to prove that every finite semigroup whose
idempotents form a subsemigroup divides a finite orthodox semigroup.
These results show that the so-called "Type II Conjecture" stated by
J. Rhodes and B. Tilson in 1972 holds for finite monoids whose idempo
tents form a submonoid. B. Tilson wrote an article, specifically for
this volume, clarifying the proof of the original result that led to the
Type II Conjecture. This article is included in the Proceedings even
though it was not presented at the conference.
The papers in this volume are written in a variety of forms. Some
represent extensive surveys, others are written as regular research
articles. A number of contributions are research announcements whose
proofs should appear elsewhere. And several papers represent a mixture
of all three categories of articles. There is no need to discuss here
the contents of each contribution. The reader is advised to read them
all.
PREFACE
This conference would never have taken place if it had not been
generously financed by the Office of the Provost, the Graduate School
and the College of Natural Sciences of the California State University,
Chico and the Pacific Gas and Electric Company. The support of Provost
G. R. Stairs and Associate Vice President for Research J. S. Morgan was
especially vital. R. J. Bakke from the Graduate School worked hard
trying to raise external funds to support the conference and B. L. Lundy
from the College of Natural Sciences did a lot of work as the conference
accountant. T. A. McCready, Chairman of the Department of Mathematics,
constantly encouraged and supported me during the preparation of the
conference and the editing of its Proceedings. A. Bowman, S. Jones and
B. Stansbury, Mathematics Department secretaries, worked tirelessly
typing various materials related to the conference as well as parts of
this volume. The work and help of all these individuals is greatly
appreciated.
Shortly after the first announcement about the conference had been
mailed, my colleague and friend, P. M. Higgins, left Chico to take an
appointment at Deakin University. In spite of the great distance be
tween Chico and Geelong, Peter's interest in the conference never weak
ened. He arrived in Chico a week before the meeting and was eager to
help with the final preparations. For all that and for his willingness
to help me with editing some of the papers for this volume lowe him my
gratitude.
Last but not least I am especially grateful to my wife Faina for
her encouragement and understanding during the preparation of the meet
ing and the editing of this volume and for being such a gracious hostess
to the conference participants.
Chico, California, December 26, 1986 Simon M. Goberstein
LIST OF PARTICIPANTS
J. Almeida (Braga, Portugal)
J.-C. Birget (Lincoln, Nebraska)
R. Brandon (La Grande, Oregon)
S. Bulman-Fleming (Waterloo, Ontario)
K. Byleen (Milwaukee, Wisconsin)
D. Cowan (Burnaby, British Columbia)
M. Drazin (West Lafayette, Indiana)
M.D. Gass (Collegeville, Minnesota)
J.A. Gerhard (Winnipeg, Manitoba)
S.M. Goberstein (Chico, California)
M. Gould (Nashville, Tennessee)
T.E. Hall (Clayton, Victoria)
H. Hamilton (Sacramento, California)
D. Hardy (Fort Collins, Colorado)
K. Henckell (Willits, California)
P.M. Higgins (Chico, California and Geelong, Victoria)
J.A. Hildebrant (Baton Rouge, Louisiana)
J.M. Howie (St. Andrews, Scotland)
J. Iskra (Macon, Georgia)
P.R. Jones (Milwaukee, Wisconsin)
K. Johnston (Charleston, South Carolina)
x LIST OF PARTICIPANTS
G. Karpilovsky (Johannesburg, South Africa)
R.J. Koch (Baton Rouge, Louisiana)
W. Lex (Clausthal, West Germany)
K.D. Magill (Buffalo, New York)
E. Manes (Amherst, Massachusetts)
S. Margolis (Lincoln, Nebraska)
K. McDowell (Waterloo, Ontario)
J. Meakin (Lincoln, Nebraska)
D.W. Miller (Lincoln, Nebraska)
P. Misra (Staten Island, New York)
W.D. Munn (Glasgow, Scotland)
G. Naber (Chico, California)
W. Nico (Hayward, California)
T.D. Parsons (Chico, California)
F. Pastijn (Milwaukee, Wisconsin)
M. Petrich (Burnaby, British Columbia)
J.-E. Pin (Paris, France)
N.R. Reilly (Burnaby, British Columbia)
L. Renner (London, Ontario)
J. Rhodes (Berkeley, California)
H.E. Scheiblich (Columbia, South Carolina)
R. Spake (Davis, California)
J. Stephen (Lincoln, Nebraska)
H. Straubing (Boston, Massachusetts)
T. Tamura (Davis, California)
M.C. Thornton (Lincoln, Nebraska)
LlST OF PARTICIPANTS xi
B. Tilson (Sausalito, California)
E. Vought (Chico, California)
S. Wismath (Lethbridge, Alberta)
K. Yamaoka (Davis, California)