Table Of ContentOrdered Algebraic Structures
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
CenJrefor Matherrultics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
F. CALOGERO, Universita degli Studi di Roma, Italy
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands
G.-C. ROTA, M.l.T., Cambridge, Mass., U.S A.
Volume 55
Ordered
Algebraic Structures
Proceedings of the
Caribbean Mathematics Foundation Conference
on Ordered Algebraic Structures,
Cura<;ao, August 1988
edited by
Jorge Martinez
Department of Mathematics, University of Florida,
Gainesville, U.S.A.
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
CarIbbean Mathematics Foundation Conference on Ordered Algebraic
Structures (1988 Princess Beach Hotel. Cura,ao)
Ordered algebraic structures: proceedings of the Caribbean
Mathematics Foundation Conference on Ordered Algebraic Structures.
Cura,ao. August 1988 / edited by Jorge Martinez.
p. cm. -- (Mathematics and Its applications)
1. Ordered algebraic structures--Congresses. I. Martinez. Jorge.
1945- II. Title. III. Series: Mathematics and its applications
(Kluwer Academic Publishers)
OA1n.C37 1988
512' .2--dc20 89-24396
ISBN-13: 978-94-010-7615-9 e-ISBN-13: 978-94-009-2472-7
DOl: 10.1 007/978-94-009-2472-7
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No part of the material protected by this copyright notice may be reproduced or
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This volume is dedicated to the memory of my father,
who would have enjoyed this event more than anyone else.
SERIES EDITOR'S PREFACE
'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the
je n'y serais point alit.' human race. It has put common sense back
Jules Verne where it belongs. on the topmost shelf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bcll
o. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nOD
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vii
viii SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series still aims at books dealing with:
a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
Now that a great deal of mathematics has been developed for all kinds of 'single' structures, the
time seems to have come to pay a good deal of attention to the case of 'composite' structures; i.e.
sets equipped with two (or more) different structures which are compatible in a suitable way. In the
present case an order structure together with an algebraic one, usually a group or ring.
In the preface of a greatly related volume in this series (A.M.W. Glass and W.Ch. Holland
(Eds.), Lattice-ordered Groups, KAP, 1989) I made a number of remarks on 'composite' structures
in algebra and noted that, though in most cases exploration of the consequences of compatibility
has only just begun, the case of lattice-ordered algebraic structures has reached a certain plateau of
sophistication and maturity. That does not imply in any way that nothing important is happening,
as the present result of a conference in Cura~o amply testifies.
One aspect of 'composite' structures seems to be that the compatibility tends to force all of the
components to be more regular, beautiful, symmetric. In these terms the question has, it seems to
me, hardly been explored at all. However, in the case that one of the components is order, it has
certainly become clear, at the Cura~o conference for instance, that the phenomena become very
different from what they would have been without such a relation imposing order.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
I'occasion de r':soudre des problemes ... eUe right than other people, but to be wrong for
nous fai t presscntir la solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, September 1989 Michiel Hazewinkel
CONTENTS
Preface xi
Conference Program xvii
CHAPTER ONE: GROUPS
Lattice-Ordered Groups of Divisibility: An Expository Introduc
tion.
MARLOW ANDERSON 3
Universal Aspects of the Theory of Lattice-Ordered Groups.
WAYNE B. POWELL 11
Recent Results on the Free Lattice-Ordered Group over a Right
Order able Group.
MICHAEL R. DARNEL 51
Non-Amalgamation and Conjugation for €-Metabelian Lattice
Ordered Groups.
A. M. W. GLASS 59
Periodic Extensions of Ordered Groups.
A. H. RHEMTULLA 65
The Gate Completion G: of a Lattice-Ordered Permutation Group.
RICHARD N. BALL
STEPHEN H. McCLEARY 71
Telling Left from Right.
STEPHEN H. McCLEARY 93
Applications of Spaces with Filters to Archimedean €-Groups with
Weak Unit.
RICHARD N. BALL
ANTHONY W. HAGER 99
Prime Ideal and Sikorski Extension Theorems for some €-Groups.
DANIEL GLUSCHANKOF 113
Some Applications of Definable Spine Analysis in Ordered Abelian
Groups.
F. LUCAS 123
x CONTENTS
CHAPTER TWO: RINGS
Ordered Algebraic Structures in Analysis.
W. A. J. LUXEMBURG 131
Rings of Continuous Functions from an Algebraic Point of View.
MELVIN HENRIKSEN 143
On Some Classes of Lattice-Ordered Algebras.
S. J. BERNAU
C. B. HUIJSMANS 175
Primary i-ideals in a class of f-rings.
SUZANNE LARSON 181
Epimorphisms of f-rings.
NIELS SCHWARTZ 187
Representation of a Real Polynomial f(X) as a Sum of 2m-th Pow
ers of Rational Functions.
M. BRADLEY
A. PRESTEL 197
Partially Ordered *-Division Rings.
JOHN DAUNS 209
Lattice-Ordered Division Rings Exist.
JOHN DAUNS 229
Vector-Lattices and a Problem in Geometry.
JAMES J. MADDEN 235
Banaschewski Functions and Ring-Embeddings.
R. H. REDFIELD 247
On Orderpotent Rings.
PIOTR WOJCIECHOWSKI 257
A Sampling of Open Problems 263
Subject Index 275
ORDER IN CURAQAO: A PREFACE
On the eighth of August, 1988, nearly three dozen mathematicians gathered
in Curac;ao, in the Netherlands Antilles, for a conference on Ordered Algebraic
Structures. For five consecutive days 26 speakers presented papers on various
aspects of that brand of mathematics which commonly falls under the label that
gave its title to the conference.
During these lectures a number of mathematical ideas and topics were dis
cussed; lattices, groups, rings of continuous functions, totally ordered groups,
lattice-ordered groups, f-rings, ordered fields, orderability of groups, groups
of order-preserving permutations, model-theoretic aspects of totally ordered
groups, amalgamations of ordered groups, structure topologies on ordered groups,
Riesz spaces and the theory of positive linear operators. The preceding list
should not be taken as complete; it is haphazard in its assemblage, by design.
There were mathematicians present who are frequently considered analysts; oth
eres who are categorized as topologists; still another group commonly thought of
as algebraists. A large number of the participants would, I should like to think,
resist being labelled by any of the three traditional disciplines of mathematics.
There were five problem sessions during the conference, which were reason
ably lively, and during which no evidence surfaced that the participants were
handicapped by the differences in their specialties or preferences. Instead, these
individuals extended themselves beyond the particularity and slant of their train
ing to converse intelligently with their collegues.
All of which suggest - in addition to the obvious fact that the conference
had brought together some outstanding men and women - that a number of
assumptions commonly made about Ordered Algebra are wrong, chief among
them the belief that the "field" is not going anywhere. Secondly, I should think
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