Ordered Algebraic Structures: The 1991 Conrad Conference PDF
Preview Ordered Algebraic Structures: The 1991 Conrad Conference
ORDERED ALGEBRAIC STRUCTURES Ordered Algebraic Structures The 1991 Conrad Conference Edited by J. MARTINEZ Caribbean Mathematics Foundation, Department of Mathematics, Vniversity of Florida, Gainesville, Florida, V.SA. and C. HOLLAND Department of Mathematics and Statistics, Bowling Green State Vniversity, Bowling Green, Ohio, V.SA. SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-4755-5 ISBN 978-94-011-1723-4 (eBook) DOI 10.1007/978-94-011-1723-4 Printed an acid-free paper Ali Rights Reserved © 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover l st edition 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner. CONTENTS Preface vii Conference Program xi PART ONE: GROUPS AND VECTOR SPACES Isotone Projection Cones S. J. BERNAU 3 Torsion Classes of Vector Lattices P. F. CONRAD, S. M. LIN & D. G. NELSON 11 Disjoint Conjugate Chains MICHAEL R. DARNEL 31 Big Subgroups of Automorphism Groups of Doubly Homogeneous Chains A. M. W. GLASS & STEPHEN H. McCLEARY 51 Orderable Groups Satisfying an Engel Condition Y. K. KIM & A. H. RHEMTULLA 73 On Covers in the Lattice of Quasivarieties of I-Groups N. YA. MEDVEDEV 81 PART TWO: RINGS Ordered Rings of Generalized Power Series A. BENHISSI & P. RIBENBOIM 99 Natural Partial Orders on Division Rings with Involution JOHN DAUNS 111 Functorial Rings of Quotients I ANTHONY W. HAGER & JORGE MARTINEZ 133 Semiprime f-Rings that are Subdirect Products of Valuation Domains MELVIN HENRIKSEN & SUZANNE LARSON 159 Piecewise Polynomial Functions NIELS SCHWARTZ 169 Central f-Elements in Lattice-Ordered Algebras STUART A. STEINBERG 203 vi Archimedean Almost f-Algebras that Arise as Generalized Semigroup Rings PIOTR J. WOJCIECHOWSKI 225 A Characterization of Local-Global f-Rings SCOTr D. WOODWARD 235 Index 251 PRE F ACE From the 12th through the 14th of December, 1991, there was a conference on Ordered Algebraic Structures, at the University of Florida, Gaines ville, to celebrate the 70th birthday of Paul F. Conrad. The mathemati cians who came together for this event shared, in addition to their affinity for the mathematics of ordered algebra, a deep sense of friend ship and respect for Professor Conrad. Over the years, he influenced the lives of so many of those who were there, with his example and his refreshing directness. The papers which appear in these proceedings are taken from pre sentations made at the conference. The participants who joined me at the conference were: Marlow Anderson, Richard N. Ball, Simon Bernau, Roger Bleier, Ludmilla Bobek, Y.-Q. Chen, Alfred H. Clifford, Michael Darnel, John Dauns, Yi Dong, Laszlo Fuchs, Michele Giraudet, Daniel G1uschankof, Stephen Gunhouse, Anthony W. Hager, John Harvey, James B. Hart, Melvin Henriksen, W. Charles Holland, Herbert Hollister, David B. Kenoyer, V. M. Kopytov, Sunil Koswatta, Suzanne Larson, F. Lucas, James J. Madden, Steve Mc Cleary, N. Ya. Medvedev, David Nelson, Wayne B. Powell, Sybilla Priess Crampe, Robert H. Redfield, Akbar Rhemtulla, Paulo Ribenboim, Niels Schwartz, Stuart Steinberg, Constantine Tsinakis, Piotr Wojciechowski, Scott Woodward. Those are the facts. What remains to be related, in a preface which would do justice to the man to whom this volume is dedicated, is a very personal and biased history. Paul Conrad is singularly responsible for giving impetus to the study of ordered algebraic structures, from the point of view which took the ordering imposed on the structure as the focal point of the investi gation. It is fair to say that the paper on the Hahn Embedding Theorem for abelian lattice-ordered groups (with John Harvey and W. Charles Holland,) published in 1963, was seminal in the development of the theory of lattice-ordered groups and rings. Many of the concepts and ideas that Professor Conrad and several students and colleagues went on to exploit in the succeeding years find their first and full expo sition in this paper. But it was Paul Conrad himself who developed the technique of investigating a lattice-ordered group by studying its lattice of convex I-subgroups. In this line of research came the results on lex-sums, finite-valued I-groups and hyper-archimedean I-groups. During the first half of the seventies came the papers which systematic ally studied the various hulls of lattice-ordered groups. His papers on viii lateral completions certainly catalyzed the work of Roger Bleier, as well as the formidable article on the subject by Simon Bernau. Not to be forgotten in this resume of his accomplishments are the striking results on how the additive structure of a lattice-ordered group determines the vector-space scalar multiplications and ring-theoretic products one can define. The culminating effort in this line of research is the astonishing article on uniqueness of addition, with Michael Darnel. It would be sufficient to remember Paul Conrad for these accomplish ments, if there were no more to say. Except that he had students who pushed the investigation on ordered algebraic structures in a variety of directions: the study of order-preserving permutation groups on a poset; the study of universal algebraic concepts in ordered algebra; the appli cation of order-algebraic techniques in contexts where topology and analysis playa prominent role. In each of these excursions the ideas developed by Professor Conrad were crucial in defining their courses. If it is true that ordered algebra has natural origins in a variety of traditional disciplines, then several individuals who were influenced by his ideas, were instrumental in reconnecting ordered algebra to its origins, greatly enriched. Others yet who, from their own origins in mathematics, were drawn to the study of ordered algebra, if not with Professor Conrad himself, then with students and colleagues of his, borrowed liberally from the ideas he developed, and in the process showed us new and interesting connections between ordered algebra and not-so-traditional disciplines, notably model theory and theoretical computer science. Rather than cite his many co-authors, rather than give an ex haustive, but likely-as-not incomplete (and therefore unfair) list of those who were influenced by Paul Conrad, let this record, instead, be a witness to that influence, and to the diversity of the subject today. Several of us who have spent a good part of our mathematical lives studying ordered algebraic structures have at times remarked that ours is a reasonably congenial group, as they go in the mathematical commun ity. It has been suggested that Paul Conrad is an important reason for that congeniality, for his example as a teacher and colleague. Probably so. But if not, then surely it ~s true that we get together (for the mathematics, of course,) but also to propagate the lore of Conrad stories; (and embellish it, why not?) And since he is likely to keep friend and unsuspecting foe on their toes for years to come, there will be plenty of occasions for order-algebrists to gather; occasions to complement the self-evident mathematical imperatives, to be sure. I am pleased to acknowledge the debt owed to my institution, the Univer sity of Florida, for the financial support needed to realize the confer ence. To my department, a great vote of thanks, for the support of my colleagues, and the good will and graciousness of the staff. In parti cular, I wish to recognize JoAnne McLeary, who went the extra mile in making certain that the required paperwork was prepared on time. A heartfelt thanks to the participants and contributors, who made both the conference and this volume possible. To Kluwer Academic Publishers profound gratitude for undertaking to publish the proceedings of the conference, and for being helpful and congenial in overseeing the pre paration of the book. Finally, to my co-organizer and co-editor, W. Charles Holland, a salute, for being the perfect interlocutor and proving that this job can be done, simultaneously, with grace and dispatch. Jorge Martinez Gainesville, Florida December, 1992 ORDERED ALGEBRAIC STRUCTURES A Conference Dedicated to Professor Paul Conrad On the Occasion of his 70th Birthday 12th, 13th & 14th of December, 1991 Sponsored by the University of Florida Gainesville, Florida THURSDAY, 12TH DECEMBER Morning Session Alfred H. Clifford, Chair. 8.30 8.40 Introduction. 8.45 9.30 Paul Conrad, University of Kansas. TORSION CLASSES OF VECTOR LATl'ICES 9.45 - 10.30 Richard N. Ball, University of Denver. ORDER OUT OF CHAOS: 1-GROVPS IN TOPOLOGICAL DYNAMICS 10.45 - 11.30 Niels Schwartz, UniversitHt Passau. PIECEWISE POLYNOMIAL FUNCTIONS 11.45 - 12.05 James J. Madden, Louisiana State University. SEPARATING IDEALS AND THE REAL SPECTRUM Afternoon Session Laszlo Fuchs, Chair. 2.30 2.50 Suzanne Larson, Loyola Marymount University. f-RINGS IN WHICH EVERY PSEUDOPRlME IDEAL IS AN I-IDEAL 3.00 3.20 Scott Woodward, University of Florida. ON LOCAL-GLOBAL RINGS VS. ZERO-DIMENSIONAL SPACES Break 3.30 - 4.00 4.00 4.20 Robert H. Redfield, Hamilton College. EXPONENTIAL ALGEBRAIC STRUCTURES 4.30 - 4.50 John Dauns, Tulane University. PARTIALLY ORDERED DOMAINS 5.00 - 5.20 Paul Ribenboim, Queen's University. ORDER IN GENERALIZED POWER SERIES RINGS xi xii FRIDAY. 13TH DECEMBER Morning Session John Harvey. Chair. 8.30 9.15 Stephen H. McCleary. Bowling Green State University. SUBGROUPS OF A(~) 9.30 - 10.15 Michael R. Darnel, Indiana University at South Bend. THE CENTRAL ROLE OF THE REPRESENTABLE VARIETY OF LATl'ICE-DRDERED GROUPS 10.30 - 11.15 N. Ya. Medvedev, Altai State University. ON COVERS IN THE LATTICE OF QUASI-VARIETIES OF 1- GROUPS 11.30 - 11.50 Akbar H. Rhemtulla. University of Alberta. ORDERED ENGEL GROUPS Afternoon Session Roger Bleier. Chair. 2.00 - 2.20 Daniel Gluschankof, Universidad de Buenos Aires/Univer site de Paris VII. CYCLIC ORDERED GROUPS AND MV-ALGEBRAS 2.30 2.50 F. Lucas, Universite d'Angers. HYPER-REGULAR LATl'ICE-DRDERED GROUPS 3.00 3.20 Michele Giraudet. Universite du Mans. AUTOMORPHISM GROUPS OF CYCLIC ORDERS Break 3.30 - 4.00 4.00 - 4.20 Piotr Wojciechowski, University of Texas at El Paso. ARCHIMEDEAN ALMOST f-ALGEBRAS THAT ARISE AS GENERALIZED SEMIGROUP RINGS 4.30 - 4.50 Mel Henriksen, Harvey Mudd College. MORE ON RINGS IN WHICH EVERY ORDER IDEAL IS A RING IDEAL 5.00 - 5.20 Sybilla Priess-Crampe, UniversitHt Mnnchen. A FIXED-POINT THEOREM FOR ULTRAMETRIC SPACES WITH PARTIALLY ORDERED VALUE-SETS
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