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The Theory ofLattice-Ordered Groups Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 307 The Theory of Lattice-Ordered Groups by V. M. Kopytov Institute ofMathematics, RussianAcademyof Sciences, Siberian Branch, Novosibirsk, Russia and N. Ya. Medvedev Altai State University, Bamaul, Russia Springer-Science+Business Media, B.Y A C.I.P. Catalogue record for this book is available from the Library ofCongress. ISBN 978-90-481-4474-7 ISBN 978-94-015-8304-6 (eBook) DOI 10.1007/978-94-015-8304-6 Printed on acid-free paper All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994. Softcover reprint ofthe hardcover Ist edition 1994 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 retrie val system, without written permission from the copyright owner. Contents Preface IX Symbol Index Xlll 1 Lattices 1 1.1 Partially ordered sets 1 1.2 Lattices .. ..... 3 1.3 Properties of lattices 5 1.4 Distributive and modular lattices. Boolean algebras 6 2 Lattice-ordered groups 11 2.1 Definition of the l-group 11 2.2 Calculations in I-groups 15 2.3 Basic facts . 22 3 Convex I-subgroups 31 3.1 The lattice of convex l-subgroups .......... .. 31 3.2 Archimedean o-groups. Convex subgroups in o-groups. 34 3.3 Prime subgroups 39 3.4 Polars ... ..................... 43 3.5 Lattice-ordered groups with finite Boolean algebra of polars ....................... 47 4 Ordered permutation groups 51 4.1 Holland's embedding theorem . 51 4.2 Groups of order permutations of totally ordered sets . 53 4.3 Transitive ordered permutation groups . 57 v VI 4.4 Doubly transitive ordered permutation group s 63 4.5 O-primitive ordered permutation l-groups , 67 4.6 Simple lattice-ordered groups 77 4.7 Examples of simple I-groups 86 5 Right-ordered groups 91 5.1 Definition of right-ordered group . 91 5.2 Representation of right-ordered groups by order permu- tations ................... 95 5.3 Convex subgroups in right-ordered group 99 5.4 Locally nilpotent right-ordered groups . 102 5.5 Right orderable groups . 107 6 Totally ordered groups 111 6.1 Tests for orderability of groups . 111 6.2 Weakly abelian total orders . 115 6.3 Total orders on Abelian groups of finite rank . 121 6.4 Orderable groups with finite number of total orders . 123 6.5 Fully orderable groups ................ 127 7 Embeddings of lattice-ordered groups 133 7.1 Embeddings into lattice-ordered groups . 133 7.2 Vector groups. Projectable l-groups . 137 7.3 Divisibility in l-groups . 144 7.4 Embeddings into simple l-groups . 148 7.5 Free products of I-groups . 150 7.6 Amalgamations of I-groups . 156 8 Lattice properties in lattice-ordered groups 161 8.1 Order completeness and Archimedean property in groups 161 8.2 Completely distributive l-groups , . 167 8.3 Compactness in I-groups . 177 8.4 Descending chain condition for positive elements .. 181 8.5 Lattice-ordered groups with finite orthogonal rank .. 182 9 Varieties of lattice-ordered groups 187 9.1 I-Varieties . 187 VII 9.2 Normal-valued I-groups . 195 9.3 Representable l-groups . .202 9.4 Weakly abelian l-groups .211 9.5 Torsion and radical classes .218 9.6 Finite basis problem .225 10 Free I-groups 237 10.1 Description of free l-groups . 237 10.2 Approximability of free I-groups . .244 10.3 Direct indecomposability of free l-groups .250 11 The semigroup of I-varieties 255 11.1 The wreath product . .255 11.2 Wreath products and extensions . 261 11.3 Wreath products and the semigroup of l-varieties . 268 11.4 The order on the semigroup of l-varieties ... .. 276 12 The lattice of I-varieties 283 12.1 Fundamental properties . 283 12.2 Identities for product and join of l-varieties . .284 12.3 The small l-varieties . .287 12.4 Solvable l-varieties . 305 12.5 Covers in the lattice of l-varieties . .311 12.6 Covers for l-variety of Abelian I-groups .319 12.7 Independent axiomatization of l-varieties .333 13 Ordered permutation groups and I-varieties 335 13.1 l-Varieties generated by A(O) . 335 13.2 l-Varieties generated by right-ordered groups . 338 13.3 l-Varieties generated by simple l-groups . .342 14 Quasivarieties of lattice-ordered groups 345 14.1 l-Quasivarieties .............. 345 14.2 The lattice of l-quasivarieties . .347 14.3 The universal theory of Abelian l-groups .357 14.4 Covers of A in the lattice of l-quasivarieties . 361 14.5 Covers in the lattice of l-quasivarieties ... .374 Vlll 14.6 Non-axiomatizability of lattice-ordered groups . 376 Bibliography 379 Index 397 Preface A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat­ ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al­ gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc­ tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R.Dedekind, a.Holder, D.Gilbert, B.Neumann, A.I.Mal'cev, P.Hall, G.Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam­ ple, partially ordered groups with interpolation property were intro­ duced in F.Riesz's fundamental paper [1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G.Birkhoff, H.Nakano and P.Lorenzen the theory of lattice-ordered groups became the subjec­ t of the study in its own rights. Recently, lattice-ordered groups have cropped up in many other areas of mathematics, for instance, in the theory of Bezout domains, ordered fields, and universal algebra. In 1963 W.Ch.Holland proved that every lattice-ordered group can IX x be represented as a group of order-preserving permutations of a totally ordered set . This result is the cornerstone in the theory of lattice­ ordered groups since it is the only tool available for studying arbitrary lattice-ordered groups. This theorem also provides an application of group-theoretical methods to l-groups. The purpose of this book is to make the deep results of theory of lattice-ordered groups accessible to mathematicians generally, to por­ tray its structure and indicate some of its applications to the group theory. This book is self contained for anyone familiar with the basic results on group theory. Graduate students and researchers in ordered structures and group theory will find this book valuable both an intro­ duction into the theory of partially ordered groups, and as presentation of new ideas and results in this theory. Chapters 1-3 contain preliminary material needed for reading the main body of the book. Here a great part of concepts and their basic properties is introduced. In Chapter 4, the theory of ordered permutation lattice-ordered groups is considered. Chapter 5 is devoted to the theory of right-ordered groups. In Chapter 6, the basic facts of the theory of totally ordered groups are proved. Some applications to group theory are mentioned here too. In Chapter 7 a lot of embedding theorems of lattice-ordered groups are proved. It is established that each lattice-ordered group G can be embedded into a divisible lattice-ordered group. Free products and amalgamation property of l-groups are dwelt with. Chapter 8 is concerned with the properties of lattice of lattice­ ordered groups such as: complete distributivity or compactness. Chapters 9 - 13 represent a systematic study of the structure of the lattice and the semigroup of varieties of lattice-ordered groups. The final Chapter 14 provides an exposition of the theory of quasi­ varieties of lattice-ordered groups. The statements are denoted by three numbers, e.g., Theorem 3.1.2. means: Chapter 3, Section 1, Theorem 2. They are referred to in full except when they appear in the same chapter; in such case the first two numbers are omitted. It is a pleasure to acknowledge the influence of research which stim­ ulated our interest in and research on ordered groups: A.I.Mal'cev's, xi M.I.Kargapolov's and A.I.Kokorin's works on totally ordered group­ s, W.Ch.Holland's and S.H.McCleary's works on ordered permutation lattice-ordered groups, P.Conrad's works on lattice-ordered groups, D.M.Smirnov's and P.Conrad's works on right-ordered groups. We found it expedient to use A.M.W.Glass's notation from his book [ 6 ], since it is best suited for our purposes. May 9 1993 Valeri M. Kopytov Nikolai Ya. Medvedev Institute of Mathematics Altai State University Siberian Branch RAN Novosibirsk Barnaul Russia Russia Symbol Index Symbol (Listed in order of appearance) Section IT Ma cartesian product of po-sets, Ma (Q; E 1) 1.1 aE! --+ IT Ma lexicographic product
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