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Home , Lattice (group), Lattice (order), Partially ordered group

The Theory ofLattice-Ordered Groups and Its Applications

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 307 The Theory of -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- 11 2.2 Calculations in I-groups 15 2.3 Basic facts . . . . 22

3 Convex I- 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 permutations of totally ordered sets . 53 4.3 Transitive ordered permutation groups ...... 57

v VI

4.4 Doubly transitive ordered 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 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 of I-varieties 255 11.1 The . .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 is an algebraic object having the structure of a group and the structure of a 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 led to the definition of ordered groups and ordered fields, the construc­ tion of non-archimedean 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 , for instance, in the theory of Bezout domains, ordered fields, and . 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 . 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 of po-sets, Ma (Q; E 1) 1.1 aE! --+ IT Ma lexicographic product of po-sets MaE! 1.1 aE! xVy the least upper bound of x, y 1.2 xl\y the greatest lower bound of x, y 1.2 A(!1) the set of all order permutations of !1 1.2 Z o-group of 2.1 Q o-group of rationals 2.1 R o-group of reals 2.1 x+ xVe 2.2 x xl\e 2.2 Ixl x V X-I 2.2 G~H l-groups G and H are l-isomorphic 2.3 G/H 2.3 AxB direct product of I-groups A and B 2.3 --+ IT Aa lexicographic product of I-groups 2.3 aE! IT Ga cartesian product of l-groups 2.3 aE! IT o, direct productof l-groups 2.3 aE! R(G :H) set of right of H in G 2.3 (M)[ convex l- of G generated M 2.3 C(G) the set of convex I-subgroups of G 3.1

Xlll xiv

Symbol (Listed in order of appearance) Section

c c y, /\ lattice operations in C(G) 3.1 SA(M) A-invariant subsemigroup 3.2 n(A) normalizer of A 3.2 c(A) centralizer of A 3.2 [A ,B] commutator subgroup A and B 3.2 [a, b] commutator a and b, a-1b-1ab 3.2 Va(g) value of 9 3.3 Va(g) (Va(g) , g)1 3.3 g.1 polar of element 9 3.4 Xl. polar of set X 3.4 P(G) boolean algebra of polars of G 3.4 'P 'P y,/\ lattice operations in P( G) 3.4 0 totally ordered set 4.1 A(O) the group of order permutations of 0 4.1 (G,O) ordered permutations group 4.1 convex G-congruence on 0 4.2 ~ G-block 4.2 LlJ the lazy subgroup 4.2 G(t>.) stabilizer of ~ 4.2 Gt>. stabilizer of ~ 4.2 Ga stabilizer of a 4.2 n Dedekind completion of n 4.2 supp(g) support of 9 4.3 ((k,(k) covering pair of G-congruences 4.5 R(O) elements of A(O) that live on the right 4.6 L(O) elements of A(O) that live on the left 4.6 B(O) bounded elements of A(O) 4.6 sgr(M) subsemigroup generated by M 5.1 K[G] group G 5.2 gp(M) subgroup or group generated by M 5.5 S(M) normal subsemigroup generated by M 6.1 x«y z" < y for all n EZ 6.2 xv

Symbol (Listed in order of appearence) Section t: class of I-groups 7.1 FdG, 71") lC-free I-group over po-group G 7.1 G[F] SP-hull of G 7.2 AWrB wreath product of groups A and B 7.3 AwrB restricted wreath product 7.3 +- XUY the order of po-sets X and Y 7.4 * I1 c'y free product of groups G-y(, E f) 7.5 -yEr /-* I1 Go l-free product of l-groups Go EA 7.5 oEA U(X) set of upper bounds of X 8.1 D(G) the distributive radical of G 8.2 xT thread of G 8.3 A l-variety of all abelian I-groups 9.1 n k I-variety of k-nilpotent I-groups 9.1 L the lattice of all I-varieties 9.1 Wa l-variety of weakly abelian I-groups 9.1 N l-variety of normal-valued l-groups 9.1 n I-variety of representable l-groups 9.1 var/G I-variety generated by l-group G 9.1 var/lC l-variety generated by class lC 9.1 locn class of locally nilpotent l-groups 9.5 Wp / q Feil I-variety 9.6 (G,D)Wr(H,T) wreath product of (G, D) and (H, T) 11.1 (G,D)wr(H, /IT) restricted wreath product of 11.1 Wrn(G,D) nth Wreath product of (G, D) 11.2 wrn(G, D) nth wreath product of (G, D) 11.2 e trivial l-variety 11.2 c I-variety of all l-groups 11.2 dimU of l-variety U 11.2

S·p Scrimger I-variety 12.3 Mo,M+,M- Medvedev's l-varieties 12.4 A the lattice of all l-quasivarieties 14.1 xvi

Symbol (Listed in order of appearance) Section

l-quasivariety generated by l-group G 14.1 the l-quasivariety defined by the implication (z" = yn) '* X = Y 14.2 Thv universal theory of G 14.3