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Universitext Universitext Editorial Board (North America): S. Axler K.A. Ribet T.S. Blyth Lattices and Ordered Algebraic Structures T.S. Blyth Emeritus Professor School of Mathematics and Statistics Mathematical Institute University of St Andrews North Haugh St Andrews KY16 9SS UK Editorial Board (North America): S. Axler K.A. Ribet Mathematics Department Department of Mathematics San Francisco State University University of California at Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Mathematics Subject Classification (2000): 06-01, 20-01, 06A05, 06A06, 06A12, 06A15, 06A23, 06A99, 06B05, 06B10, 06B15, 06C05, 06C10, 06C15, 06D05, 06D10, 06D15, 06D20, 06E05, 06E10, 06E15, 06E20, 06E25, 06E30, 06F05, 06F15, 06F20, 06F25, 12J15, 20M10, 20M15, 20M17, 20M18, 20M19, 20M99 British Library Cataloguing in Publication Data Blyth, T. S. (Thomas Scott) Lattices and ordered algebraic structures 1. Lattice theory 2. Ordered algebraic structures I. Title 511.3′3 ISBN 1852339055 Library of Congress Cataloging-in-Publication Data Blyth, T. S. (Thomas Scott) Lattices and ordered algebraic structures / T.S. Blyth. p. cm. — (Universitext) Includes bibliographical references and index. ISBN 1-85233-905-5 1. Ordered algebraic structures. 2. Lattice theory. I. Title. QA172.B58 2005 511.3′3—dc22 2004056612 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case or reprographic reproduc- tion in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning repro- duction outside those terms should be sent to the publishers. ISBN 1-85233-905-5 Springer Science+Business Media springeronline.com © Springer-Verlag London Limited 2005 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera-ready by author Printed in the United States of America 12/3830-543210 Printed on acid-free paper SPIN 10994870 Preface The notion of an order plays an important rˆole not only throughout mathe- matics but also in adjacent disciplines such as logic and computer science. The purpose of the present text is to provide a basic introduction to the theory of ordered structures. Taken as a whole, the material is mainly designed for a postgraduate course. However, since prerequisites are minimal, selected parts of it may easily be considered suitable to broaden the horizon of the advanced undergraduate. Indeed, this has been the author’s practice over many years. A basic tool in analysis is the notion of a continuous function, namely a mapping which has the property that the inverse image of an open set is an open set. In the theory of ordered sets there is the corresponding concept of a residuated mapping, this being a mapping which has the property that the inverse image of a principal down-set is a principal down-set. It comes there- fore as no surprise that residuated mappings are important as far as ordered structures are concerned. Indeed, albeit beyond the scope of the present ex- position, the naturality of residuated mappings can perhaps best be exhibited using categorical concepts. If we regard an ordered set as a small category then an order-preserving mapping f : A → B becomes a functor. Then f is residuated if and only if there exists a functor f + : B → A such that (f,f+) is an adjoint pair. Residuated mappings play a central rˆole throughout this exposition, with fundamental concepts being introduced whenever possible in terms of natural properties of them. For example, an order isomorphism is precisely a bijection that is residuated; an ordered set E is a meet semilattice if and only if, for every principal down-set x↓, the canonical embedding of x↓ into E is residu- ated; and a Heyting algebra can be characterised as a lattice-based algebra in which every translation λx : y → x ∧ y is residuated. The important notion of a closure operator, which arises in many situations that concern ordered sets, is intimately related to that of a residuated mapping. Likewise, Galois connections can be described in terms of residuated mappings, and vice versa. Residuated mappings have the added advantage that they can be composed to form new residuated mappings. In particular, the set Res E of residuated mappings on an ordered set E forms a semigroup, and here we include de- scriptions of the types of semigroup that arise. VI Preface A glance at the list of contents will reveal how the material is marshalled. Roughly speaking, the text may be divided into two parts though it should be stressed that these are not mutually independent. In Chapters 1 to 8 we deal with the essentials of ordered sets and lattices, including boolean algebras, p-algebras, Heyting algebras, and their subdirectly irreducible algebras. In Chapters 9 to 14 we provide an introduction to ordered algebraic structures, including ordered groups, rings, fields, and semigroups. In particular, we in- clude a characterisation of the real numbers as, to within isomorphism, the only Dedekind complete totally ordered field, something that is rarely seen by mathematics graduates nowadays. As far as ordered groups are concerned, we develop the theory as far as proving that every archimedean lattice-ordered group is commutative. In dealing with ordered semigroups we concentrate mainly on naturally ordered regular and inverse semigroups and provide a unified account which highlights those that admit an ordered group as an im- age under a residuated epimorphism, culminating in structure theorems for various types of Dubreil-Jacotin semigroups. Throughout the text we give many examples of the structures arising, and interspersed with the theorems there are bundles of exercises to whet the reader’s appetite. These are of varying degrees of difficulty, some being designed to help the student gain intuition and some serving to provide further examples to supplement the text material. Since this is primarily designed as a non-encyclopaedic introduction to the vast area of ordered structures we also include relevant references. We are deeply indebted to Professora Doutora Maria Helena Santos and Professor Doutor Herberto Silva for their assistance in the proof-reading. The traditional free copy is small recompense for their labour. Finally, our thanks go to the editorial team at Springer for unparalleled courtesy and efficiency. St Andrews, Scotland T.S. Blyth Contents 1 Ordered sets; residuated mappings......................... 1 1.1 Theconceptofanorder.................................. 1 1.2 Order-preserving mappings ............................... 5 1.3 Residuatedmappings.................................... 6 1.4 Closures................................................ 10 1.5 Isomorphismsoforderedsets.............................. 12 1.6 Galoisconnections....................................... 14 1.7 Semigroupsofresiduatedmappings........................ 15 2 Lattices; lattice morphisms ................................ 19 2.1 Semilattices and lattices .................................. 19 2.2 Down-set lattices ........................................ 23 2.3 Sublattices ............................................. 26 2.4 Lattice morphisms ....................................... 28 2.5 Complete lattices ........................................ 29 2.6 Baersemigroups......................................... 35 3 Regular equivalences ...................................... 39 3.1 Orderingquotientsets ................................... 39 3.2 Stronglyupperregularequivalences........................ 41 3.3 Lattice congruences ...................................... 45 4 Modular lattices ........................................... 49 4.1 Modularpairs;Dedekind’smodularitycriterion.............. 49 4.2 Chainconditions........................................ 54 4.3 Join-irreducibles......................................... 58 4.4 Baersemigroupsandmodularity .......................... 61 5 Distributive lattices ....................................... 65 5.1 Birkhoff’s distributivity criterion .......................... 65 5.2 Moreonjoin-irreducibles................................. 69 5.3 Primeidealsandfilters................................... 72 5.4 Baer semigroups and distributivity ........................ 74 VIII Contents 6 Complementation; boolean algebras ....................... 77 6.1 Complementedelements.................................. 77 6.2 Uniquely complemented lattices ........................... 78 6.3 Booleanalgebrasandbooleanrings........................ 82 6.4 Booleanalgebrasofsubsets............................... 86 6.5 The Dedekind–MacNeille completion of a boolean algebra .... 90 6.6 Neutralandcentralelements.............................. 92 6.7 Stone’s representation theorem ............................ 95 6.8 Baer semigroups and complementation ..................... 97 6.9 Generalisationsofbooleanalgebras........................101 7 Pseudocomplementation; Stone and Heyting algebras ......103
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