Fundamental Structures of Algebra and Discrete Mathematics

361 Pages·1994·3.98 MB·English

by  Stephan Foldes

FUNDAMENTAL STRUCTURES
OF ALGEBRA AND DISCRETE
MATHEMATICS
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FUNDAMENTAL
STRUCTURES OF
ALGEBRA AND DISCRETE
MATHEMATICS
STEPHAN FOLDES
A Wiley-lnterscience Publication
JOHN WILEY & SONS, INC.
New York / Chichester / Brisbane / Toronto / Singapore
This text is printed on acid-free paper.
Copyright ©1994 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work
beyond that permitted by Section 107 or 108 of the
1976 United States Copyright Act without the permission
of the copyright owner is unlawful. Requests for
permission or further information should be addressed to
the Permissions Department, John Wiley & Sons, Inc.
Library of Congress Cataloging in Publication Data:
Foldes, Stephan, 1951-
Fundamental structures of algebra and discrete mathematicsrt>y
Stephan Foldes.
p. cm.
"A Wiley-Interscicncc publication."
Includes bibliographical references and index.
ISBN 0-471-57180-6 (cloth)
1. Algebra. I. Title.
QA155.F65 1993
512'.02-dc20 93-8787
10 9 8 7 6 5 4 3 21
In token of my respect and gratitude
this book is dedicated to
llonka and Sandor Szasz
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CONTENTS
ILLUSTRATIONS xi
PREFACE xiii
I. SETS 1
1. Elementary Constructions and Axioms, 1
2. Cardinal and Ordinal Numbers, 11
3. Intersections, 21
Bibliography, 28
II. ORDERED SETS 31
1. Relations, Orders, and Zorn's Lemma, 31
2. Lattices and Closures, 43
3. Covering Relations, 49
4. Intersecting Convex Sets, 54
Bibliography, 56
vii
Vill CONTENTS
III. GROUPS 59
1. Binary Operations, Homomorphisms, and
Congruences, 59
2. Permutation Groups, 68
3. Integers and Cyclic Groups, 75
4. Alternating Groups, 82
Bibliography, 90
IV. RINGS 93
1. Ideals, 93
2. Polynomials, 104
3. Factorization and the Euclidean Algorithm, 111
Bibliography, 123
V. FIELDS 125
1. Rational and Real Numbers, 125
2. Galois Groups and Imaginary Roots, 137
Bibliography, 154
VI. VECTOR SPACES 157
1. Bases, 157
2. Linear Maps and Equations, 167
3. Affine and Projective Geometry, 176
4. Hyperplanes in Linear Programming, 183
5. Time and Speed in Special Relativity, 187
Bibliography, 194
VII. GRAPHS 197
1. Trees and Median Graphs, 197
2. Games, 202
CONTENTS iX
3. Chromatic Polynomials, 207
Bibliography, 210
VIII. LATTICES 213
1. Complements and Distributivity, 213
2. Boolean Algebra, 230
3. Modular and Geometric Lattices, 237
Bibliography, 242
IX. MATROIDS 245
1. Linear and Abstract Independence, 245
2. Minors and Tutte Polynomials, 252
3. Greedy Optimization Procedures, 262
Bibliography, 268
X. TOPOLOGICAL SPACES 269
1. Filters, 269
2. Closure, Convergence, and Continuity, 272
3. Distances and Entourages, 281
Bibliography, 288
XI. UNIVERSAL ALGEBRAS 291
1. Homomorphisms and Congruences, 291
2. Algebra of Syntax, 295
3. Truth and Formal Proof, 300
Bibliography, 308
XII. CATEGORIES 311
Bibliography, 324
INDEX OF DEFINITIONS 327
INDEX OF NOTATION 339
INDEX OF THEOREMS 343