Post-Modern Algebra

by Smith, Jonathan D. H.; Romanowska, Anna B.

• ISBN13: 9780471127383
• ISBN10: 0471127388
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-02-02
• Publisher: Wiley-Interscience

Summary

Advanced algebra in the service of contemporary mathematical research-- a unique introduction. This volume takes an altogether new approach to advanced algebra. Its intriguing title, inspired by the term postmodernism, denotes a departure from van der Waerden's Modern Algebra--a book that has dominated the field for nearly seventy years. Post-Modern Algebra offers a truly up-to-date alternative to the standard approach, explaining topics from an applications-based perspective rather than by abstract principles alone. The book broadens the field of study to include algebraic structures and methods used in current and emerging mathematical research, and describes the powerful yet subtle techniques of universal algebra and category theory. Classical algebraic areas of groups, rings, fields, and vector spaces are bolstered by such topics as ordered sets, monoids, monoid actions, quasigroups, loops, lattices, Boolean algebras, categories, and Heyting algebras. The text features: * A clear and concise treatment at an introductory level, tested in university courses. * A wealth of exercises illustrating concepts and their practical application. * Effective techniques for solving research problems in the real world. * Flexibility of presentation, making it easy to tailor material to specific needs. * Help with elementary proofs and algebraic notations for students of varying abilities. Post-Modern Algebra is an excellent primary or supplementary text for graduate-level algebra courses. It is also an extremely useful resource for professionals and researchers in many areas who must tackle abstract, linear, or universal algebra in the course of their work.

Author Biography

JONATHAN D. H. SMITH is Professor of Mathematics at Iowa State University. His research interests comprise algebra, combinatorics, and information theory, with applications in computer science, complex systems, physics, and biology. He has published more than sixty research papers and written or edited six books. ANNA B. ROMANOWSKA is Professor of Mathematics at Warsaw University of Technology. Her research interests include universal algebra, lattice theory, and logic, with applications in computer science and music theory. She has published fifty-five research papers and written or edited three books.

Table of Contents

PREFACE	ix
0 INTRODUCTION	1	(28)
1. Modern and Post-Modern Algebra	1	(1)
2. Algebra: The Central Discipline of Mathematics	1	(2)
3. Sets with Structure and Sets without Structure	3	(14)
3.1 Set Mappings	4	(3)
3.2 Cartesian Products and Disjoint Unions	7	(3)
3.3 Relations, Good Definition, and the First Isomorphism Theorem	10	(2)
3.4 Exponentiation, Quantifiers, and Negation	12	(3)
3.5 Ordered Sets and Induction	15	(2)
4. Semigroups and Monoids	17	(12)
4.1 Free Monoids and Codes	20	(2)
4.2 Dynamical Systems and Cyclic Monoids	22	(2)
4.3 Semilattices and Ordered Sets	24	(1)
4.4 Monoids of Relations	25	(4)
I GROUPS AND QUASIGROUPS	29	(82)
1. Monoid Actions	29	(22)
1.1 Automata	33	(3)
1.2 The Class of All Actions	36	(5)
1.3 Group Actions	41	(4)
1.4 Free Groups	45	(4)
1.5 Free Commutative Monoids and Partitions	49	(2)
2. Groups and Quasigroups	51	(21)
2.1 Multiplication Groups of Quasigroups	55	(4)
2.2 Divisions and Quasigroup Homomorphisms	59	(4)
2.3 Restriction and Induction	63	(2)
2.4 Quasigroup and Group Conjugacy Classes	65	(3)
2.5 Stability and the Class Equation	68	(4)
3. Symmetry	72	(14)
3.1 Permutation Groups	74	(4)
3.2 Symmetric and Alternating Groups	78	(4)
3.3 Sylov's Theorem, p-Groups and Simplicity	82	(4)
4. Loops, Nets and Isotopy	86	(25)
4.1 Inverse Properties and Moufang Loops	91	(5)
4.2 Loop Isotopes and Bol Loops	96	(4)
4.3 Right Loops and Loop Transversals	100	(4)
4.4 Loop Transversal Codes	104	(7)
II LINEAR ALGEBRA	111	(80)
1. General Algebra and Linear Algebra	111	(18)
1.1 Products and Coproducts of Abelian Groups	114	(4)
1.2 Matrices	118	(4)
1.3 Unital and Non-Unital Rings	122	(4)
1.4 Ideals, Fields and Domains	126	(3)
2. Vector Spaces and Modules	129	(29)
2.1 Duality and Transposed Matrices	133	(4)
2.2 Solving Linear Equations	137	(7)
2.3 Bases and Free Modules	144	(6)
2.4 Determinants and Dimension	150	(8)
3. Commutative Algebra	158	(33)
3.1 Fractions	164	(4)
3.2 Factors	168	(6)
3.3 Modules over Principal Ideal Domains	174	(7)
3.4 Linear Dynamical Systems	181	(6)
3.5 Elementary Field Theory	187	(4)
III CATEGORIES AND LATTICES	191	(90)
1. Posets, Monoids and Categories	191	(22)
1.1 Diagonalization and Large Categories	196	(4)
1.2 Functors and Concrete Categories	200	(5)
1.3 Commuting Diagrams, Epimorphisms and Monomorphisms	205	(4)
1.4 Natural Transformations and Functor Categories	209	(4)
2. Limits and Lattices	213	(31)
2.1 Products and Coproducts	218	(5)
2.2 Slice Categories	223	(5)
2.3 Equalizers and Pullbacks	228	(7)
2.4 Groups in Categories	235	(4)
2.5 Limits	239	(5)
3. Adjoint Functors	244	(37)
3.1 Adjunctions	250	(4)
3.2 Equivalence and Boolean Algebras	254	(6)
3.3 Galois Connections and Galois Theory	260	(6)
3.4 Continuity and Topology	266	(6)
3.5 Existence of Adjoints	272	(5)
3.6 Tensor Products of Modules	277	(4)
IV UNIVERSAL ALGEBRA	281	(78)
1. Sets with Operations	281	(22)
1.1 Operations and Types	282	(3)
1.2 The Isomorphism Theorems	285	(6)
1.3 Word Algebras	291	(5)
1.4 Universal Geometry	296	(4)
1.5 Clones and Relations	300	(3)
2. Varieties	303	(20)
2.1 Replication and Prevarieties	304	(6)
2.2 Bicompleteness	310	(3)
2.3 Satisfaction and Varieties	313	(5)
2.4 Entropic Algebras and Tensor Products	318	(5)
3. Algebraic Theories	323	(23)
3.1 Set-Valued Functors	327	(4)
3.2 Limits of Functors	331	(5)
3.3 Finitary Algebraic Theories	336	(4)
3.4 Theory Maps	340	(6)
4. Monads	346	(13)
4.1 Monads and Their Algebras	348	(4)
4.2 Monadic Adjunctions	352	(7)
INDEX	359