Universal Algebra, Algebraic Logic, and Databases

444 Pages·1994·29.04 MB·English

by  B. Plotkin (auth.)

Universal Algebra, Algebraic Logic, and Databases
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 272
Universal Algebra,
Algebraic Logic,
and Databases
by
B. Plotkin
Hebrew University,
Jerusalem, Israel
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Plotkin. B. 1. (Boris Isaakovich)
[ Un i ve r saI 'n a fa a 19 e b r a. a 19 e b rai c h e s k a fa log i k a iba z y dan n y k h .
Engl ishl
Universal algebra. algebraic logic. and databases I by B.
Plotkin.
p. cm. -- (Mathematics and its appl ications v. 272)
Includes bibliographical references and index.
ISBN 978-94-010-4352-6 ISBN 978-94-011-0820-1 (eBook)
DOI 10.1007/978-94-011-0820-1
1. Algebra. Universal. 2. Algebraic logic. 3. Data bases.
1. Title. II. Series: Mathematics and its applications (Kluwer
Academic Publ ishers) ; v. 272.
QA251.P6213 1994
005.74'01'512--dc2Q 93-44246
CIP
ISBN 978-94-010-4352-6
This is an updated and revised translation of the original Russian work
Universal Algebra, Algebraic Logic, and Databases,
1991 Nauka, Moscow,
Translated by 1. Cirulis, A. Nenashev and V. Pototsky
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
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,
inc1uding photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner.
Contents
PREFACE xiii
INTRODUCTION 1
o GENERAL VIEW ON OBJECTIVES AND CONTENTS
OF THE BOOK 3
1 Preliminary information on databases. Examples . . . . " 3
1.1 Examples and suggestive considerations. . . . . . .. 3
1.2 Setting of main problems. The relational approach.
Application of algebra. . . . . . . . . . . 9
2 Further examples. Network model .... . . . 14
2.1 Other examples of relational databases. 14
2.2 Network databases. . . . 16
2.3 Axioms of states. . . . . 19
3 Contents of the book: a review 20
3.1 Notes to Part 1. .... 20
3.2 The content of Part 2. . 21
3.3 Part 3. The model of a database. 22
I UNIVERSAL ALGEBRA 27
1 SETS, ALGEBRAS, MODELS 29
1 Sets .............. . 29
1.1 Sets, subsets and mappings. 29
1.2 Multiplication of mappings. 30
1.3 Cartesian product of sets. . 30
1.4 Free sum of sets. . ...... . 32
1.5 Characteristic functions of sets. 32
1.6 Binary relations. . . . . . . . . 33
1. 7 Equivalences........... 33
1.8 Quotient sets. . . . . . . . . . . 34
1.9 Fundamental mapping theorem. 34
1.10 Cardinality of a set. . ..... . 34
v
VI CONTENTS
1.11 Fuzzy sets. ............ . 35
2 Algebras and models . . . . . . . . . . . . 36
2.1 Algebraic operations and relations. 36
2.2 Algebras............... 36
2.3 Models and algebraic systems. 37
2.4 Homomorphisms between algebras and models. 37
2.5 Quotient algebras and models. 38
2.6 Homomorphism theorem ... 39
2.7 Sub algebras and submodels .. 39
2.8 Cartesian products. 40
2.9 Remak theorem. ... . . . . 40
2.10 Algebra of terms. ...... . 40
2.11 Generators and defining relations. 41
2.12 Classes of algebras and models; their axioms. 42
3 Many-sorted systems. ............ . 42
3.1 Set complexes. ....... . . . . . . 42
3.2 Many-sorted operations and relations. 44
3.3 Many-sorted algebras and models. 44
3.4 Algebras of many-sorted terms. 45
2 FUNDAMENTAL STRUCTURES 47
1 Definitions and examples. 47
1.1 Semigroups. . . . . . . . . . 47
1.2 Groups............ 48
1.3 Origins of groups and semigroups. 49
1.4 Quasigroups and loops. 49
1.5 Rings................. 50
1.6 Fields and skew-fields. . . . . . . . 51
1. 7 More examples ofrings and fields. The origins. 51
1.8 Linear spaces and modules. . . . . . . . 52
1.9 Associative linear algebras. ....... 54
1.10 Group algebras and semigroup algebras. 54
1.11 Other structures. . . . . . . 55
2 Homomorphisms. Free Algebras. . 56
2.1 Definition of a free algebra. 56
2.2 Semigroups. . 56
2.3 Groups............ 57
2.4 Rings............. 58
2.5 Linear spaces and modules. 59
2.6 Linear algebras. . . . . . . . 59
3 Some many-sorted structures . . . 60
3.1 Representations of groups and semigroups. 60
3.2 Linear representations. . . . . . . . 60
3.3 Automata.............. 61
3.4 Affine spaces and affine automata. 62
CONTENTS Vll
3 CATEGORIES 65
1 General information and examples 65
1.1 Definition of a category. .. 65
1.2 Examples........... 66
1.3 Subcategories........ 67
1.4 Monomorphisms, epimorphisms and isomorphisms. 67
1.5 Duality......................... 69
1.6 Functors... '. . . . . . . . . . . . . . . . . . . . . 69
1. 7 Natural transformations of functors. Categories of
functors .......... . 71
1,8 Equivalence of categories. . . . . . . . . . . . . . . . 73
2 Some technical notions . . . . . . . . .. . . . . . . . . . . . 73
2.1 Universal objects.. ................. . 73
2.2 Direct and free products (products and coproducts). 74
2.3 Other examples of universal objects. 76
2.4 Tensor products of modules .. 77
2.5 Adjoint functors. . ...... , .. . 79
2.6 Cones, equalizers, and limits. . .. . 83
4 THE CATEGORY OF SETS. TOPOl. FUZZY SETS 85
1 Further general concepts. ............... . 85
1.1 Remarks on the category of sets. . , .... , . 85
1.2 Amalgams and coamalgams. Cartesian squares. 86
1.3 Completeness and co completeness. 90
1.4 Exponentiation ....... . 90
1.5 Cartesian closed categories. 93
2 Topoi, .... , ......... . 94
2.1 Subobjects.. ....... . 94
2.2 Elements. Names of arrows. 95
2.3 Subobject classifier. ..,. 96
2.4 Definition of a topos .... . 99
2.5 Power objects. . ..... . 99
2.6 General remarks on topoi. Well-pointed topoi. 101
2.7 Examples of topoi ................ . 102
2.8 Operations with subobjects. Heyting algebras. 106
2.9 The Heyting algebra of subobjects. Boolean topoi. 107
3 Fuzzy sets. Miscellany . . . . . . . . . . . 108
3.1 Fuzzy sets and fuzzy quotient sets, 108
3.2 The category of fuzzy sets. 110
3.3 The top os of fuzzy sets. . ..... 111
3.4 Remarks on the foundations. History. . 113
5 VARIETIES OF ALGEBRAS. AXIOMATIZABLE CLA-
SSES 115
1 Varieties., ..... ,.......... 115
1.1 Closed classes and free algebras. 115
1.2 Classes and identities. 118
1.3 Birkhoff theorem. . 120
1.4 Verbal functions. . , . 120
Vlll CONTENTS
2 Some constructions .................... . 123
2.1 Free products in varieties. ............ . 123
2.2 Amalgams...................... 123
2.3 Epimorphisms and monomorphisms in varieties. 124
3 Axiomatic classes of algebras '" . . . . 125
3.1 General remarks. ................. . 125
3.2 Reduced products. . . . . . . . . . . . . . . . . . 126
3.3 Quasivarieties and pseudovarieties ........ . 127
6 CATEGORY ALGEBRA AND ALGEBRAIC THEO-
RIES 129
1 Clones, and clones of operations . 129
1.1 Clones of operations. .. . 129
1.2 Abstract clones. . ... . 131
1.3 Clones and free algebras. . 131
1.4 Representations of clones and varieties. 133
2 Algebraic theories. . . . . . . . . . . . . . . . . 139
2.1 Clones and categories. . ........ . 139
2.2 Algebras as functors. . . . . . . . . . .. . 140
2.3 Algebraic theories and varieties of algebras. 143
2.4 Additional remarks. . ........... . 147
II ALGEBRAIC LOGIC 153
7 BOOLEAN ALGEBRAS AND PROPOSITIONAL CAL-
CULUS 155
1 Boolean algebras. . . . . . . . . . . . . . . . 155
1.1 Boolean algebras, rings and lattices. 155
1.2 Homomorphisms, ideals and lattices. 159
1.3 }<ree Boolean algebras. . . . . . . . . 163
1.4 Finite Boolean algebras. . . . . . . . 165
2 Propositional calculus and Boolean algebras . 166
2.1 Propositional calculus. . . . . . . . . . . 166
2.2 The Lindenbaum-Tarski algebra of propositional cal-
culus. . ........................ . 167
2.3 Consistence, compatibility and models ........ . 169
8 HALMOS ALGEBRAS AND PREDICATE CALCULUS 173
1 Halmos algebras .............. . 173
1.1 Quantifiers and quantifier algebras. . 173
1.2 Halmos algebras. . . . . . . . . . . . 177
1.3 Supports of elements. ....... . 180
2 Halmos algebras of predicate calculus. . . . 182
2.1 Predicate calculus ........... . 182
2.2 Halmos algebra of predicate calculus .. 184
3 Halmos equality algebras. Cylindric algebras 189
3.1 Equality in Halmos algebras. 189
3.2 Cylindric algebras. . ......... . 193
CONTENTS IX
4 Homomorphisms and structure of Halmos algebras. Addi-
tional remarks ................... 194
4.1 Homomorphisms, ideals and filters. . . . . 194
4.2 Simple algebras. Semisimplicity theorem. 196
4.3 Constants, terms and predicates. 197
4.4 Other remarks. . . . . . . . . . . . . . . . 199
9 SPECIALIZED HALMOS ALGEBRAS 201
1 Halmos algebras over a variety of universal algebras 201
1.1 Axiomatics and examples. . . . . 201
1.2 General information. . . . . . . . . . . . 204
1.3 Equality in specialized algebras. .... 209
2 Halmos algebra over a free algebra of a variety 209
2.1 Supports of elements of a free algebra. . 209
2.2 Specialized algebra of formulas. . . . . . 212
2.3 Halmos algebra over the free algebra of a variety. 215
3 Modification ofthe scheme .......... 219
3.1 Changing the variety. ......... 219
3.2 The kernel of a passage to subvariety. 224
3.3 Additional remarks . . . . . . . . . . . 228
10 CONNECTIONS WITH MODEL THEORY 229
1 Existence of models .... 229
1.1 Preliminaries.... 229
1.2 The main theorem. . 232
1.3 Additional remarks . 234
2 Miscellany . . . . . . . . . . 238
2.1 Consistency, compatibility and completeness. 238
2.2 Certain applications of the model existence theorem. 240
2.3 Classes and filters. The knowledge base of a model. 241
11 THE CATEGORIAL APPROACH TO ALGEBRAIC LO-
GIC 247
1 Relation algebras . . . . . . . . . . . . 247
1.1 Notes on quantifiers. . . . . . . 247
1.2 Definition of relation algebras. 250
1.3 Another approach. . . . . . . . 255
2 Relational algebras associated with Halmos algebras 258
2.1 The principal construction. 258
2.2 Algebras of relations. . 267
2.3 Additional remarks. . . . . 269
3 Generalizations........... 270
3.1 Generalized relational algebras. 270
3.2 Intuitionistic logic and models in toposes. 271
x CONTENTS
III DATABASES - ALGEBRAIC ASPECTS 275
12 ALGEBRAIC MODEL OF A DATABASE 277
1 Universal databases ..... :-. . . . . . 277
1.1 Definition of a universal database. 277
1.2 Functor property. . . . . . . . . . . 279
1.3 Additional notes. . . . . . . . . . . 281
2 The model. . . . . . . . . . . . . . . . . . 284
2.1 Definition of a database. . . . . . . . . . . . . . 284
2.2 Homomorphisms and the category of databases. 288
3 Dynamic databases . . . . . . . . . 295
3.1 Preliminary notes. . . . . . 295
3.2 Dynamic Halmos algebras. 297
3.3 Dynamic databases. .... 298
4 Generalizations........... 301
4.1 Non-deterministic action.. ..... ....... 301
4.2 Databases based on cylindric and relational algebras. 302
4.3 Databases with fuzzy information. . . . . . . . . .. 303
13 EQUIVALENCE AND REORGANIZATION OF DATABA-
SES 305
1 More about homomorphisms ............. 305
1.1 Replacement of a scheme. . . . . . . . . . . . 305
1.2 Canonical decomposition of homomorphism. . 309
1.3 Additional comments. . . 311
2 Equivalence and reconstruction . . . 311
2.1 Basic information. . . . . . . 311
2.2 Equivalence of databases. . . 312
2.3 Reorganization of a database. 313
2.4 Modification of axioms. " . 315
2.5 Reconstruction of scheme. . . 316
3 Functional dependencies of relations 319
3.1 Structure of functional dependences. 319
3.2 Modification of the attribute set. . . . 321
3.3 Axioms and functional dependencies. . 322
4 Changing states. Storage and cleaning. . 323
4.1 Preliminary remarks. . . . . . . . 323
4.2 Relations and partial multimaps. . . 324
4.3 Storage................. 325
4.4 Cleaning................ 325
4.5 Interaction of cleaning and storage. . 327
14 SYMMETRIES OF RELATIONS AND GALOIS THEORY
OF DATABASES 329
1 The Galois theory of relational algebras. Preliminaries 330
1.1 Automorphisms of a relational algebra. 330
1.2 Galois connection. . . . . . . . . 334
1.3 Basic results. . . . . . . . . . . . 335
2 Proofs in the pure Halmos algebra case. . . . . 336