Lukasiewicz-Moisil Algebras (eBook)

V. Boicescu, A. Filipoiu, G. Georgescu, S. Rudeanu (Autoren)

eBook Download: PDF

1991 | 1. Auflage
582 Seiten
Elsevier Science (Verlag)
978-0-08-086789-2 (ISBN)

Infos

The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.

This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr,-valued algebras are presented, as well as &THgr,-algebras with negation.

Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.

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The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and -valued algebras are presented, as well as -algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.

Inhalt

Front Cover 1
Lukasiewicz-Moisil Algebras 4
Copyright Page 5
Table of Contents 10
Preface 6
List of Symbols 14
Chapter 1. Lattices, Universal Algebra and Categories 18
$1. Posets and lattices 18
$2. Distributive lattices, De Morgan and Boolean algebras 35
$3. Filters, ideals and congruences in lattices 48
$4. Monadic and polyadic Boolean algebras 60
$5. Universal algebra 71
$6. Categories and functors 90
Chapter 2. Topological Dualities in Lattice Theory 100
$1. The Stone duality of distributive lattices 100
$2. The Stone duality of De Morgan algebras 109
$3. The Priestley duality of distributive lattices 112
$4. The Priestley duality of De Morgan algebras 119
Chapter 3. Elementary Properties of Lukasiewicz-Moisil Algebras 122
$1. Basic concepts 122
$2. Axiomatics of v-valued and n-valued algebras 141
$3. Axiomatics of three-valued algebras 148
$4. lrredundant n-valued Moisil algebras 172
Chapter 4. Connections With Other Classes of Lattices 182
$1. Post algebras 182
$2. Axled Lukasiewicz-Moisil algebras 194
$3. Heyting algebras 218
$4. Pseudocomplemented lattices 230
$5. Completeness properties and atomicity 245
$6. m-algebras 258
Chapter 5. Filters, Ideals and v-Congruences 264
$1. v-filters, v-ideals and v-congruences 264
$2. Prime filters 282
$3. The Cignoli problem 296
Chapter 6. Representation Theorems and Duality for Lukasiewicz-Moisil-Algebras 302
$1. The representation theorem of Moisil 302
$2. Applications of the representation theorem 334
$3. The representation of Lukasiewicz-Moisil algebras by continuous functions 334
$4. The representation of Lukasiewicz-Moisil algebras by Moisil fields of sets 343
$5. The Stone duality of v-valued Lukasiewicz-Moisil algebras 349
$6. The Stone duality of & valued Lukasiewicz-Moisil algebras with negation
$7. The Priestley duality of v-valued Lukasiewicz-Moisil algebras 358
$8. The Priestley duality of v-valued Lukasiewicz-Moisil algebras with negation 362
$9. The representation of n-valued Moisil algebras by three-valued algebras 366
Chapter 7. Categorical Properties of Lukasiewicz-Moisil Algebras 376
$1. Some adjoint functors 376
$2. Injective Lukasiewicz-Moisil algebras and injective hulls 388
$3. Free Lukasiewicz-Moisil algebras 397
$4. Epimorphisms and projective Lukasiewicz-Moisil algebras 406
$5. Direct sums 414
$6. Free Post and m-Post extensions 427
Chapter 8. Monadic and Polyadic Lukasiewicz-Moisil Algebras 434
$1. Monadic Lukasiewicz-Moisil algebras 434
$2. Modal operators on Lukasiewicz-Moisil algebras 445
$3. A construction of a three-valued Moisil algebra from a monadic Lukasiewicz-Moisil algebra 458
$4. Polyadic Lukasiewicz-MoisiI aIgebras 469
Chapter 9. Lukasiewicz Logics 476
$1. The Wajsberg axiomatization of the three-valued Lukasiewicz logic 476
$2. The Cignoli axiomatization of the n-valued Lukasiewicz logic 488
$3. The v-valued propositional calculus 504
$4. Analytic tableaux for the v-valued propositional calculus 522
$5. The v-valued predicate calculus 530
$6. Kripke-style semantics for v-valued predicate logics 548
Appendix: Applications to Switching Theory 556
Rrferences 568
Author Index 592
Subject Index 596

Produkt-Details

• Erscheint lt. Verlag: 13.5.1991
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Technik
• ISBN-10: 0-08-086789-8 / 0080867898
• ISBN-13: 978-0-08-086789-2 / 9780080867892