Relation algebras

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Relation algebras / Roger D. Maddux

版:: 1st ed
資料種別:: 図書
出版情報:: Amsterdam ; Tokyo : Elsevier, 2006
形態:: xxvi, 731 p. ; 24 cm
シリーズ名:: Studies in logic and the foundations of mathematics ; v. 150 <BA00099329>
著者名:: Maddux, Roger D., 1948- <DA0448175X>
ISBN:: 9780444520135 [0444520139]
書誌ID:: BA77754096

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注記:

Includes bibliographical references (p. 713-722) and index

主題:

代数学

言語:

英語

目次情報:

Preface

List of Figures

List of Tables

Calculus of relations / Chapter 1：

De Morgan, Peirce, and Schroder / 1：

Binary relations / 2：

Complement and converse / 3：

Union and intersection / 4：

Relative multiplication and addition / 5：

More operations / 6：

Four distinguished relations / 7：

Axiomatization of the calculus of relations / 8：

Definitions of relation algebras / 9：

Undecidability and inexpressibility / 10：

Incompleteness / 11：

Representability / 12：

Weakened associativity / 13：

Set theory / Chapter 2：

Classes, equality, membership, sets, and proper classes

Language of set theory

An axiomatization of set theory

Axiom of Extensionality

Virtual classes, names, and notational concerns

Axiom of the Empty Set

Axiom of Complementation

Axiom of Intersection

Calculus of classes

Axiom of Unordered Pairs

Axiom of Relative Product

Axiom of Converse

Axiom of the [epsilon]-Relation

Axioms of the calculus of relations / 14：

Kinds of relations / 15：

Coextensivity / 16：

Functional and injective relations / 17：

Functional and injective parts / 18：

Projection functions / 19：

Boolean and relative operations on sets / 20：

Relation Existence Theorem / 21：

Axiom of Singletons / 22：

Class Union Axiom / 23：

Class Existence Theorem / 24：

Lifting relations to sets / 25：

Replacement Axiom / 26：

Set Union Axiom / 27：

Powerset Axiom / 28：

Partial orderings, meets, joins, and lattices / 29：

Axiom of Infinity / 30：

Axiom of Choice / 31：

Axiom of Regularity / 32：

Ordinals and cardinals / 33：

Dedekind-MacNeille completion / 34：

General algebra / Chapter 3：

Algebraic structures

Subalgebras

Congruence relations and quotients

Homomorphisms

Filters and ideals

Products of algebras

Operators S, H, I, P, Up

Assembly Lemma

Clones

Free algebras

Algebras of sets and relations

Proper relation algebras and RRA

Closure of RRA under subalgebras and products

Relational ideals

S and H commute on proper relation algebras

Peircean ideals

Closure of RRA under homomorphisms

Logic with equality / Chapter 4：

Syntax

Semantics

Axiomatization and formalisms

Formalisms of Tarski-Givant

Soundness

Deduction theorem

Implicational fragment

Completeness of (HI), (HII), (HIII')

Completeness of (LI)-(LIII)

Quantifier axioms

Equality axioms

Axioms for a binary relational language

Quotients of interpretations

Consistent and complete theories

Witnesses

Completeness and compactness

Boolean algebras / Chapter 5：

Axioms R[subscript 1]-R[subscript 3]

Partial orderings, completeness, atoms, density

Meets and joins of subsets

Ideals, filters, and ultrafilters

Functions between Boolean algebras

Congruence relations, ideals, filters, and homomorphisms

Complete additivity and multiplicativity

Completeness and atoms

Duals and conjugates

Regular-open BA of a closure operator

Regular-open BA of a topological closure operator

Topological spaces and closure operators

Complex algebra of a binary relation

Complete BA of a partial ordering

Completion of a BA

Perfect extension of a BA

Summary of constructions

Extending Boolean operators

Composing extended Boolean operators

Extending operators within a BA

Preservation theorems for complete extensions

Relation algebras / Chapter 6：

Boolean relation algebras

Group relation algebras

NA, WA, and SA

Special kinds of elements

Axioms R[subscript 7], R[subscript 8]

Axiom R[subscript 5]

Axioms R[subscript 7], R[subscript 8], R[subscript 9]

Axioms R[subscript 5], R[subscript 7], R[subscript 8], R[subscript 9]

Axioms R[subscript 6], R[subscript 7], R[subscript 9]

Axioms R[subscript 6], R[subscript 7], R[subscript 8], R[subscript 9]

Axioms R[subscript 5], R[subscript 6], R[subscript 7], R[subscript 9]

Axioms R[subscript 5], R[subscript 6], R[subscript 7], R[subscript 8], R[subscript 9]

Axiom R[subscript 10] with others

Theorem K and the cycle law

Special elements in NA

Characterizations of NA and RA

Duality for NA

Completions

Perfect extensions

Matrices of elements

Bases

Elementary arithmetic in WA

Properties of bases

n-dimensional relation algebras

Cycles of atoms

Complex algebras of ternary relations

The very nonassociative algebra in NA [similar] WA

McKinsey's algebra in WA [similar] SA

An algebra in SA [similar] RA

Lyndon's nonrepresentable algebras in RA [similar] RRA

Jonsson's algebras from projective geometries

Lyndon's algebras from projective geometries

McKenzie's nonrepresentable algebra

Allen's interval algebra

Cycle structures of complex algebras / 35：

Representation by complex algebras / 36：

Elementary arithmetic in SA / 37：

Associativity in groupoids / 38：

Independence of seven weak associative laws / 39：

Consequences of 4-associativity / 40：

Relativization / 41：

Ideals / 42：

Ideal elements, relativization, and homomorphisms / 43：

Simplicity / 44：

Direct products / 45：

Necessary subalgebras of SAs / 46：

Elementary arithmetic in RA / 47：

Functional elements / 48：

Transitive and equivalence elements / 49：

Forbidden matrices / 50：

Equational basis for RA[subscript n] / 51：

Equational basis for RRA / 52：

Representation theorems / 53：

Cycles in structures / 54：

Classification of simple finite algebras / 55：

Finite integral relation algebras with 0, 1, 2, or 3 atoms / 56：

Finite integral relation algebras with 4 or 5 atoms / 57：

Cycles of the algebras 1[subscript 37]-37[subscript 37] / 58：

Multiplication tables for algebras 1[subscript 37]-37[subscript 37] / 59：

Diversity cycles for the algebras 1[subscript 65]-65[subscript 65] / 60：

Multiplication tables for the algebras 1[subscript 65]-65[subscript 65] / 61：

Diversity cycles of the algebras 1[subscript 83]-83[subscript 83] / 62：

Multiplication tables for algebras 1[subscript 83]-83[subscript 83] / 63：

Failures of (J), (L), (M) among 1[subscript 1]-1[subscript 83] / 64：

Independence of (J), (L), and (M) / 65：

5-dimensional relational basis data for 198 algebras / 66：

Algebras of every dimension / 67：

Flexible atoms / 68：

Finite algebras with many automorphisms / 69：

Splitting atoms / 70：

RRA is not finitely based / 71：

The number of finite integral relation algebras / 72：

Many nonrepresentable relation algebras / 73：

Algebras with few subalgebras / 74：

Non-embeddable relation algebras / 75：

Complex algebras of cycle structures / 76：

Flexible systems of atoms / 77：

Trails of matrices / 78：

Singletons and twins in a simple SA / 79：

Algebras from modular lattices / 80：

Factor algebras / 81：

A characterization of representability / 82：

Complete representability / 83：

RRAs with no complete representations / 84：

Point-density and pair-density / 85：

Simple pair-dense algebras / 86：

Complete representability results / 87：

Algebraic logic / Chapter 7：

Equipollence of L and L[superscript +]

Inequipollence of L[superscript x] and L[superscript +]

Finite-variable formalisms

Algebras of formulas

Free RRAs of formulas

SAs and RAs of formulas

Algebraic semantics

Algebraic satisfaction and substitution

Algebraic soundness

Free SAs and RAs of formulas

Formalizing set theory in L[superscript x]

4329 finite integral relation algebras / Chapter 8：

Cycles of algebras 1[subscript 1316]-1316[subscript 1316]

Cycles of algebras 1[subscript 3013]-3013[subscript 3013]

Failures of (J), (L), (M) among 1[subscript 1316]-1316[subscript 1316] and 1[subscript 3013]-3013[subscript 3013]

5-dimensional basis data for 1[subscript 1316]-1316[subscript 1316] and 1[subscript 3013]-3013[subscript 3013]

Bibliography

Index

Preface

List of Figures

List of Tables

Calculus of relations / Chapter 1：

De Morgan, Peirce, and Schroder / 1：

Binary relations / 2：

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