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Relation Algebras, Volume 150 [Kõva köide]

(Department of Mathematics, Iowa State University, Ames, Iowa, 5001, USA)
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Relation Algebras, Volume 150Suurem pilt
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Püsilink: https://www.kriso.ee/db/9780444520135.html
Märksõnad:
The modern theory of algebras of binary relations, reformulated by Tarski as an abstract, algebraic, equational theory of relation algebras, has considerable mathematical significance, with applications in various fields: e.g., in computer science---databases, specification theory, AI---and in anthropology, economics, physics, and philosophical logic.

This comprehensive treatment of the theory of relation algebras and the calculus of relations is the first devoted to a systematic development of the subject.

Key Features:
- Presents historical milestones from a modern perspective
- Careful, thorough, detailed guide to understanding relation algebras
- Provides a framework and unified perspective of the subject

The modern theory of algebras of binary relations, reformulated by Tarski as an abstract, algebraic, equational theory of relation algebras, has considerable mathematical significance, with applications in various fields: e.g., in computer science---databases, specification theory, AI---and in anthropology, economics, physics, and philosophical logic.



This comprehensive treatment of the theory of relation algebras and the calculus of relations is the first devoted to a systematic development of the subject.



Key Features:



- Presents historical milestones from a modern perspective.


- Careful, thorough, detailed guide to understanding relation algebras.


- Includes a wealth of scholarly material --- 100 years of work by a research community --- presented in book form for the first time.


- Provides a framework and unified perspective of the subject.


- Roger D. Maddux is one of the world's leading experts in the field of relation algebras.

Key Features:



- Presents historical milestones from a modern perspective.


- Careful, thorough, detailed guide to understanding relation algebras.


- Includes a wealth of scholarly material --- 100 years of work by a research community --- presented in book form for the first time.


- Provides a framework and unified perspective of the subject.


- Roger D. Maddux is one of the world's leading experts in the field of relation algebras.

Arvustused

"An indispensable tool for scholars and research workers in mathematics and the mathematical sciences." --Mathmatical Reviews, 2007

"The book is an introduction to the calculus of relations and the theory of relation algebras (r.a.s): the reader need not have any preliminary knowledge of the subject [ T]he book contains very extensive material (the bibliography, in particular) both on relation algebras and from related areas and may serve as a handbook for a researcher." --ZentralblattMATH

Muu info

A modern perspective on the historical milestones of Relation Algebra
Preface vii
List of Figures
xxiii
List of Tables
xxv
Calculus of relations
1(34)
De Morgan, Peirce, and Schroder
1(3)
Binary relations
4(1)
Complement and converse
5(1)
Union and intersection
6(1)
Relative multiplication and addition
7(4)
More operations
11(1)
Four distinguished relations
12(2)
Axiomatization of the calculus of relations
14(6)
Definitions of relation algebras
20(5)
Undecidability and inexpressibility
25(1)
Incompleteness
25(4)
Representability
29(2)
Weakened associativity
31(4)
Set theory
35(86)
Classes, equality, membership, sets, and proper classes
35(1)
Language of set theory
35(1)
An axiomatization of set theory
36(3)
Axiom of Extensionality
39(1)
Virtual classes, names, and notational concerns
40(1)
Axiom of the Empty Set
41(1)
Axiom of Complementation
42(1)
Axiom of Intersection
43(1)
Calculus of classes
44(3)
Axiom of Unordered Pairs
47(2)
Axiom of Relative Product
49(3)
Axiom of Converse
52(4)
Axiom of the e-Relation
56(2)
Axioms of the calculus of relations
58(1)
Kinds of relations
58(4)
Coextensivity
62(4)
Functional and injective relations
66(3)
Functional and injective parts
69(6)
Projection functions
75(9)
Boolean and relative operations on sets
84(1)
Relation Existence Theorem
85(5)
Axiom of Singletons
90(2)
Class Union Axiom
92(1)
Class Existence Theorem
93(1)
Lifting relations to sets
94(1)
Replacement Axiom
95(1)
Set Union Axiom
96(2)
Powerset Axiom
98(1)
Partial orderings, meets, joins, and lattices
98(2)
Axiom of Infinity
100(1)
Axiom of Choice
101(1)
Axiom of Regularity
102(1)
Ordinals and cardinals
102(1)
Dedekind-MacNeille completion
102(19)
General algebra
121(46)
Algebraic structures
121(2)
Subalgebras
123(1)
Congruence relations and quotients
124(1)
Homomorphisms
125(2)
Filters and ideals
127(1)
Products of algebras
128(2)
Operators S, H, I, P, Up
130(1)
Assembly Lemma
131(1)
Clones
132(1)
Free algebras
133(8)
Algebras of sets and relations
141(2)
Proper relation algebras and RRA
143(5)
Closure of RRA under subalgebras and products
148(2)
Relational ideals
150(6)
S and H commute on proper relation algebras
156(2)
Peircean ideals
158(1)
Closure of RRA under homomorphisms
159(8)
Logic with equality
167(66)
Syntax
167(10)
Semantics
177(9)
Axiomatization and formalisms
186(3)
Formalisms of Tarski-Givant
189(5)
Soundness
194(1)
Deduction theorem
194(2)
Implicational fragment
196(4)
Completeness of (HI), (HII), (HIII')
200(5)
Completeness of (LI)--(LIII)
205(3)
Quantifier axioms
208(4)
Equality axioms
212(2)
Axioms for a binary relational language
214(3)
Quotients of interpretations
217(2)
Consistent and complete theories
219(2)
Witnesses
221(9)
Completeness and compactness
230(3)
Boolean algebras
233(56)
Axioms R1--R3
233(3)
Partial orderings, completeness, atoms, density
236(1)
Meets and joins of subsets
237(4)
Ideals, filters, and ultrafilters
241(1)
Functions between Boolean algebras
242(2)
Congruence relations, ideals, filters, and homomorphisms
244(1)
Complete additivity and multiplicativity
244(3)
Completeness and atoms
247(5)
Duals and conjugates
252(4)
Regular-open BA of a closure operator
256(1)
Regular-open BA of a topological closure operator
257(6)
Topological spaces and closure operators
263(2)
Complex algebra of a binary relation
265(2)
Complete BA of a partial ordering
267(3)
Completion of a BA
270(1)
Perfect extension of a BA
271(2)
Summary of constructions
273(1)
Extending Boolean operators
274(5)
Composing extended Boolean operators
279(4)
Extending operators within a BA
283(2)
Preservation theorems for complete extensions
285(4)
Relation algebras
289(238)
Boolean relation algebras
291(1)
Group relation algebras
292(1)
NA, WA, and SA
293(1)
Special kinds of elements
294(2)
Axioms R7, R8
296(5)
Axiom R5
301(1)
Axioms R7, R8, R9
302(1)
Axioms R5, R7, R8, R9
303(1)
Axioms R6, R7, R9
304(1)
Axioms R6, R7, R8, R9
305(1)
Axioms R5, R6, R7, R9
306(1)
Axioms R5, R6, R7, R8, R9
306(1)
Axiom R10 with others
307(2)
Theorem K and the cycle law
309(4)
Special elements in NA
313(6)
Characterizations of NA and RA
319(3)
Duality for NA
322(1)
Completions
323(1)
Perfect extensions
324(2)
Matrices of elements
326(4)
Bases
330(1)
Elementary arithmetic in WA
331(7)
Properties of bases
338(7)
n-dimensional relation algebras
345(5)
Cycles of atoms
350(4)
Complex algebras of ternary relations
354(3)
The very nonassociative algebra in NA ~ WA
357(1)
McKinsey's algebra in WA ~ SA
357(1)
An algebra in SA ~ RA
358(1)
Lyndon's nonrepresentable algebras in RA ~ RRA
358(1)
Jonsson's algebras from projective geometries
359(1)
Lyndon's algebras from projective geometries
359(1)
McKenzie's nonrepresentable algebra
360(2)
Allen's interval algebra
362(2)
Cycle structures of complex algebras
364(1)
Representation by complex algebras
365(1)
Elementary arithmetic in SA
366(4)
Associativity in groupoids
370(3)
Independence of seven weak associative laws
373(2)
Consequences of 4-associativity
375(4)
Relativization
379(2)
Ideals
381(1)
Ideal elements, relativization, and homomorphisms
382(1)
Simplicity
383(4)
Direct products
387(2)
Necessary subalgebras of SAs
389(4)
Elementary arithmetic in RA
393(1)
Functional elements
394(2)
Transitive and equivalence elements
396(2)
Forbidden matrices
398(3)
Equational basis for RAn
401(7)
Equational basis for RRA
408(3)
Representation theorems
411(6)
Cycles in structures
417(3)
Classification of simple finite algebras
420(3)
Finite integral relation algebras with 0, 1, 2, or 3 atoms
423(12)
Finite integral relation algebras with 4 or 5 atoms
435(1)
Cycles of the algebras 1/37--37/37
436(1)
Multiplication tables for algebras 1/37--37/37
437(2)
Diversity cycles for the algebras 1/65--65/65
439(1)
Multiplication tables for the algebras 1/65--65/65
440(4)
Diversity cycles of the algebras 1/83--83/83
444(2)
Multiplication tables for algebras 1/83--83/83
446(6)
Failures of (J), (L), (M) among 1/1--1/83
452(2)
Independence of (J), (L), and (M)
454(1)
5-dimensional relational basis data for 198 algebras
454(2)
Algebras of every dimension
456(2)
Flexible atoms
458(2)
Finite algebras with many automorphisms
460(11)
Splitting atoms
471(2)
RRA is not finitely based
473(3)
The number of finite integral relation algebras
476(4)
Many nonrepresentable relation algebras
480(2)
Algebras with few subalgebras
482(1)
Non-embeddable relation algebras
482(4)
Complex algebras of cycle structures
486(2)
Flexible systems of atoms
488(1)
Trails of matrices
489(8)
Singletons and twins in a simple SA
497(4)
Algebras from modular lattices
501(1)
Factor algebras
502(4)
A characterization of representability
506(6)
Complete representability
512(3)
RRAs with no complete representations
515(2)
Point-density and pair-density
517(1)
Simple pair-dense algebras
518(3)
Complete representability results
521(6)
Algebraic logic
527(56)
Equipollence of L and L+
527(3)
Inequipollence of Lx and L+
530(5)
Finite-variable formalisms
535(3)
Algebras of formulas
538(4)
Free RRAs of formulas
542(8)
SAs and RAs of formulas
550(7)
Algebraic semantics
557(2)
Algebraic satisfaction and substitution
559(5)
Algebraic soundness
564(10)
Free SAs and RAs of formulas
574(4)
Formalizing set theory in Lx
578(5)
4329 finite integral relation algebras
583(130)
Cycles of algebras 1/1316--1316/1316
583(29)
Cycles of algebras 1/3013--3013/3013
612(37)
Failures of (J), (L), (M) among 1/1316--1316/1316 and 1/3013--3013/3013
649(31)
5-dimensional basis data for 1/1316--1316/1316 and 1/3013--3013/3013
680(33)
Bibliography 713(10)
Index 723