RELATION ALGEBRAS. Roger D. MADDUX. Department of Mathematics Iowa State University Ames, Iowa USA ELSEVIER
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1 RELATION ALGEBRAS Roger D. MADDUX Department of Mathematics Iowa State University Ames, Iowa USA ELSEVIER AMSTERDAM. BOSTON HEIDELBERG LONDON NEW YORK. OXFORD PARIS SAN DIEGO. SAN FRANCISCO. SINGAPORE. SYDNEY. TOKYO
2 Contents Preface List of Figures List of Tables vii xxiii xxv Chapter 1. Calculus of relations 1 1. De Morgan, Peirce, and Schroder 1 2. Binary relations 4 3. Complement and converse 5 4. Union and intersection 6 5. Relative multiplication and addition 7 6. More operations Four distinguished relations Axiomatization of the calculus of relations Definitions of relation algebras Undecidability and inexpressibility Incompleteness Representability Weakened associativity 31 Chapter 2. Set theory 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 e-relation Axioms of the calculus of relations 58
3 xviii CONTENTS 15. Kinds of relations Coextensivity Functional and injective relations Functional and injective parts Projection functions Boolean and relative operations on sets Relation Existence Theorem Axiom of Singletons Class Union Axiom Class Existence Theorem Lifting relations to sets Replacement Axiom Set Union Axiom Powerset Axiom Partial orderings, meets, joins, and lattices Axiom of Infinity Axiom of Choice Axiom of Regularity Ordinals and cardinals Dedekind-MacNeille completion 102 Chapter 3. General algebra 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 159 Chapter 4. Logic with equality Syntax Semantics Axiomatization and formalisms Formalisms of Tarski-Givant Soundness 194
4 CONTENTS xix 6. 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 230 Chapter 5. Boolean algebras Axioms R1-R 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 l Preservation theorems for complete extensions 285 Chapter 6. Relation algebras Boolean relation algebras Group relation algebras NA, WA, and SA Special kinds of elements Axioms R 7, R Axiom R Axioms R7, Axioms Rs, Axioms Re, Axioms R6 Rs, Rg R 7, Rs, R7, Rg, R.7, R-8 Rg, R
5 CONTENTS 11. Axioms R 5, Re, R7, R Axioms R 5, R 6, R7, Rs, Rg Axiom Rio 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~WA McKinsey's algebra in WA ~ SA An algebra in SA~ RA Lyndon's nonrepresentable algebras in RA ~ 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 Representation by complex algebras Elementary arithmetic in SA Associativity in groupoids Independence of seven weak associative laws Consequences of 4-associativity Relativization Ideals \ Ideal elements, relativization, and homomorphisms Simplicity Direct products Necessary subalgebras of SAs Elementary arithmetic in RA Functional elements Transitive and equivalence elements Forbidden matrices Equational basis for RA n Equational basis for RRA Representation theorems Cycles in structures Classification of simple finite algebras 420
6 CONTENTS xxi 56. Finite integral relation algebras with 0, 1, 2, or 3 atoms Finite integral relation algebras with 4 or 5 atoms Cycles of the algebras 1, Multiplication tables for algebras I Diversity cycles for the algebras Multiplication tables for the algebras Diversity cycles of the algebras lg Multiplication tables for algebras ls Failures of (J), (L), (M) among li-l Independence of (J), (L), and (M) dimensional relational basis data for 198 algebras Algebras of every dimension Flexible atoms Finite algebras with many automorphisms Splitting atoms RRA is not finitely based The number of finite integral relation algebras Many nonrepresentable relation algebras Algebras with few subalgebras Non-embeddable relation algebras Complex algebras of cycle structures Flexible systems of atoms Trails of matrices Singletons and twins in a simple SA Algebras from modular lattices Factor algebras A characterization of representability Complete representability '~ RRAs with no complete representations Point-density and pair-density Simple pair-dense algebras Complete representability results \ 521 Chapter 7. Algebraic logic Equipollence of C and C Inequipollence of x and 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 x 578
7 xxii CONTENTS Chapter finite integral relation algebras Cycles of algebras Ii3i6-1316i3i Cycles of algebras I3013-3OI33013, Failures of (J), (L), (M) among Ii3i6-1316i3i6 and I3013-3OI dimensional basis data for Ii3i6-1316i3i6 and I3013-3OI Bibliography 713 Index 723
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