RELATION ALGEBRAS Roger D. MADDUX Department of Mathematics Iowa State University Ames, Iowa 50011 USA ELSEVIER AMSTERDAM. BOSTON HEIDELBERG LONDON NEW YORK. OXFORD PARIS SAN DIEGO. SAN FRANCISCO. SINGAPORE. SYDNEY. TOKYO
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 11 7. Four distinguished relations 12 8. Axiomatization of the calculus of relations 14 9. Definitions of relation algebras 20 10. Undecidability and inexpressibility 25 11. Incompleteness 25 12. Representability... 29 13. Weakened associativity 31 Chapter 2. Set theory 35 1. Classes, equality, membership, sets, and proper classes. 35 2. Language of set theory 35 3. An axiomatization of set theory 36 4. Axiom of Extensionality 39 5. Virtual classes, names, and notational concerns 40 6. Axiom of the Empty Set 41 7. Axiom of Complementation 42 8. Axiom of Intersection 43 9. Calculus of classes 44 10. Axiom of Unordered Pairs 47 11. Axiom of Relative Product 49 12. Axiom of Converse 52 13. Axiom of the e-relation 56 14. Axioms of the calculus of relations 58
xviii CONTENTS 15. Kinds of relations 58 16. Coextensivity 62 17. Functional and injective relations 66 18. Functional and injective parts 69 19. Projection functions 75 20. Boolean and relative operations on sets 84 21. Relation Existence Theorem 85 22. Axiom of Singletons 90 23. Class Union Axiom 92 24. Class Existence Theorem 93 25. Lifting relations to sets 94 26. Replacement Axiom 95 27. Set Union Axiom 96 28. Powerset Axiom 98 29. Partial orderings, meets, joins, and lattices 98 30. Axiom of Infinity 100 31. Axiom of Choice 101 32. Axiom of Regularity 102 33. Ordinals and cardinals 102 34. Dedekind-MacNeille completion 102 Chapter 3. General algebra 121 1. Algebraic structures 121 2. Subalgebras 123 3. Congruence relations and quotients 124 4. Homomorphisms 125 5. Filters and ideals 127 6. Products of algebras - 128 7. ' -Operators S, H, I, P, Up 130 8. Assembly Lemma 131 9. Clones 132 10. Free algebras ; 133 11. Algebras of sets and relations ' 141 12. Proper relation algebras and RRA 143 13. Closure of RRA under subalgebras and products 148 14. Relational ideals 150 15. S and H commute on proper relation algebras 156 16. Peircean ideals 158 17. Closure of RRA under homomorphisms 159 Chapter 4. Logic with equality 167 1. Syntax 167 2. Semantics 177 3. Axiomatization and formalisms 186 4. Formalisms of Tarski-Givant 189 5. Soundness 194
CONTENTS xix 6. Deduction theorem 194 7. Implicational fragment 196 8. Completeness of (HI), (HII), (HIII') 200 9. Completeness of (LI)-(LIII) 205 10. Quantifier axioms 208 11. Equality axioms 212 12. Axioms for a binary relational language 214 13. Quotients of interpretations 217 14. Consistent and complete theories 219 15. Witnesses 221 16. Completeness and compactness 230 Chapter 5. Boolean algebras 233 1. Axioms R1-R3 233 2. Partial orderings, completeness, atoms, density 236 3. Meets and joins of subsets 237 4. Ideals, filters, and ultrafilters 241 5. Functions between Boolean algebras 242 6. Congruence relations, ideals, filters, and homomorphisms 244 7. Complete additivity and multiplicativity 244 8. Completeness and atoms 247 9. Duals and conjugates 252 10. Regular-open BA of a closure operator 256 11. Regular-open BA of a topological closure operator 257 12. Topological spaces and closure operators 263 13. Complex algebra of a binary relation 265 14. Complete BA of a partial ordering 267 15. Completion of a BA - 270 16. Perfect extension of a BA 271 17. Summary of constructions 273 18. Extending Boolean operators, 274 19. Composing extended Boolean operators \ 279 20. Extending operators within a BA l 283 21. Preservation theorems for complete extensions 285 Chapter 6. Relation algebras 289 1. Boolean relation algebras 291 2. Group relation algebras 292 3. NA, WA, and SA 293 4. Special kinds of elements 294 5. Axioms R 7, R 8 296 6. Axiom R 5 301 7. 8. 9. 10. Axioms R7, Axioms Rs, Axioms Re, Axioms R6 Rs, Rg R 7, Rs, R7, Rg, R.7, R-8 Rg, R-9 302 303 304 305
CONTENTS 11. Axioms R 5, Re, R7, R9 306 12. Axioms R 5, R 6, R7, Rs, Rg. 306 13. Axiom Rio with others 307 14. Theorem K and the cycle law 309 15. Special elements in NA 313 16. Characterizations of NA and RA 319 17. Duality for NA 322 18. Completions 323 19. Perfect extensions 324 20. Matrices of elements 326 21. Bases 330 22. Elementary arithmetic in WA 331 23. Properties of bases 338 24. n-dimensional relation algebras 345 25. Cycles of atoms 350 26. Complex algebras of ternary relations 354 27. The very nonassociative algebra in NA~WA 357 28. McKinsey's algebra in WA ~ SA 357 29. An algebra in SA~ RA 358 30. Lyndon's nonrepresentable algebras in RA ~ RRA 358 31. Jonsson's algebras from projective geometries 359 32. Lyndon's algebras from projective geometries 359 33. McKenzie's nonrepresentable algebra 360 34. Allen's interval algebra 362 35. Cycle structures of complex algebras 364 36. Representation by complex algebras 365 37. Elementary arithmetic in SA 366 38. Associativity in groupoids - 370 39. Independence of seven weak associative laws 373 40. Consequences of 4-associativity 375 41. Relativization 379 42. Ideals \ 381 43. Ideal elements, relativization, and homomorphisms 382 44. Simplicity 383 45. Direct products 387 46. Necessary subalgebras of SAs 389 47. Elementary arithmetic in RA 393 48. Functional elements. 394 49. Transitive and equivalence elements 396 50. Forbidden matrices 398 51. Equational basis for RA n 401 52. Equational basis for RRA 408 53. Representation theorems 411 54. Cycles in structures 417 55. Classification of simple finite algebras 420
CONTENTS xxi 56. Finite integral relation algebras with 0, 1, 2, or 3 atoms 423 57. Finite integral relation algebras with 4 or 5 atoms 435 58. Cycles of the algebras 1,37-3737 436 59. Multiplication tables for algebras I37-3737 437 60. Diversity cycles for the algebras 165-6565 439 61. Multiplication tables for the algebras 165-6565 440 62. Diversity cycles of the algebras lg3-8383 444 63. Multiplication tables for algebras ls3-8383 446 64. Failures of (J), (L), (M) among li-l 83 452 65. Independence of (J), (L), and (M) 454 66. 5-dimensional relational basis data for 198 algebras 454 67. Algebras of every dimension 456 68. Flexible atoms 458 69. Finite algebras with many automorphisms 460 70. Splitting atoms 471 71. RRA is not finitely based 473 72. The number of finite integral relation algebras 476 73. Many nonrepresentable relation algebras 480 74. Algebras with few subalgebras 482 75. Non-embeddable relation algebras 482 76. Complex algebras of cycle structures 486 77. Flexible systems of atoms 488 78. Trails of matrices 489 79. Singletons and twins in a simple SA 497 80. Algebras from modular lattices 501 81. Factor algebras 502 82. A characterization of representability 506 83. Complete representability '~ 512 84. RRAs with no complete representations 515 85. Point-density and pair-density 517 86. Simple pair-dense algebras 518 87. Complete representability results \ 521 Chapter 7. Algebraic logic 527 1. Equipollence of C and C + 527 2. Inequipollence of x and + 530 3. Finite-variable formalisms. 535 4. Algebras of formulas 538 5. Free RRAs of formulas 542 6. SAs and RAs of formulas 550 7. Algebraic semantics 557 8. Algebraic satisfaction and substitution 559 9. Algebraic soundness 564 10. Free SAs and RAs of formulas 574 11. Formalizing set theory in x 578
xxii CONTENTS Chapter 8. 4329 finite integral relation algebras 583 1. Cycles of algebras Ii3i6-1316i3i6 583 2. Cycles of algebras I3013-3OI33013,612 3. Failures of (J), (L), (M) among Ii3i6-1316i3i6 and I3013-3OI33013 649 4. 5-dimensional basis data for Ii3i6-1316i3i6 and I3013-3OI33013 680 Bibliography 713 Index 723