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References 547 87. Monk, J. D.: On representable relation algebras. Michigan Mathematical Journal 11 (1964), pp. 207 210. 88. Peirce, C. S.: Note B. The logic of relatives. In: C. S. Peirce (ed.) Studies in logic by members of the Johns Hopkins University, Little, Brown, and Company, Boston, 1883, pp. 187 203. [Reprinted by John Benjamins Publishing Company, Amsterdam, 1983.] 89. Pixley, A. F.: Functionally complete algebras generating distributive and permutable classes. Mathemamtische Zeitschrift 114 (1970) pp. 361-372. 90. Riguet, J.: Relations binaires, fermetures, correspondances de Galois. Bulletin de la Société Mathématique de France 76 (1948), pp. 114 155. 91. Riguet, J.: Quelques propriétès des relations difonctionnelles. Comptes Rendus Mathématique de l Académie des Sciences 230 (1950), pp. 1999 2000. 92. Russell, B.: The principles of mathematics. Cambridge University Press, 1903. [Reprinted by Allen & Unwin, London, 1948.] 93. Sain, I.: Strong amalgamation and epimorphisms of cylindric algebras and Boolean algebras with operators. Preprint #1982/17, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1982, 44 pp. 94. Sain, I.: Weak products for universal algebra and model theory. Diagrammes, vol. 8 (1982), pp. S2 S15.: 95. Schmidt, G.: Relational mathematics. Encyclopedia of Mathematics and its Application, vol. 132, Cambridge University Press, Cambridge, 2011, xiii +566 pp. 96. Schmidt, G.: Relational concepts in social choice. In: T. G. Griffin and W. Kahl (eds.) Relational and algebraic methods in computer science. 13th interational conference on relational and algebraic methods in computer science 2012, Cambridge, UK, September 17 20, 2012. Proceedings, Lecture Noties in Computer Science, vol. 7650, Springer Verlag, Berlin, 2012, pp. 278 293. 97. Schmidt, G. and Ströhlein, T.: Relations and graphs Discrete mathematics for computer scientists. EATCS Monograph on Theoretical Computer Science, Springer-Verlag, Berlin, 1993, ix + 301. 98. Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik). Dritter Band. Algebra und Logik der Relative. Erste Abteilung. B. G. Teubner, Leipzig, 1895. [Reprinted by Chelsea Publishing Company, New York, 1966.] 99. Sikorski, R.: Cartesian products of Boolean algebras. Fundamenta Mathematicae 37 (1950), pp. 125 136.
548 References 100. Sikorski, R.: Boolean algebras. Second edition. Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 25, Springer-Verlag, Berlin, 1964, x + 237 pp. 101. Skolem, T.: Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen. Skrifter utgitt av Videnskapsselskapet i Kristiania, I, Matematisk-naturvidenskabelig klasse 4 (1920), 36 pp. 102. Tarski, A.: Bemerkung der Redaktion. Fundamenta Mathematicae 23 (1934), p. 161. 103. Tarski, A.: Grundzüge des Systemenkalküls. Fundamenta Mathematicae 26 (1936), pp. 283 301. 104. Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6 (1941), pp. 73 89. 105. Tarski, A.: Lecture notes on the theory of relation algebras. Taken by B. Jónsson, University of California at Berkeley, some time during the period 1942 to 1945. 106. Tarski, A.: Manuscript of a book containing some of Tarski s early contributions to the theory of relation algebras, written during the period 1943 to 1945. The book was never published, but most of the results in the book were later included in [113]. 107. Tarski, A.: Some metalogical results concerning the calculus of relations. Journal of Symbolic Logic 18 (1953), pp. 188 189. 108. Tarski, A.: A formalization of set theory without variables. Journal of Symbolic Logic 18 (1953), p. 189. 109. Tarski, A.: An undecidable system of sentential calculus.journal of Symbolic Logic 18 (1953), p. 189. 110. Tarski, A.: Contributions to the theory of models, III. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences 58 (=Indagationes Mathematicae 17) (1955), pp. 56 64. 111. Tarski, A.: Equationally complete rings and relation algebras. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences 59 (=Indagationes Mathematicae 18) (1955), pp. 39 46. 112. Tarski, A.: Lecture notes on the theory of relation algebras. Taken by S. Givant, University of California at Berkeley, 1970. 113. Tarski, A. and Givant, S.: A formalization of set theory without variables. Colloquium Publications, vol. 41, American Mathematical Society, Providence RI, 1987, xxi + 318 pp.
References 549 114. Tarski, A. and Vaught, R.: Arithmetical extensions of relational systems. Compositio Mathematica 13 (1957), pp. 81 102. 115. Weinstein, J. M.: First-order properties preserved by direct products. Doctoral dissertation, University of Wisconsin, Madison WI, 1965, 162 pp. 116. Werner, H.: A Mal cev condition for admissible relations. Algebra Universalis 3 (1973), pp. 263. 117. Werner, H.: Diagonal products and affine completeness. Algebra Universalis 4 (1974), pp. 269 270. 118. Werner, H.: Congruences on products of algebras and functionally complete algebras. Algebra Universalis 4 (1974), pp. 99 105. 119. Werner, H.: Discriminator algebras. Studien zur Algebra und ihre Andwendungen, vol. 6, Akademie Verlag, Berlin, 1978. 120. Winter, M.: Dependencies in relational models of databases. Toappear in: Festschrift J.N. Oliveira of JLAMP, September 01, 2015. 121. Wostner, U.: Finite relation algebras. Notices of the American Mathematical Society 23, (1976), A-482. 122. Zierer, H.: Relation algebraic domain constructions. Theoretical Computer Science 87 (1991), pp. 163 188..
Index 0, 38, 133 1, 38, 133 2, 16 1, 36 0, 41, 133 n, 133 ι, 79, 82, 5, 16, 36, 38, 58, 133,, 133, 36, 58, 133 +, 5, 16, 36, 58, 133, 39, 144, 16, 38,39 ;, 36, 58,+, 41, 38, 20, 22, 38 <, 38, 2 U U, 2 id U,2 di U,2, 6 1,6,79, 5, 5, 6, 6,7,79, 6, 133, 1 2,47, 2 =, 2, 58, 2, 230, 2, 230, 2, 2, 58, 59, 88, 337, 369, 380, 59, 88, 337, 369, 380, 58, 59, 59, 58, 337, 337 =, 285 n, 18 r/m, 331 r M, 330 r n, 144 rs, 82 r s mod M, 331 Springer International Publishing AG 2017 S. Givant, Introduction to Relation Algebras, DOI 10.1007/978-3-319-65235-1 551
552 Index r s mod Θ, 313 r/θ, 319 v 0,v 1,...,58 x, y, z, 58 A/M, 331 Eq(K), 63 M R,23 M, 19 M + N, 19 M N, 22 M N, 20 M T,21 N/M, 363 R M,23 R 0, R 1,...,64 Th(K), 63 L, 58 L,64 L 3, 106 S, 63 Mo(S), 63 A, B,...,37 A/M, 331 A/Θ, 319 B C, 419 B C, 436 B I, 445 Cm(G), 80 Cm(L), 89 Cm(P ), 83 M 0, M 1, M 2, M 3,73 Ma(U), 74 Re(E), 72 Re(U), 71 U, V,...,65 (α, β), 1 α 0, α 1,...,65 ϕ(m), 332 ϕ 1 (N), 332 σ(v 0,...,v n 1 ), 59 Δ, 59 Γ,59 Γ (v 0,...,v n 1 ), 60 Θ, 313 abacus of binary relatives, 218 abelian group, 299, 487 relation algebra, 143, 158, 277, 299, 304, 311, 321, 423, 448, 489 Abian, Alexander Smbat, 541 absolute operation, see Boolean operation absorption law for lattices, 88 91 addition, 16 25, 32 34, 36, 114, see also matrix addition, notation, relational addition, relative addition of functions, 17 18 symbol, 58 additive operation, see distributive operation affine representation, 311 algebra, 36, see also Boolean algebra, Boolean algebra with operators, relation algebra of relations, see set relation algebra operations of, see fundamental operations of an algebra universe of, see universe of an algebra amalgamation of homomorphisms external, 485, 510 515 internal, 484 486 ample internal product, 507 510, 516, 518 Andréka, Hajnal Ilona, vii, 105, 219 221, 264, 372, 398, 488, 541, 543, 545 annihilator of an ideal, 374 375 anthropology, xiv antisymmetric relation, 230 Aristotle, xiv arithmetic, see laws arrow diagram, 356, 366, 438 Arrow, Kenneth Joseph, vi, 541 542 associative law for addition, 36, 112
Index 553 for composition, 10 for direct products, 490, 493 for groups, 79 for lattices, 88 91 for relational addition, 10 for relative addition, 119, 134 for relative multiplication, 36, 134 infinite, see infinite associative law atom, 40, 68 69, 80 81, 186 188, 195, 214 216, 227, 277, 286, 288 296, 305 306, 346 347, 354 355, 376, 392 394, 410 412, 414, 418, 427, 443, 449, 464, 507, 530 535 Atomic Decomposition Theorem, 473 475, 488 Isomorphism Theorem, 288 291, 303, 308 309, 533 Monomorphism Theorem, 292 296, 303, 309 311 Subalgebra Theorem, 245 248 atomic Boolean algebra, 46, 48 57, 68, 346 347, 376, 470 475 formula in language of relation algebras, 59 in language of theory of relations, 65 relation algebra, 40, 71 74, 80 104, 286, 288 296, 307, 346 347, 354 355, 376, 410 411, 418, 427, 449, 464, 473 474, 507, see also finite relation algebra, small relation algebra subalgebra, 243 253, 266 270 term, 58 atomless Boolean algebra, 68, 474 480, 488 of ideal elements, 476 480 relation algebra, 40, 107, 270, 277, 286, 305, 307, 376 Augenquaderrelativ, 220 automorphism, 299 300, 303, 311, see also Boolean automorphism, inner automorphism, trivial automorphism axioms, see also associative law, Boolean axioms, commutative law, distributive law, distributivity axiom, Huntington s law, identity law, independence of axioms, involution law, logical axiom, Pasch Axiom, Tarski s law of projective geometry, 81 of relation algebra, 35 41, 51 57, 69, 76 103, 112, 138 139, 230, 321, 373, 404 405, 489, 493 of theories of minimal relation algebras, 538 base case of definition by induction, 58 of proof by induction, 59 clause, see base case isomorphic relation algebras, 287 isomorphism, 286 287, 303, 308, set of an algebra, 72 van Benthem, Johannes Franciscus Abraham Karel, 541 bifunctional element, see bijection bijection, 216 217, 304, 395 396 bijectional element, see bijection binary operation on relations, see operation relation, see relation tree, 476 477 Bird, Richard Simpson, 541 Birkhoff, Garrett, 487, 516, 542 Boole, George, xi, xiv xvi, 27, 542 Boolean addition, see addition algebra, xi xiv, 16 27, 32, 42 57, 77 79, 113, 376, 496, 516, 518,
554 Index see also atomic Boolean algebra, atomless Boolean algebra, Boolean part of a relation algebra of ideal elements, 181 182, 344 353, 370, 376 378, 470 480, 488, 492 of right-ideal elements, 172 175, 214 216 with complete operators, 50, 245 248, 262, 288 296, 303 with normal operators, 401 with operators, xvii, 41 50, 381 383 with quasi-complete operators, 50, 267 automorphism, 40, 114 115, 142 143 axioms, 51 element, see ideal element epimorphism, 40, 406 407 filter, 367, 370 flag, 87 88, 104 group, 38, 143, 318 homomorphism, 40, 273 ideal, 325, 344 346, 350 352, 376, 386 inequality, see partial order isomorphism, 40 law, 113, 134 matrix, see matrix monomorphism, 40 multiplication, see multiplication operation, 5 8, 36, 280, see also addition, multiplication, complement, equivalence, implication, intersection, one, subtraction, symmetric difference, union, zero part of a relation algebra, 37 relation algebra, 77 79, 103, 107, 182, 265, 299, 389, 417, 519 520, 522, 530, 536 subalgebra, 376, 382 bounded relation, 30 Boyd, John Paul, 542 Brams, Steven J., 542 Brink, Chris, 542 Burris, Stanley Neal, 542 calculus of relations, x xvii, 1 32, 65 66 cancellation law for direct products, 496 canonical homomorphism, see projection isomorphism, 436 439, 460 463, 491, 495, 501 502, 517 cardinality of a relation algebra, 37, 235 236 Cartesian factor, see factor product, see direct product Cayley Arthur, xii representation, 296, 303 chain of congruences, 317, 372 of elementary subalgebras, 260 of ideals, 350 of subalgebras, 237, 240, 479 480 Chang, Chen-Chung, 487, 542 characterization of equivalence elements, 149 152, 155 157, 218 of functions, 198 200, 208 213, 221 of ideal elements, 178 179, 182 184, 219 of rectangles, 190 192, 220 of right-ideal elements, 166 167, 176 177, 219 of squares, 197 198, 220 of symmetric elements, 141 142 of transitive elements, 144 146 Chin, Louise Hoy, xvii xix, 66, 218 222, 542 closure, see also one-step closure equivalence, see equivalence closure of an element
Index 555 properties of the set of equivalence elements, 152 153, 161 162, 218 of functions, 201 208, 221 of ideal elements, 180 182, 219 220, 225 of rectangles, 192 196, 220 of right-ideal elements, 169 174, 219 of symmetric elements, 142 143, 217, 222 of transitive elements, 147 149, 217, 222 223 symmetric-transitive, see equivalence closure of an element transitive, see transitive closure of an element under operations, 229 cofinite set, 264 cokernel coset, 368, 378 of a congruence, 368, 378 of a homomorphism, 368, 380 collinear points, 82 86 collineation, 309 column of a matrix, 18 Comer, Stephen Daniel, 104, 398, 542 543 common refinement, 496 commutative law, x for addition, 36, 112 for direct products, 491, 493 for lattices, 88 91 for relative multiplication, 83 87, 276 relation algebra, see abelian relation algebra compact element, 238 239, 317, 342, 346, 370, 373, 432, 491 compactly generated lattice, 239, 317, 319, 343, 346, 373, 432, 491 Compactness Theorem, 63, 258 complement, ix, 6, 16 20, 24, 32 34, 38, 306 of a function, 17 18 relative, see relative complement symbol, 58 complementation, see complement complete distributivity law for converse over addition, 114 115 for converse over multiplication, 114 115 for functions, 210 211 for relative multiplication over addition, 124 125 embedding, see complete monomorphism epimorphism, 407, 450 equational theory, 536, 538 filter, 380 generators, 240, 285 homomorphic image, 283 homomorphism, 283 285, 307, 378, 426, 429, 449, 480, 485 ideal, 378, 380 lattice, 236, 239, 319, 337, 343, 346 monomorphism, 283 285, 293 296, 302 operation, see completely distributive operation relation algebra, 39, 286, 307, 411, 428, 450, 458, 464, 473 476 subalgebra, 240 241, 262, 265 266, 517 subuniverse, 240, 285 Complete Decomposition Theorem, 474 476, 488 completely distributive operation, 44 57 for atoms, 48 57 representable relation algebra, 103 Completeness Theorem for equational logic, 64 for first-order logic, 63 complex algebra
556 Index of a geometry, 81 88, 92 93, 103 108, 143, 224, 264 265, 267, 307, 309 311, 395, 400, 405 406 of a group, 79 81, 103, 158, 218, 248 249, 264 265, 267 269, 286, 307, 309, 395, 400, 405 of a lattice, 88 94, 104, 108, 224, 309, 400, 417 inverse, 80 multiplication, 80 product of filters, 369 of ideals, 337 338 sum, 369 of ideals, 337 338 component, 436, 459 composition, see functional composition, relational composition of homomorphisms, 272 computer science, xiii conditional equation, 60, 422, 447 448, 515 516 equational theory, 63 congruence, 313 331, 366 368, 378, 381, 450 454 class, 319 323, 366 368 generated by a set, see generators of a congruence induced by a homomorphism, 322 relation, see congruence conjunction symbol, 59 constant almost everywhere, 500 507 term, 500 507 conventions, see order of operations converse, ix, 36, see also definability of converse, notation, relational converse complement, 30, 133 symbol, 133 symbol, 58 conversion, see converse coordinate sequence, 420, 446 coplanar lines, 82 Correspondence Theorem, 365, 371, 381, 391 coset, 330, 408 countably complete relation algebra, 39, 147, 223 generated ideal, 375 subalgebra, 235 Couturat, Louis, xvi cycle law, 41, 68, 120 121 cylinder, 168, 172 Decomposition Theorem for Re(E), 473 Dedekind, Julius Wilhelm Richard, 371, 543 definability of complement, 280 of converse, 123 124, 135, 280, 302 of identity element, 123, 138 of inequality, 276 definition by induction on formulas, 60 on terms, 59 degenerate relation algebra, 40, 74, 105, 139, 306, 386, 433, 445, 448, 491, 520, 528 De Morgan Augustus, xiv, 27, 28, 217, 543 Tarski laws, 11 12, 41, 68, 119 122, 124 132, 135 136, 138, 164, 213 214, 221 dense linear order, 106 without endpoints, 105 relation, 222 Desargue s theorem, xviii Déscartes, René, 486 diagonal embedding, 510 difference, see subtraction, symmetric difference
Index 557 direct decomposition external, 429 433, 450 458 internal, 439 443, 464 476, 483, 525 528, 537 of homomorphisms, 480 484 power, 16 18 product, 314, 385, 419 517, 527 528 binary external, 435, 443 444 binary internal, 435 444, 491 492 general external, 444 458 general internal, 459 476, 480 488 of set relation algebras, 433 435, 459, 472 473, 491, 494 sum, 516, see also weak direct product directed edge, 5 graph, 4 set of elements, 148 149, 204 system of congruences, 317, 372 373 of elementary subalgebras, 259 260 of elementary substructures, 263 of filters, 370 of homomorphisms, 305 of ideals, 341 343 of regular subalgebras, 242 of subalgebras, 237 240, 265 triangle, 121 122 directly indecomposable algebra, 399, 401 relation algebra, 396 397, 433, 492 Dirichlet, Johann Peter Gustav Lejeune, 486, 543 discriminator, see also unary discriminator function, 399 term, 401 variety, 399, 401, 497 disjoint domains, 204 205 elements, 40, 160 162 rectangles, 193 194 set, 40 system of elements, 40 disjunction symbol, 59 dissimilar homomorphisms, 400 distinguished constant, 5 distributive lattice, 340 344, 346 law, x, 376, see also complete distributivity law, general complete distributivity law, general finite distributivity law, general quasi-complete distributivity law for bijections, 216 for composition over union, 11 for converse, 112 for converse over addition, 36, 114, 134 for converse over complement, 114 for converse over intersection, 11 for converse over multiplication, 134 for converse over subtraction, 114 for converse over symmetric difference, 114 for converse over union, 11 for direct product over intersection, 491 for direct product over join, 491 for functions, 208 210, 221 for ideal elements, 182 184, 220, 225, 282 for relational addition over intersection, 11 for relative addition over multiplication, 119, 134 for relative multiplication over addition, 36, 68, 112, 116, 134
558 Index for relative multiplication over multiplication, 276 277 operation, 42 50 distributivity axiom, 41, 51 diversity element, 41, 131 132, 153, see also notation matrix, 19 relation, 2, 10, 72 74 domain algebra, 271 of a function, 203 206, 214, 216 of a relation, 164 of an element, 185 190, 220, 226, 392 394, 412 downward Löwenheim-Skolem-Tarski Theorem, see Löwenheim- Skolem-Tarski Theorem dual filter, 370 371, 379 380 ideal, 370 371, 379 of symmetric difference, 367 368 duality principle, 134, see also first duality principle, second duality principle, third duality principle Düntsch,Ivo,543 economics, xiv element, see compact element, diversity element, domain, equivalence element, function, identity element, left-ideal element, range, rectangle, reflexive element, right-ideal element, square, subdiversity element, subidentity element, symmetric element, transitive element field of, see field of an element elementary diagram, 258 embedding, 301 302, 304, 307 extension, 254, 258 language, see first-order language monomorphism, see elementary embedding subalgebra, 254 262, 270, 301 302, see also directed system of elementary subalgebras substructure, 263, see also directed system of elementary substructures theory, 63 of a class of models, 63 elimination of quantifiers, 263 embedding, 272, see also complete monomorphism, diagonal embedding, elementary embedding, monomorphism emotional relation, ix empty relation, 2, 10 set, 39 subspace, 92 entry of an array, 18 epimorphism, 271, 278 280, 305, 321 323, 353, 374, see also Boolean epimorphism equality between relations, 2 symbol, 58 equation, 59, 390 391, 515 516, 518 equational theory, 63 64, see also complete equational theory, inconsistent equational theory of a class of algebras, 63 of relation algebras, 538 equivalence, 29 class, 32, 178, 319 closure of an element, 152 element, 149 164, 197 198, 218 219, 222 228, 276, 304, 403 418, 424, 449, 469, 489, 495, 536, 538 539, see also characterization of equivalence elements, closure properties of the set of equivalence elements,
Index 559 modular law for equivalence elements, reflexive equivalence element, type of an equivalence element logical, see logical equivalence modulo a set of formulas, 75 relation, 32, 72, 149, 177, 223, 281 282, 285, 313, 316, 319, 327, 405, 472 473, 526, 528 symbol, 59 equivalent relation algebras, see base isomorphic relation algebras Exchange Principle, 258, 297 299, 303, 311, 362, 439 Existence Theorem for internal products, 463 binary version, 439 for weak internal products, 501 502 existential formula, 60 quantification symbol, 59 sentence, 60 exponentiation, 144, see also first law of exponentiation for direct powers,secondlawofexponentiation for direct powers, third law of exponentiation for direct powers, expression, 58 extension, see also regular extension of a homomorphism, 272 of a relation algebra, 230, 297 299 external product, see direct product factor external, 420, 444 445 homomorphism, 449, 481 internal, 436 439, 459 field of a relation, 148, 185 of an element, 185 filter, 366 372, 378 381, see also Boolean filter, proper filter element, 369 generated by a set, see generators of a filter finite Boolean algebra, 94 100, 109 111 cofinite subalgebra, 264 join property, 374, 377 meet property, 379 381 relation algebra, 94 100, 109 111, 458, 474, 537, see also small relation algebra finitely generated Boolean algebra, 236 congruence, 316 317 filter, 369, 379 ideal, 333 335 relation algebra, 234 subalgebra, 234, 237 239 First Homomorphism Decomposition Theorem, 483 484, 488, 497 Isomorphism Theorem, 362, 371, 381, 385 386, 407 408 for congruences, 322 323, 430, 451 for ideals, 355 356, 512 first dual of a law, 115 116, 164 of a notion, 179 duality principle, 115 116 involution law, 112 law of exponentiation for direct powers, 494 first-order language, see language of relation algebras, language of relations logic, 58 65 theory, see elementary theory Fishburn, Peter C., 542 formula, see also atomic formula, conditional equation, equation, existential formula, open formula, positive formula, quantifier-free formula, uni-
560 Index versal Horn formula, universal formula in language of relation algebras, 59 64 in language of relations, 65 relation algebra, 75 77, 103, 105 107, 400 Frias, Marcelo Fabián, 543 Frobenius algebra, see complex algebra of a group Ferdinand Georg, 486, 543 full set relation algebra on a set, 71 72, 74 75, 103, 287, 389, 395, 405, 522, 532 536, 539 on an equivalence relation, 72, 389, 405, 472 473, 526 527 function, 32, 198 217, 221 222, 226 228, 276, 304, 392 394, 424, 449, 489, 495, see also characterization of functions, closure properties of the set of functions, distributive law for functions, modular law for functions inverse, 304 functional composition, 7, 205, 304 element, see function part of an element, 201 fundamental operations of an algebra, 36 Galvin, Frederick William, 488, 543 general complete distributivity law, 44 50 finite distributivity law, 42 43 monotony law, 43 quasi-complete distributivity law, 44 45 generalized relativization, 414 418 generated, see also finitely generated ideal element, see ideal element generated by an element right-ideal element, see right-ideal element generated by an element generating set, see generators generators complete, see complete generators of a congruence, 316 317 of a filter, 369, 379 of a relation algebra, 233 239 of a subalgebra, 234 239, 278 281, 286, 305, 358 361, 389 390, 502 509 of an ideal, 333 343, 357, 374 376 geometric complex algebra, see complex algebra of a geometry relation algebra, see complex algebra of a geometry geometry, 81 88, 224 order of, see order of a geometry Givant, Steven Roger, 66 67, 105, 113, 136, 217 221, 262 264, 303, 371 372, 398, 416, 487 488, 516, 536, 541 543, 548 Goldblatt, Robert Ian, vii graph of a relation, 3 10, 28 31 Grätzer, George, 543 greatest lower bound, see infimum Griffen, Timothy G., 547 group, 40 41, 79 81, 216 217, 299, 395 396, 487, 500, 516, see also abelian group, Boolean group complex algebra, see complex algebra of a group isomorphism, 292 Halmos, Paul Richard, 516, 543 544 Henkin, Leon Albert, 303, 544 Hirsch, Robin David, xii xiii, 262, 302, 544 Hodkinson, Ian Martin, vii, xii xiii, 262, 302, 544 homomorphic image, 272, 280, 304 305, 321 323, 355 356, 385, 408 410, 414 415, 418, see
Index 561 also complete homomorphic image Homomorphism Decomposition Theorem, see First Homomorphism Decomposition Theorem, Second Homomorphism Decomposition Theorem Extension Theorem, 361 363, 383, 399 Theorem, 371, 381 for congruences, 322 for ideals, 355 homomorphism, 271 311, 321 322, 353 356, 373 374, 386, 399 400, 406 410, 425 426, 449, 480 486, 490, 497, see also Boolean homomorphism, complete homomorphism, complete epimorphism, complete monomorphism, epimorphism, dissimilar homomorphisms, extension of a homomorphism, isomorphism, monomorphism, Peircean homomorphism, restriction of a homomorphism on an algebra, 272 Huntington s law, 36, 112 Huntington, Edward Vermilye, xvi ideal, 164, 220, 313, 325 366, 374 383, 385 386, 408 410, 414, 416 418, 427, 490, 511 515, see also annihilator of an ideal, Boolean ideal, complete ideal, dual ideal, finitely generated ideal, improper ideal, interval of ideals, maximal ideal, non-principal ideal, non-trivial ideal, orthogonal pair of ideals, orthogonal system of ideals, principal ideal, proper ideal, trivial ideal element, 177 184, 219 220, 222, 225, 228, 276, 282, 304, 334 337, 342 343, 369 370, 374, 382, 386 389, 405 408, 413 418, 423 424, 432 433, 439 443, 449, 454 458, 463 464, 478 479, 502 509, see also Boolean algebra of ideal elements, characterization of ideal elements, closure properties of the set of ideal elements, distributive law for ideal elements, modular law for right-ideal elements, orthogonal system of ideal elements atom, 351 353, 413 414, 470 476, 507 generated by an element, 179 generated by a set, see generators of an ideal idempotent element, 175 law for lattices, 88 identity, 61 element, 36, 115, 228, 392 396, see also definability of identity element, notation law for composition, 10 for direct products, 491 for groups, 79 for relational addition, 10 for relative addition, 119, 135 for relative multiplication, 36, 112, 116, 135 matrix, 19 relation, 2, 315, 317, 372 singleton, 530 534, 539 symbol, 58 image, see also homomorphic image, inverse image set algebra, 272 set, 277 280, 288 289, 332, 363 365, 368, 374 implication, 29 symbol, 58
562 Index improper filter, 369 ideal, 333, 335, 348, 374, 386 subalgebra, 230, 234 incidence relation, 81 inclusion between algebras, see subalgebra between relations, see inequality between relations proper, see inequality between relations inconsistent equational theory, 538 independence of axioms, 112 independent axiom, 100 set of axioms, 100 individual constant symbol, 64 induction base case, see base case clause, see induction step definition, see definition by induction on formulas, 60, 255 on positive formulas, 274 275 on quantifier-free formulas, 231 232 on terms, 59, 231, 273 274, 314 315, 421 422, 446 447 step, 59 inequality, see also partial order between relations, 2, 12 infimum, 39, 67, 427, see also meet infinitary associative law, 67 distributive law, see complete distributivity law injection, see bijection injectional element, see bijection inner automorphism, 300, 311 cylindrification, 172 ideal closure, 181 integral relation algebra, 104, 107 108, 270, 386, 391 396, 398 400, 411 412, 423, 489 Integrality Theorem, 392 396, 412, 533 internal product, see direct product interpretation in language of relation algebras, 61 62 in language of relations, 65 intersection, ix, 5, 10 of a system of congruences, 315 317 of filters, 379 of ideals, 333, 374 of subalgebras, 233 234 of lines, 82 interval of ideals, 363 365 inverse, see also converse complex, see complex inverse image ideal, 374 image set, 277 278, 305, 332, 363 365, 368, 374 law for groups, 79 of an isomorphism, 272 Inversion Theorem, 135 involution, 40 law first, 10, 36, 40, 134 for relative addition, 119 second, 10 11, 36, 40, 135, 272 isomorphic algebras, 272 image, 272 isomorphism, 115, 118 119, 271 272, 285 299, 302, 304, 307 309, 322 323, 355 366, see also Atomic Isomorphism Theorem, automorphism, base automorphism, base isomorphism, Boolean isomorphism, Correspondence Theorem, First Isomorphism Theorem, group isomorphism, lattice isomorphism, Peircean isomorphism, Second Isomorphism Theorem, Third Isomorphism Theorem type, 285
Index 563 Jevons, William Stanley, xiv xvi, 27 Jipsen, Peter, vii, 105, 398, 416, 541, 544 join, 88 of a system of congruences, 317 319 of filters, 370 of ideals, 337 339, 341 of subalgebras, 236 239 Jónsson, Bjarni, vii, xii xiii, xvii xix, 66, 104, 218, 263, 302, 398 399, 416, 487 488, 516, 536, 542, 544 545, 548 Kahl, Wolfgang, 542, 545, 547 Kamel, Hyman, 105, 136, 545 kernel of a congruence, 324, 343 344, 381 of a homomorphism, 331, 354 366, 490 kinship relation, ix x Korselt, Alwin Reinhold, 28 Köthe, Gottfried Maria Hugo, 516, 545 Kramer, Richard Lynn, vii, 67, 417, 545 Kronecker, Leopold, 486 Lambek, Joachim, 545 Langford, Cooper Harold, 545 language, see also expression, formula, sentence, term of relation algebras, 57 65, 113, 254 262, 301 302, 386, 390 391, 500 of relations, 64 65, 75, 400 Lattice of Ideals Theorem, 346, 350 352, 371 372, 380, 386, 518 lattice, 88, see also compactly generated lattice, complex algebra of a lattice, complete lattice, distributive lattice, modular lattice isomorphism, 343, 370 371 of Boolean filters, 370, 380 of Boolean ideals, 344 346, 386 of congruences, 317 319, 343 344 of filters, 370 371, 380 of ideals, 337 371, 385 386, 391 of subalgebras, 236 240 law, x, 10 12, 28, 31, 37, 113 139, 139, 141 228, see also absorption law for lattices, associative law, Boolean law, commutative law, complete distributivity law, cycle law, De Morgan- Tarski laws, distributive law, first law of exponentiation for direct powers, general complete distributivity law, general monotony law, general quasi-complete distributivity law, Huntington s law, identity law, idempotent law for lattices, infinitary associative law, involution law, modular law, second law of exponentiation for direct powers, Tarski s law, third law of exponentiation for direct powers least upper bound, see supremum left ideal, 164 element, 164, 178 179, 191, 220 Lewis, Clarence Irving, 28, 545 line, 81 linguistics, xiv Liu, Kexin, vii logic, see first-order logic logical axiom, 63 equivalence, 75 symbol, 58, 64 Lovász, László, 487, 545 Löwenheim Leopold, xv, 28, 263, 545 Skolem-Tarski Theorem, 254 258, 263 relational version, 270 Lukács, Erzsbébet, 398, 544
564 Index Lyndon algebra, see complex algebra of a geometry Roger Conant, xii, xvii xix, 103 104, 302 303, 398, 488, 545 546 Maddux algebra, see complex algebra of a lattice Roger Duncan, vii, xii xiii, 28, 104, 220, 263, 398, 416 417, 488, 536, 541, 544, 546 maps, 205 206 Marx, Maarten Johannes, 417, 546 material relation, ix matrix, 16, 18 27, 32 34 addition, 21 22, 33 algebra, 74 75, 105 on a set, 74 associated with a relation, 23 27, 33 multiplication, 20 21, 33 notation, 18 19 transposition, 21, 24, 33 maximal congruence, 518 equational theory, see complete equational theory filter, 381 ideal, 347 353, 376 377, 385, 391, 490, 515 in an ideal, 347 351 Maximal Ideal Theorem, 350 351, 372, 376 377, 381 McCoy, Neal Henry, 516, 546 McKenzie, Ralph Nelson Whitfield, vii, xviii xix, 104, 398, 546 McKinsey, John Charles Chenoweth, xii, 103 104, 220, 398, 487, 546 McNulty, George Frank, 546 meet, 88 of a system of congruences, 317 of filters, 370 of ideals, 337 339 of subalgebras, 236 Mikulás, Szabolcs, 372, 541 minimal relation algebra, 72 74, 105, 131, 389, 395, 494, 519, 522 524, 527 529, 536 538 subalgebra, 234, 249 253, 278, 500, 517 model, 62 modular lattice, 88 94, 157 158, 224, 319, 417, 432, 491 law, 129, see also standard modular law, strong modular law for equivalence elements, 155 158, 182, 218, 225 for functions, 211 212, 221, 227 for lattices, 88 91, 129 for multiplication over relative multiplication, 131 for right-ideal elements, 176 177, 182, 219, 225 Monk, James Donald, vii, xviii xix, 221, 303, 516, 544 547 monoid, 40 monomorphism, 272, 301, 354 355, 386, see also Atomic Monomorphism Theorem, Boolean monomorphism monotone operation, 43 monotony law, see also general monotony law for converse, 114 for relative addition, 119 for relative multiplication, 117 de Moor, Oege, 541 multiplication, 16, 20, 24, 32 34, 38, 114, 315, see also matrix multiplication, notation, relative multiplication of functions, 17
Index 565 negation symbol, 58 Németi, István, vii, 105, 264, 372, 541, 545 Noether, Amalie Emmy, 371, 516 non-degenerate relation algebra, 40, 386, 391, 396 397, 433, 476, 512 514, 520, 522 524 non-desarguesian plane, 103 non-logical symbol, 58, 64 non-principal ideal, 351 353 non-trivial filter, 369 ideal, 332, 375 normal operation, 45 46, 50, see also Boolean algebra with normal operators subgroup, 158 notation, 5 7, 36 41, 133 134 nullary operation on relations, see operation one, 38, 114, see also notation one-step closure, 235, 279 280 extension, 336 337 open formula, see quantifier-free formula Horn formula, 61, 422 423, 447 448 operation, ix x, 5 10, 19 28, 36 58, 61, see also Boolean operation, completely distributive operation, distributive operation, monotone operation, normal operation, operator, order of operations, Peircean operation, preservation of operations, quasi-completely distributive operation operator, 50, see also Boolean algebra with complete operators, Boolean algebra with operators, Boolean algebra with quasi-complete operators, operation order, see also partial order of a geometry, 82, 107 108 of logical connectives, 59 60 of operation symbols, 58 of operations, 9, 37 partial, see partial order ordered pair, 1 Orlowska,Ewa,545 orthogonal pair of congruences, 430 432, 453 of ideals, 432 433 system of congruences, 451 453 of ideal elements, 457 458, 465 476, 482 483, 526 of ideals, 453 456 outer cylindrification, 168 ideal closure, 179 pair, see ordered pair parentheses, 58, see also order of operations partial order, 38, 230 partition of an element, 40 Pasch Axiom, 82 Peirce, Charles Sanders, xi, xiv xvi, 27 28, 133 135, 219, 398, 547 Peircean homomorphism, 273 isomorphism, 273 operation, 5 28, 36, 280, see also composition, converse, converse-complement, diversity element, diversity relation, identity element, identity relation, relational addition, relative addition, relative multiplication part of a relation algebra, 40 permutation, 106, 216 217, 221, 300, 304
566 Index permutational element, see permutation Pixley, Alden French, 399, 547 plane, 82 point, 81, see also identity singleton polynomial, 61 62, 142, 146, 152, 167, 179, 192, 201, see also projection positive formula, 60, 274 277 sentence, 60 power of a relation algebra, 445, 476 480 of an element, see exponentiation preservation of operations, 271, 313 315, 319 320, 323, 327 330, 343, 373, 406 407 on atoms, 288 296 of properties under congruences, 314 315 under direct products, 420 428, 445 450, 487 488 under homomorphisms, 273 281 under isomorphisms, 286 under subalgebras, 230 233 under subdirect products, 515 516, 518, 535 of the supremum property, 481 484 principal filter, 369 370, 379 ideal, 335, 339 343, 346, 351 353, 432 433, 454 456, 513 514 principle of induction, see definition by induction, induction product, see also direct product, infimum, multiplication, subdirect product, weak direct product homomorphism, 449, 481 484 sequence, 420, 446 Product Decomposition Theorem, 475, 487 488, 492, 495, 503, 519 binary external version, 433 binary internal version, 439 443 general external version, 458 general internal version, 464 468, 470 471, 482, 526 projection, 429 430, 450 451, 485 486, 491, 494, 510 515 of a homomorphism, 486 projective geometry, see geometry proper filter, 369, 379 ideal, 333, 336 337, 357, 374, 376 377 relation algebra, see set relation algebra subalgebra, 230 quantifier elimination, see elimination of quantifiers quantifier-free formula, 60, 390 391, 400 401 quasi-atom, 46 48, 308 309 quasi-complete operation, see quasi-completely distributive operation operator, see quasi-completely distributive operation quasi-completely additive operation, see quasicompletely distributive operation distributive operation, 43 50, 267, 308 309 for quasi-atoms, 46 50 quasi-dense set, 375 quotient algebra, 320 323, 330 331, 353 366, 368, 385, 391, 407 408, 414, 416, 430 433, 451 454, 490, 511 515 homomorphism, 321 322, 331, 353 355, 358, 373 374, 378, 431 432, 451 452, 511 515 range algebra, see image algebra
Index 567 of a function, 205 206 of a relation, 164 of an element, 185 190, 214, 220, 392, 412 Rauszer, Cecilia, 544 rectangle, 190 198, 220, 226, 276, 304, 388, 489, 531 535, see also characterization of rectangles, closure properties of the set of rectangles, symmetric rectangle refinement property, 487, 496 reflexive element, 144, 149 equivalence element, 149, 157 158, 162, 222 223, 411 412, 539 relation, 144, 230, 316 transitive closure, 147 regular extension, 241 subalgebra, 241 245, 262, 266, 284, 408 409, 489, 493, see also directed system of regular subalgebras relation, ix xi, 1 32, see also antisymmetric relation, bounded relation, calculus of relations, dense relation, diversity relation, emotional relation, empty relation, equivalence relation, equality between relations, graph of a relation, identity relation, incidence relation, inequality between relations, kinship relation, material relation, partial order, reflexive relation, symmetric relation, ternary relation, transitive relation, universal relation associated with a matrix, 23 27, 34 field of, see field of a relation symbol, 64 relation algebra, x xxv, 36, passim, see also abelian relation algebra, atomic relation algebra, atomless relation algebra, axioms of relation algebra, Boolean part of a relation algebra, Boolean relation algebra, cardinality of a relation algebra, complete relation algebra, completely representable relation algebra, complex algebra, countably complete relation algebra, degenerate relation algebra, directly indecomposable relation algebra, extension of a relation algebra, finite relation algebra, finitely generated relation algebra, integral relation algebra, minimal relation algebra, non-degenerate relation algebra, Peircean part of a relation algebra, representable relation algebra, restricted formula relation algebra, set relation algebra, simple relation algebra, small relation algebra, subdirectly irreducible relation algebra, symmetric relation algebra relational addition, ix, 6, 10 composition, ix x, 6 9, 287, 317 319 converse, 6 structure, 65 relative addition, 41, 206 208, 315, 329 330, see also notation symbol, 41 complement of an element, 404 multiplication, 36, passim, see also notation symbol, 58 operation, see Peircean operation