6.5 X 10.5 Long Title.P65

Cambridge University Press 978-0-521-76268-7 - Relational Mathematics Gunther Schmidt Index More information

Index

1-graph, 21 block-transitive, 243, 312, 352 absorbant set, 103, 104, 427 factorization, 243 adjacency, 23 Boole, George, 1, 36, 538, 540 edge-, 228, 333 bound greatest lower, 189 affine incidence plane, 452 least upper, 189 antichain, 196 lower, 185 antisymmetric, 63 upper, 185 Aristotle, 532 Byron, Ada, 543 assignment, 428, 436 associated Cantor’s diagonal enumeration, 111 ordering, 67, 68 cardinality preorder, 148 relation, 21 Cardoso, Rodrigo, 122, 152 strictorder, 67, 68 carré(French),96 association rule, 229 category, 158 asymmetric, 63, 70, 351 Cayley, Arthur, 543 attractor, 420 chain, 196 attribute dependency system, 461 chainable, 228, 231, 403 Aumann contact relation, 280, 281 Chipman’s condition, 306 Aumann, Georg, 280 Choquet integral, 379 Choquet, Gustave, 378 baseset, 2, 5 chord, 312, 344, 345, 347, 348 ground, 8 chordal graph, 344 basin, 420 chordless, 348 Bayesian measure, 373 circuit, 93, 344 BDD, 16 clique, 97, 265, 337, 338 belief clique, maximal, 205, 337, 338 mapping, 372 closure, 276 overcredulous, 373 contact, 170, 281 vacuous, 373 difunctional, 227, 228, 407 below, eventually, 503 rectangular, 97 Bentham, George, 535, 539, 540 transitive, 76 Berghammer, Rudolf, 1, 152, 199, 409 weakorder, 320, 321, 357, 358 bijection by concepts, 257 codomain, 91 bijection by maxcliques, 264 cogredient permutation, 101, 304 bijective, 57 column binary decision diagram, 16 -is-contained preorder, 71 biorder, 232, 309 equivalence, 71, 221, 238, 324, 355 bipartitioned graph, 20 mask, 166 Birkhoff–Milgram Theorem, 324 space, 253 bisimulation, 81, 494, 498 column set intersection, 255 block, 96, 200 column union space, 251 block-diagonal form, 47, 74, 213, 216, 238 commonality, 387

562 Index

comparability graph, 337 diclique matching, 257 complement, 145, 388, 392 difunctional, 74, 224, 232, 404 complement expansion, 102 closure, 227, 228, 407 complete, 62 factorization, 225 order, 68 relation, 74, 220, 221, 226, 228, 230–232, 352 strictorder, 68 Dilworth Theorem, 199, 314 strongly, 62 direct complete quadrangle, 20, 452 power, 146 completion product, 107, 117 by cuts, 294, 297 sum, 127, 131 by ideals, 294 directed graph, 22 component, strongly connected, 78 DirPow, 141, 512 composition, 40 DirPro, 108, 512 concept bijection, 257 DirSum, 128, 512 concept lattice, 268–270 discharge, fully, 503 cone, 185 disjoint union, 127 lower, 187 domain, 91 upper, 187 dotted arrow convention, 52, 159, 234, 305, cone functional, 168 306, 397 congruence, 80, 229, 500, 520 doubly stochastic, 403 connex, 62, 234, 351 down sets, 295 connex preorder, 527 draw, 424 conorm, t-, 388 dual, 43, 68, 350 conorm, relational, 392 Duns, John (Scotus), 532 consecutive 1 s property, 346 dynamics, 501 contact, 281 echelon form, 233 infectivity, 280 edge-adjacency, 228, 333 contact closure, 170, 278 Egli–Milner contact relation, 280, 281 order, 494 continuous, 193, 373, 487 preorder, 493 contraction, 100, 101, 397, 507 element converse, 39 greatest, 187 coproduct, 131 in powerset, 11 counter-enforce, 461 least, 187 covering of graphs, 445 enforce, 461 covering pair of sets, 98 equivalence, 8, 65, 66, 70, 76, 131, 133, 175 covering set, 98, 206 left, 77 cryptography via projective planes, 456 of columns, 71, 221, 238, 324 cut, 23, 295 of rows, 71, 221, 238, 324 cut completion, 294, 297 partial, 528 cycle, 93 right, 77 cyclic, 396, 400 eventually below, 503 dashed arrow convention, 52, 159, 234, 305, existential image, 145, 484, 504 306, 397 expanding, 277 De Morgan extension complement, 388, 392 Ferrers, 245 triples, 387 linear, 317 De Morgan, Augustus, 536, 537, 540 Szpilrajn, 80, 245, 316, 317, 351, 526, 527 decision diagram, 16 externally stable set, 104 decomposable, partly, 402 Extrude, 136, 512 Dedekind rule, 42, 60, 158, 159, 518, 520 extrusion, 8, 135, 137, 513, 522 dense, 212, 324 factorization dependency system, 461 into dicliques, 260 diagonal enumeration, 111 into maxcliques, 265, 267 diclique, 200 of a block-transitive relation, 243 factorization, 260 of a difunctional relation, 225

Index 563

of a Ferrers relation, 237 ground set, 6 Farkas’s Lemma, 459, 460 ordering of, 6 feedback vertex set, 414 ground type, 107 Ferrers Hahn, Hans, 541 closure, 245 Hajós, G., 345 extension, 245 Hall condition, 208, 403, 432 relation, 69, 215–217, 230, 232–234, 237, Hamilton, Sir William, 536 239, 241, 304, 309, 310, 328, 352, 363 Hamilton, Sir William Rowan, 541, 543 strongly, 235 hammock, 413 Ferrers relation factorization, 237 Haskell, xii, 6, 25, 33, 34, 127, 511, 513 Ferrers, Norman Macleod, 232 Hasse Fishburn, Peter C., 309, 323 diagram, 67 flexible availability, 469 relation, 67 forbid, 461 Hausdorff, Felix, 457 fork operator, 115 heterogeneous, 16 forward simulation, 496 homogeneous, 16 fringe, 211, 213, 214, 216, 217 homomorphism, 84 fringe-partial homomorphism theorem, 446 column equivalence, 217 hypergraph, 20 row equivalence, 217 ideal completion, 294 Frobenius–König Theorem, 437 idempotent, 70, 277, 528 fully discharge, 503 symmetric, 217 fully indecomposable, 402 identity, 40 function, 26, 49, 50 immediate set-valued, 15, 16 predecessor, 67 fuzzy measure, 372 successor, 67 Galois correspondence, 463, 485 implication structure, 461 Galois iteration, 415 incidence, 20, 452 games, 422 incomparability, 358, 360, 361 generalized inverse, 127, 210, 247 indecomposable geometry fully, 402 affine, 452 totally, 402 projective, 455 independent pair of sets, 98, 206 Ghouilà-Houri, Alain, 345 independent set, 206, 231 glb, 188, 189 indifference, 352–355, 358, 359, 361, 363 glbR, 189 strict, 361 Gödel, Kurt, 541 infectivity of contact, 280 Gram matrix, 267 infimum, 189 Gram, Jørgen Pedersen, 267 initial part, 418 graph Inject, 136, 512 1-, 21 injection, 131, 135 bipartitioned, 20 natural, 137, 522 chordal, 344 injective, 50, 57 comparability-, 337 integral directed, 22 Choquet, 379 intersection-, 333 relational, 376 interval-, 333, 337 Sugeno, 371, 376, 378, 379 simple, 23 intermediate point theorem, 182, 493 triangulated, 344 intermediate vector theorem, 493 graph covering, 445 internally stable, 348 gre, 187 internally stable set, 104 greatest intersection graph, 333 element, 187 interval lower bound, 189 (strict)order, 309 ground, 9 graph, 333, 337 baseset, 8 order, 61, 308, 309, 323, 330, 332, 352

564 Index

inverse, 248 mapping, 57 generalized, 127, 210 Marczewski, Edward (see Szpilrajn), 69 Moore–Penrose, 210 mask, 166 Thierrin–Vagner, 247, 249 matching, 56, 74, 428, 429, 436 Iota, 128, 512 by dicliques, 257 irreducible, 396, 397, 407 by maxcliques, 264 irreflexive, 70, 351 matrix, Gramian, 267 isomorphism theorem, 446 max, 184 isotone, 84, 277 maxclique, 205, 265, 267, 337, 338 Jevons, William Stanley, 537 bijection, 264 join operator, 115 factorization, 265, 267 jump part, 386 matching, 264 measure Kappa, 128, 512 Bayesian, 373 kernel, 426 fuzzy, 372 Klein four-group, 27 relational, 371, 372, 376 knowledge acquisition, 229, 230 , 141, 512 knowledge discovery, 221 Member membership relation, 140 König–EgerváryTheorem, 436 Kowa, Seki (Kowa) Takakazu, 542 Milgram Kronecker product, 116, 127 Theorem of Birkhoff–, 324 Kuratowski Theorem, 198 min, 184 mincut, 194, 477 L -simulation, 496, 501 minorant, 185 Ladd-Franklin, Christine, 544 modular law, 158 lattice, 183, 190 monotone, 84 lattice-continuous, 193, 373 monotype, 36 lbd, 186 Moore–Penrose, 247, 249, 406 lea, 187 Moore–Penrose inverse, 210 least Morgan triples, 387 element, 187 Morgan, Augustus De, 536, 537, 540 upper bound, 189 morphism set, 158 left multi-covering, 441–443, 500 equivalence, 77 multiplication, 40 identity, 40 multirelation, 148 residual, 44 multivalent Leibniz, Gottfried Wilhelm, 533, 534, 537, 539, part, 92 542, 543 zone, 92 lexicographic order, 112 lifting property, 444 natural injection, 137, 522 line covering, 98, 436 natural projection, 131, 133 linear negation, 389 extension, 317 strict, 389 order, 68 strong, 389 strictorder, 68 negation, relational, 392 Löwenheim, Leopold, 547 negatively transitive, 306, 351 loss, 424 non-enlargeable rectangle, 200, 217 Lovelace, Ada, 543 non-value zone, 92 lower bound, 185 norm, t-, 388 cone functional, 168 norm, relational, 392 greatest, 189 NP-complete, 467, 474, 477 lower cone, 187, 295 Ockham, William of, 38, 539 lub, 188, 189 order, 61, 65, 66 lubR, 189 -dense, 324 Luce, R. Duncan, 167, 312, 353 -shape, 242 machine learning, 221, 229 Egli–Milner, 494 Maddux, Roger Duncan, 122, 139 interval, 61, 309, 323, 330, 332, 352 majorant, 185 lexicographic, 112

Index 565

linear, 68 predictive modelling, 229 semi-, 61, 321, 328 preference, 352, 359, 360 strict-, 66 weak, 360 total, 68 preference structure, 359 weak-, 61, 306 preorder, 65, 66, 71 order ideal, 295 column-is-contained, 71 ordering, associated, 67, 68 connex, 527 ordre fort, 308 Egli–Milner, 493 ordre quasi-fort, 308 row-contains, 71 Organon, 531, 532 section-, 71, 312 orientable, transitively, 345 simulation, 499 overcredulous belief, 373 primitive permutation, 400 pair of covering sets, 98, 206 relation, 396 pair of independent sets, 98, 206, 231 product parallel composition, 116 direct, 107, 117 partial Kronecker, 127 equivalence, 528 tensor, 127 preorder, 65, 66 progressively transversal, 435 bounded, 420 partition, 31 finite, 102 partly decomposable, 402 infinite, 102, 418 path equivalence, 445 Project, 132, 512 pattern projection, 117 doubly stochastic, 403 projective geometry, 455 Peano, Giuseppe, 535 pseudo availability, 469 Peirce, Charles Sanders, 544 purely additive part, 386 Penrose Moore–, 247, 249 quadrangle, complete, 20, 452 PER, 528 quasi-series, 352 percentage, 25 quasiorder, 65, 66 periodic, 503 quotient permanent, 437 set, 131, 133 PermTgt, 149, 512 symmetric, 46, 174 permutation, 13, 28, 149 QuotMod, 132, 512 cogredient, 101, 304 randomly generated relation, 25 cyclic, 400 rank, 208 primitive, 400 Schein-, 264 relation, 18 realizable, 323 subset, 13 ReArrTo, 149, 512 Pi, 108, 512 rectangle, 94, 95, 98, 200 picycle, 94, 362 around a relation, 97 point, 58 inside a relation, 97, 200 -free, 33, 50 non-enlargeable, 200, 217 Point Axiom, 181, 206, 259, 492 outside a relation, 97 power rectangular, 96 direct, 146 closure, 97 ordering, 145 relation, 95, 107 relator, 491 reducible, 99, 396, 397 transpose, 145, 484 reducing vector, 100 powerset atom, 145 reflexive, 76, 351 powerset element, 11 relation préordre associated, 21 préordre commen¸cant, 71 block-transitive, 243, 352 préordre des sections, 71 Boolean operation on a, 37 préordre finissant, 71 carré(French),96 predecessor, immediate, 67 composition of, 40

566 Index

converse of, 39 Schröder, 41, 65, 158–160, 162, 163, 167, cyclic, 396 170, 171, 182, 210, 531, 538 difunctional, 74, 220, 221, 226, 228, Tarski, 58, 181, 398 230–232, 352, 404 Russell, Bertrand, 546 dual, 43, 350 Schein, Boris M., 264 Ferrers, 69, 215–217, 230, 232–234, 237, 239, Schein-rank, 264 241, 304, 309, 328, 352 scholastics, 532, 540 heterogeneous, 16 Schröder equivalences, 41, 158 homogeneous, 16 Schröder, Ernst, 545 irreducible, 396 Scott–Suppes Theorem, 328 membership-, 140 Scotus, (John) Duns, 532 multi-, 148 section preorder, 71, 312 multiplication of, 40 Seki (Kowa) Takakazu, 542 primitive, 396 semi-connex, 62 randomly generated, 25 semi-transitive, 305, 328 rectangular, 95, 107 semiorder, 61, 308, 321, 328 reducible, 99, 396 set square-shaped, 96 absorbant, 103, 104 strongly Ferrers, 235 covering, 98, 206 transposition of, 39 externally stable, 104 type of, 35 independent, 98, 206, 231 relational internally stable, 104 integral, 376 stable, 103, 104 measure, 371, 372, 376 set-valued function, 15, 16 relational conorm, 392 sharpness, 127 relational negation, 392 shift-inverting, 177 relational norm, 392 shunting, 58, 59, 144 relational subjunction, 393 simple circuit, 344 RelView, 1, 16, 396, 409, 467, 513 simple graph, 23 representation simple support mapping, 373 basesets, 6 simulation, 81, 90, 494, 496, 501 element, 8 simulation preorder, 499 relation, 15, 18 singleton, 145 subset, 9 singular value decomposition, 260, 263 residual, 65 Skat, 7, 27, 68 left, 44 slipping below negation, 59 right, 44 sorting Rho, 108, 512 topological, 80 right source, 36 equivalence, 77 splitting, 139 identity, 40 square, 95 residual, 44 square-shaped, 96 Riguet, Jacques, 46, 95, 96, 548 stable internally, 104 ROBDD, 16 stable set, 103, 104, 427 rook move, 228 stable, internally, 348 row staircase, 232 equivalence, 71, 221, 238, 324, 355 state transition system, 495 mask, 166 strict space, 253 fork operator, 115 row set intersection, 255 join operator, 115 row union space, 251 negation, 389 row-contains preorder, 71 preference, 361 rowset-to-columnset difunctional, 274 strictorder, 61, 65, 308 rowspace-to-columnspace bijection, 274 associated, 67, 68 rule complete, 68 Dedekind, 42, 60, 158, 159, 518, 520 linear, 68

Index 567

strong negation, 389 tight availability, 469 strongly TituRel, xii, 3, 16, 33, 50, 108, 119, 128, 131, complete, 62 132, 136, 139, 141, 149, 150, 155, 173, connected, 78, 397 443, 474, 511–513, 524, 525 Ferrers, 235 Toeplitz, Otto, 543 sub-inverse, 248 topological sorting, 69, 80, 317, 351 subjunction, 389, 394 topology, 457 relational, 393 total, 57 subset, 2, 5, 135 total order, 68 Boolean operations on, 11, 13 totally indecomposable, 402 extrusion, 8, 135, 137, 513, 522 tournament, 64 permutation, 13 transitive, 64 substitution property, 82 negatively, 306 successor, immediate, 67 semi-, 305, 328 Sudoku, 479 transitive closure, 76 Sugeno integral, 371, 376, 378, 379 transitively orientable, 345 sum, direct, 127 transposition, 39 summarization, 229 transversal, partial, 435 support mapping, 373 treillis, 190 supremum, 189 triangulated graph, 344 surjective, 57 triangulation property, 344 SVD decomposition, 263 triples, De Morgan, 387 symmetric, 63, 70 tupeling, 116, 117 idempotent, 217, 528 type of a relation, 35, 107 quotient, 46, 174 U-simulation, 496 syq, 46 ubd, 186 syq properties, 172 union, disjoint, 127 system dynamics, 420, 501 Unit-1, 151 Szpilrajn UnitOb, 151 extension, 69, 80, 245, 316, 317, 351, 526, univalent, 49–51, 516 527 univalent part, 92, 414, 475 Theorem, 69, 316 univalent zone, 92 Szpilrajn (Marczewski), Edward, 69 unsharpness, 127, 157 unsupervised learning, 221, 229 t-conorm, 388 upper bound, 185 t-norm, 388 cone functional, 168 tabulation, 139 least, 189 Takakazu, Seki (Kowa), 542 upper cone, 187 target, 36 utility, 323, 349, 369 Tarski rule, 58, 181, 398 Tarski, Alfred, 117, 181, 548 vacuous belief, 373 tensor product, 116 Vagner, 247, 249 term rank, 208, 436 value zone, 92 termination, 417 vector, 58, 95 Theorem reducing, 100 Birkhoff–Milgram, 324 vectorization, 124, 348, 454 Dilworth, 199, 314 Verband, 190 Farkas, 459, 460 Warshall algorithm, 77 Frobenius–König, 437 weak preference, 359, 360 Intermediate Point, 182, 493 weakorder, 61, 306, 308–310, 526 Intermediate Vector, 493 closure, 320, 321, 357, 358 König–Egerváry, 436 underlying, 320 Kuratowski, 198 Whitehead, Alfred North, 50, 546 Scott–Suppes, 328 Wiener, Norbert, 312 Szpilrajn, 69, 316 win, 424 Thierrin–Vagner inverse, 247, 249 Zierer, Hans, 46, 152 threshold, 319–321, 326–328