@ I* Index-N52870.fm Page 793 Monday, March 26,2007 10:04 AM

INDEX

a-, see semilinear map Beltrametti, E., 70 0-1 pasting, 359 Bennett, M.K., 6 benzene ring, 338, 766 a-finite, 699 Birkhoff, G., 2, 56, 63, 75, 479, 512 additive, 140 block, 157, 344, 527, 760 adjacent, 527 blocks all-optical quantum computer, 757 of an OMP, 566 almost sharp, 132 of an equivalence relation, 565 Amemiya, I., 500 Boole, G., 1 anti-isometry, 435 Boolean antiunitary operator, 435 algebra, 58, 628, 761 Araki, H., 500 D-poset, 392 Archimedean, 400 locally, 555 Archimedean t-norm, 67 power arguesian bounded, 349 morphism, see morphism of pro- product, 571 jective geometries sheaf, 571, 579 , see projec- subalgebra, 344, 569-572, 576 tive geometry subsystem, 562, 570 Arguesian law, 763 Artin, E., 494 Cartesian product, 640 Atiyah, M., 1 category, 583 atom, 86, 153, 336, 702, 760, 762 of generalized Hilbert spaces, see atomic, 86, 400 Hilbert space atomic consequence, 702 of Hilbert geometries, see Hilbert atomicity, 759 geometry atomistic lattice, 482 of Hilbert lattices, see Hilbert lat- , 151 tice of projective geometries, see pre Baer, R., 493, 496 jective geometry BDec A of projective lattices, see projec- for a set A, 560 tive lattice for a structure A, 567 of propositional systems, see propo- Bell effect, 600, 603 sitional system Bell inequalities, 70, 195 of vector spaces, see of Bell-Wigner type, 201 center, 152 of Clauser-Horne type, 201 chain-finite effect algebra, 337 Bell-type inequality of order n, 206 Chang, C. C., 3 @ I* Index-N52870.fm Page 794 Monday, March 26,2007 10:04 AM

794 Index

charge, 97 decompositions, 570 absolutely continuous, 112 elements, 569, 573 atomic absolutely continuous, 112 experiments, 573, 574 bounded, 97 set, 572 completely-additive, 97 compatible elements, 56, 62 free, 97 of an effect algebra, 345 regular, 97, 114 of an orthoalgebra, 344 sequence of uniformly regular, 110 compatible set, 155 a-additive, 97, 114 complement, 639 a-finite, 114 complex field, 759 singular, 98 concatenation, 642, 676 uniformly absolutely continuous, concrete logic, 63 112 condensed state equation, 775 Chinese lantern M02, 763 conditional probability, 128 Chomsky normal form, 714 conformal regularity of depth-first, 672 Chovanec, F., 6 congruence, 423, 567 classical Hilbert lattice, 782 conjunction, 63 classical state, 769 connecting an atom with an element, clopen, 576 362 coexist, 142 context-free grammar, 710 collinearity relation, 480, 483 continuous lattice, 482, 483 , 496 convexity models, 592 Baer's existence theorem for cen- Con A, 567 tral, 496 correlations, 606 homothety, 497 classical, 608 translation, 497 quantum, 609 commutative covering law, 483, 503 context-freeness in the width-first Cram&-Wold theorem, 182 way, 721 cycle, 355 deterministic regularity, 660 Myhill-Nerode property, 701 D-a-poset of fuzzy sets, 371 regularity in the depth-first way, D-homomorphism, 404 653, 672 D-lattic, 381 regularity in the width-first way, D-lattice, 340 653 D-poset, 340, 375 commutator, 155 D-poset of fuzzy sets, 371 lower, 201 DAC-lattice, 462 of observables, 170 Day, A., 763 upper, 201 de Morgan triplet, 68 I'(A), 630 decomposition, 560, 581 -/(A),630 Dec A, 573 of A, 637 definite Hermitianform, 450 commute, 526 density functions, 606 compatible, 5, 129, 135, 385 density operator, 125 @ I* Index-N52870.fm Page 795 Monday, March 26,2007 10:04 AM

Index

deterministic regularity, 660 lattice of, 559 diamond, 339 permuting, 559, 562 difference, 371 regular, 565 difference poset, 6 Eq A, 559 direct limit, 347 expectation of a real observable, 177 discrete measurement, 125 expected value, 577, 579 disjoint union, 669 experimental system, 572 disjunction, 63 axioms of, 574 dispersion of a state, 196 the Hilbert space model, 577 distributive, 527 with probabilities, 576, 578 distributive triple, 150 disturbing measurement, 138 factor DP-morphism, 346 n-tuple, 562 DvureEenskij, A., 4, 64, 76 pair, 562 dynamic programming algorithm, 776 factoring numbers, 755 Fact A, 567 EA-morphism, 346 for a bounded lattice, 583 effect algebra, 6, 57, 65, 126, 140, for a Hilbert space, 568 336, 595 for a module dA, 568 chain finite, 337 for a poset, 583 effect-valued measure, 125 for a relational structure, 568 effects, 125, 140, 592 for a ring, 583 sharp, 593 for a set, 563 eikonal form, 122 for a topological space, 568 elements for a vector space, 568 f -compatible, 155 for an algebra, 567 compatible, 148 for an inner product space, 568 compatible in M, 154 Fano plane, 494 compatible pairwise, 154 Faure, C1.-A., 477, 493, 498, 515 compatible strongly, 154 field of quaternions, 460 orthogonal, 148, 339 filter, 420, 576, 577, 580 entanglement, 610 fold, 679 equivalence of categories Foulis, D. J., 5, 57 of Hilbert geometries, Hilbert lat- Foulis-Holland set, 150 tices and propositional sys- Frolicher, A., 477, 493, 515 tems, 507 frame function, 99 of projective geometries and pro- generalized, 114 jective closure spaces, 514 full fuzzy set system, 141 of projective geometries and pro- fuzziness index, 134 jective lattices, 487 fuzzy of state spaces and property lat- eigenvalues, 132 tices, 515 P-measure, 369 equivalence relation quantum logic, 61 blocks of, 565 set, 58 @ I* Index-N52870.fm Page 796 Monday, March 26,2007 10:04 AM

796 Index

set system, 141 irreducible, 511, 517 of finite rank, 503 generalized Hilbert lattice, 525 Hilbert space, 7, see generalized Hilbert generalized Hilbert space, 450, 499, space, 555, 556, 568, 577- 500 579, 581 ample unitary group axiom, 519 Hilbert space model, 75 category of -s, 505 Hilbert, D., 1 Salk's Theorem, 500, 519 Hilbert-space formulation of quantum generalized orthoarguesian equations, mechanics, 430 763 Hilbert-space-valued state, 528, 782 generators, 779 Holland, S., 512 Geneva School, 449 homogeneous coordinates, 494 Georgescu, G., 3 homomorphism Giles, R., 59, 61 a-,151 Gleason Theorem, 99, 186, 787 lattice, 151 Gleason, A. M., 76, 98 horizontal sum, 359 Gleason, W., 2 hypergraph,' 351 Godowski identity, 770 hyperplane, 496, 512 Godowski's equation, 771 Godowski, R., 769, 770 ideal, 342, 420 Greechie diagram, 161,351, 531, 760, cr-ideal, 204 766 pideal, 204 Greibach normal form, 714 prime, 204 ideal of PMV-algebra, 23 H-rich, 528 identification of atoms, 359 Hassse diagram, 761 image, 640, 737 height, 525 implication operator, 632 Heisenberg, W., 2 indefinite inner product space, 451 Hermitian form, 500, 512, 515 indefinite metric space, 451 hexagon, 766 independent, 138 hidden variables, 198 in the state W, 129 Hilbert C*-module, 454 infinite-dimensionalHilbert space, 759 Hilbert geometry, 499, 515 inner product space, 76 category of -ies, 505 intersection, 639 coproduct decomposition of, 510 property, 483 induced by a generalized Hilbert interval, 336 space, 500 effect algebras, 337 induced by a propositional sys Iorgulescu, 3 tem, 503 isometry, 435 irreducible, 510, 517 isomorphism, 140, 151, 346 of finite dimension, 503 isotropic, 140, 337 projector of, 516 index, 337, 400 representation for arguesian, 512 Hilbert lattice, 339, 501, 525, 759 Jauch-Piron condition, 198 coproduct decomposition of, 511 Jauch-Piron property, 158 @ I* Index-N52870.fm Page 797 Monday, March 26,2007 10:04 AM

Index 797

joint observable, 414 lattice, 150 joint probabibility distribution of idempotents of a ring, 568,572 of Gudder type, 174 of questions, 574, 575 regular, 151 k-homogeneous, 527 rich, 158 K&pka,F., 6 rich2, 158 Keller field, 782 separable, 152 Keller, H. A., 76, 500 unital, 158 kernel kerf, 559 Loornis-Sikorski theorem, 166 Kleene closure, 642 loop, 355, 760 Kolmogorov, A. N., 2 Loop Lemma, 356 l-valued (binary) relation, 641 Lukasiewicz, J., 55, 59, 61, 65 Lvalued (quantum) determinisitc fi- nite automaton, 655 M-symmetric, 583 l-valued (quantum) nondeterministic Mgczyliski, M. J., 55, 59, 65, 66, 70 finite automaton, 644 Mach-Zehnder inferometer, 756 l-valued context-free grammar, 710 Mackey, G., 2 l-valued finite automaton with €-moves, Mayet's Eequations, 781 661 Mayet, R., 773 l-valued interpretation, 637 McKay, B., 766 l-valued language, 642 measure, 98 l-valued pushdown automaton, 723 completely additive, 114 e-valued regular expressions, 684 measurement opeartor, 135 l-valued set, 637 Megill, N. D., 763, 773 l-valued subset construction, 655 membership function, 58 langauge L[~I(G)E L" generated Menger, K., 67 by G in the width-first way, Mesiar, R., 67, 70 718 metric operator, 452 language L[~](G)generated by G in mixed states, 127 the depth-first way, 711 lattice, 762 M02, 339 orthomodular, 339 modular, 80 lattice effect algebra, 337 lattice, 482, 483, 583 leveled set, 572 lower semi-, 483 acted on by (3, 573, 575 ortho-, 502, 516 limit upper semi-, 483 direct, 347 law, 781, 786 linear weak topology, 463 OLs, 781 Ling, C. H., 67 Molnk, L., 455 logic, 148 monomorphism, 140 AND,OR,NOT,TRUE,FALSE, 574 morphism, 140 atomic, 153 of projective geometries, 485,486, atomistic, 153 498, 508 concrete, 159 of propositional systems, 504,508 @ I* Index-N52870.fm Page 798 Monday, March 26,2007 10:04 AM

798 Index

of projective lattices, 486, 505, complementary, 180 508 function of an observable, 165 orthe, 508 functions of compatible observ- Moulton plane, 495 able~,166 Moulton, F. R., 495 fuzzy, 597 Mundici, D., 3 partially compatible, 173, 177 Myhill-Nerode property, 701 real, 165 sharp, 593 n-power, 641 sumrnable, 181 nearly sharp, 132 totally noncompatible, 177 negation, 125 OML, 526 Neumann, J. von, 2, 56, 63, 75, 479, OML law, 788 512 operational framework, 597 nOA identity law, 767 operator non-degenerate morphism, see mor- adjoint, 185, 187 phism of projective geome- bounded, 185 tries density, 186 non-disturbance, 135 domain of an operator, 187 noncommutative context-free language norm of an operator, 185 over C according to the depth- projection, 185 first principle, 712 self-adjoint, 185, 187 nondeterministic finite automaton, 643 unbounded, 187 nondeterministic pushdown automa- ordered partition, set of (3, 572 ton, 723 ordering set of states, 57 nonprincipal ultrafilter, 532 orhtoalgebra, 140 normal ideal, 23 ortharguesian law, 528 normed group with operators, 584 I-symmetry (orthesymmetry), 436 NR-substitution, 361 orthea complete logic, 5 observable, 75, 125, 165, 409 orthea-complete logic, 5 eigenvalue of a real observable, orthomodular lattice, 759 167 orthoalgebra, 5, 57, 65, 336, 583 real, 167, 168 orthoarguesian law, 763 constant, 168 orthocomplemental poset, 79 discrete, 168 orthocomplementation, 4, 56, 126 propostiion (0 - I), 168 orthogonal, 5 simple, 168 cover, 155 spectrum of a real observable, elements, 56, 339 167 joins @, 564, 568 spectrum of a spectral value, relation, 499 167 ortholattice, 80, 96, 338, 627, 766 value space of an observable, 165 orthomodular, 126 observable quantity, 576-581 a-poset, 148 observables, 589, 592 complementation, 4 compatible, 165, 177 lattice, 5, 80, 339, see modular @ I* Index-N52870.fm Page 799 Monday, March 26,2007 10:04 AM

Index

lattice, 526, 628 -valued measure, 125, 135 law, 148, 339, 555 projective closure, 482, 514 poset, 4, 56, 57, 79, 148, 762 projective geometry, 480 space, see generalized Hilbert space, arguesian, 494, 499 525 category of -ies, 485 orthoposet, 126, 338 category of arguesian -ies, 499 orthosum, 336 coproduct decomposition of, 490 orthosummation, 376 dimension of, 494 orthosupplementation, 57 discrete, 489 orthosupplement, 337 Fundamental Theorems of, 495, 498, 499 pair of dual spaces, 462 induced by a projective lattice, Pappus' Theorem, 497 484 parallel combination, 128 induced by a vector space, 480 parallel measurement, 126 irreducible, 488, 491, 517 partial diagram, 533 opposite of, 512 partially compatible subset, 169 projective lattice, 482 pasted family of Boolean algebras, 350 category of -s, 486 pasting of Boolean algebras, 350 coproduct decomposition of, 493 pasting of orthoalgebras, 358 induced by a projective geome- PaviEib, M., 763, 773 try, 482 permute, 559 induced by a vector space, 483 Peterson OML, 772 irreducible, 493, 517 phase factor, 435 phase space, 71 rank of, 494 physical symmetry, 436 projective law, 482 Piron, C., 449, 478, 500, 503, 512, projective line, 481 516 projector, 516, 784 poset with difference, 373 proper filter, 421 positive operator-valued measure, 125, proper ideal, 421 135 property lattice, 515 pre-image, 640, 683, 738 propositional system, 449, 503 probability coproduct decomposition of, 512 amplitude, 122 induced by a generalized Hilbert density current, 124 space, 503 map, 575 irreducible, 511, 517 measure, 56, 75, 577, 769 Piron's Representation Theorem velocity vector, 124 for, 512 Prob S, 575 pseudo MV-algebra, 19 product, 348, 672, 740 Ptbk, P., 3 projection map xi, 560 Pulmannovi, S., 70, 76 mapping, 526 pure state, 125 operator, 125, 141 pushout, 583 postulate, 126 Pykacz, J., 55 @ I* Index-N52870.fm Page 800 Monday, March 26,2007 10:04 AM

800 Index

quantum Santos, E., 70 bit, 757 Sasaki, U., 517 computation, 755 Sasaki-hook, 633 computing algorithms, 755 scalar product, 639 Hilbert lattice, 782 scaling, 576-580 law of total probability, 129 SEA isomorphism, 143 logic, 5, 56, 60, 61, 80, 148, 762 Segal product, 188 logic gate, 757 self-adjoint operator, 578, 579 probability, 70 semantic consequence, 638 state, 769 semi-unitary map, 514 structures, 3 semilinear, 435 quasi hidden variables, 199 semilinear map, 485, 505, 518 quasivariety, 207 sequential effect algebra, 132, 140 quaternion skew field, 759 sequential product, 128, 141 quaternionic Hilbert space, 460 series combination, 128 qubit, 757 set of states questions, 574, 578-580 order determinign, 158 ideal, 580, 581 rich, 158 unital, 158 RachBnek, J., 19 sharp, 140 Randall, C. H., 5 element, 141 random variables, 598 quantum measurement, 135 ray, 431 yes-no measurement, 125 real field, 759 Shor's algorithm, 755 real-valued state, 528 a-Riesz interpolation property, 89 recognizability recg'FS1of R by final a-effect algebra, 140 stats in the depth-first way, a-morphism, 140 724 a-orthomodular poset, 762 recognizability by R in the width-first simplex linear programming algorithm, way, 646 776 recognizability of R by final states in soft fuzzy a-algebra, 369 the width-first way, 727 Sokr, M. P., 76, 782, 500, 518 recognizability recognizability recw[Dl spectral by R in the depth-first way, measure, 186, 578 645 resolution, 187 rectangular treillis, 337 theorem, 187 regular, 379 value, 167 OMP, 572, 582 spectrum, 409 equivalence relation, 565 spectrum of an operator, 187 regularity in the depth-first way, 648 splitting subspaces, 569, 572 regularity in the width-first way, 648 standard refinement, 574 relation algebra, 582, 583 state, 5, 56, 98, 125, 140, 417, 589, rich, 528 769 Riesz interpolation property, 89 dispersion-free, 196, 590 @ I* Index-N52870.fm Page 801 Monday, March 26,2007 10:04 AM

Index

null, 576 deterministic, 198 of an experimental system, 576 imbedding, 197 on a logic, 157 Jauch-Piron, 158 Tate, J., 1 mixture, 157 trace, 457 pure, 157 triangular conorm, 67 twevalued, 157 triangular norm, 67 on the OMP Q(E), 577 Willas, E., 68 on OA, 6 on OML, 5 Uhlhorn, U., 445 on PMV-algebra, 24 ultraproduct, 532 uncertainty relation, 179, 194 subadditive, 164 uncorrelated, 129 support of, 164 union, 639 state space, 125, 500, 515 unital, 528 state-morphism on PMV-algebra, 25 unitary map, 514 Stone space, 571, 576, 579-581 unitary operator, 435 strongly compatible, 154, 390 unsharp measurement, 135 strongly regular, 379 sub-OMP,569 value space, 165 subeffect algebra, 344 variance, 178 subalgebra, 569 variety, 207, 526 sublogic Veblen, O., 478 Boolean, 151 vector space generated, 154 category of -s, 485 lattice, 151 vector state, 782 subspace, 481 von Neumann, J., 2 closed, 500, 509 dimension of, 494 wagon wheel OML, 771 maximal irreducible, 490, 510 weakly atomic orthomodular, 86 maximal strict, 496 weight, 161 substitution of an atom by an orthoal- Wigner's Theorem, 429, 514 gebra, 361 Wigner, E., 433, 514 subsystem of, 197 Wright triangle, 341 sum logic, 181 yes-no measurement, 125 of observables, 181 Young, J., 478 Summhammer, J., 756 superposition of states, 205 Zadeh, L. A., 58 superposition principle, 759 surjection, 558 Surj A, 558 symmetry, 436 symmetry group, 444 system