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Index

Acceptable operation 433 Aczel, J. 497, 498, 501 adjoints 431 agent 27

label of 27 Alexandrov, P.S. 245, 247, 248, 501 algebra

Boolean 191, 260 Heyting 257 Lukasiewicz 372 Lindenbaum-Tarski 278 Nelson 362 of sets 175 of relations 184 Post 387 pseudo-Boolean 257 Stone 367

dual 368 double 368

Wajsberg 376 alternation 96 approximation 7

R-lower 7 R-upper 7 space 23

generalized 23 approximate convergence 356

discernibility 33 Archangelsky, D.A. 402, 406, 501 Aristotle of Stagira 7, 110, 121,

122, 128, 130, 138, 148, 149, 153 Arnold, V.I. 497, 498, 501 assignment 175

elementary 275 associativity law 176 attribute 5

conditional 17 decision 16

axiom instance 278 of choice 192 of power set 183 of well-ordering 192 schema of replacement 183

of separation 182

Baire, R. 223, 224, 245, 249, 501 Balcar, B. 192, 211, 501 Bamalip 133 Banach, S. 224, 245, 249, 501 BaneDee, ~. 402, 406, 501 Barbara 125 Barbari 129 Barnsley, ~.F. 346, 358, 501 Baroco 132 basic modal logic 149 Bazan, J. 22, 33, 34, 35, 36, 37, 44,

45, 52, 324, 327, 502 Becchio, D. 153, 157, 376, 381, 406,

502 Bernays, P. 118, 119, 502 Bernstein, F. 197, 207 Birkhoff, G. 208, 209, 210, 211,

268, 270, 288, 295, 502 Birkhoff principle 194 Bocardo 133 Bochenski, I.~. 130, 135, 502 Boicescu, V. 153, 157, 374, 376,

381, 404, 406, 502 Boole, G. 19, 208, 211, 269, 271,

502 Boolean algebra 189, 260

522

Borel, E. 207, 211, 502 Borel-Lebesgue property 245 Borkowski, L. 157, 462, 463, 502 bound 187

greatest lower 187 least upper 187 lower 187 upper 187

Brouwer, L.E.J. 290, 293, 502 Buszkowski, W. 323, 327, 502

Caianiello, E.R. 483, 498, 502 Calemes 133 Calemop 133 Camestres 131 Camestrop 131 Cantor, G. 187, 198, 207, 211, 222,

245, 249, 503 Cantor cube 218 Caratheodory, C. 345, 358, 503 cardinality 196 Cartesian product 184

topological 218 category 3, 5

elementary R-exact 6 R-exact 7 R-rough 7

Cattaneo, G. 402, 406, 492, 495, 496, 498, 503

Celarent 129 Celaront 130 Cesare 131 Cesaro 131 chain 192 Chang, C.C. 415, 451, 462, 503 choice function 192 Chakraborty, M.K. 402, 406, 501 Chrysippus 110 Cignoli, R. 404, 406, 451, 462, 503 class operator 31 classification space 315 closure 226

operator 226 on semi-lattice 308

Cech 242 cluster point 230 codomain of relation 184 Comer, S. 402, 406, 503 commutativity law 176 compactness 229

of a metric space 220

INDEX

of ITo-rough set spaces 339 of a topological space 229

complement 258 Brouwer 493 Kleene-Zadeh 492 of a set 175

completeness 221

241

of 4-valued logic 151 of fuzzy sentential calculus 441 of Hausdorff-Pompeju metric

of infinite valued propositional calculus 427

344

of metric space 221 of modal system K 165 of modal system S4 167 of modal system S5 168 of modal system T 166 of predicate calculus 283 of sentential calculus 106 of sequent calculus 110 of spaces of ITo-rough sets 337

of almost ITo-rough sets

of 3-valued logic 145 strong of 3-valued logic 142

component 206 conclusion 124 congruence 306 conjunction 96 Creswell, M.J. 169, 172, 506, 507 cut 43 Cech 242, 243, 244, 245, 249, 504

closure operator 242 topology of 242

Darapti 132

INDEX

Darii 130 Datisi 130 decision 16

function 34 majority 34

logic 11 elementary formula of 11 formula of 11 meaning of a formula of 11 system 16

deterministic 16 Dedekind, R. 188, 189, 194, 195,

196, 207, 211, 504 deduction branch 109 ~-gasket 358 De Morgan laws 178 Dempster, A.P. 483, 498, 504 dependency 12

(B,C)- 12 functional 12 local12 partial12 space 308

derivation 100, 109 rule 99

descriptor 39 optimal42 range of 39 volume of 39

detachment 100 fuzzy 438

difference of sets 175 Dimatis 134 dimension 345

351

algebraic 300 fractal in information systems

Hausdorff 346 Minkowski 348 topological 345

Disamis 132 discernibility function 19

matrix 19 relative 21

523

discretization 42 distributivity law 177 domain of relation 184 Driankov, D. 450, 462, 481, 504 Dubois, D. 466, 481, 483, 485, 486,

497, 498, 504 Duentsch, I. 402, 406, 504

Element, mereological 25 dense 269 greatest 187 least 187 maximal192 minimal192 of a set 175 regular 270 R-independent 309 R-reduct 310 R-redundant 310 (R, x)-inaccessible 311 sub-reduct 311 unit 254 zero 254

elementary categories 4 R-exact 6

embedding 187 Dedekind - Mac Neille 189

Epstein, G. 407, 504 equipotency 196 equivalence relation 4, 202

class of 4, 202 fuzzy 486

extensionality axiom 181 extremal disconnectedness 232

Falconer, K.J. 246, 349, 358, 504 family of functions 302

of sets 182 centered 220 closed 187 independent 301

point separating 331 Farinas del Cerro, L. 498, 504 Faucett, W.M. 476, 479, 499, 505

524

feature 5 Fedrizzi, M. 48, 76 Felapton 132 Ferio 130 Ferison 133 Fesapo 134 Festino 132 Feys, R. 169, 171, 505 field of sets 9, 179 figure of syllogism 122 Filipoiu, A. 157, 502 filter 200

Frechet 201 maximal 200 on Boolean algebra 264 on lattice 261

residuated 434 prime 264 principal 200

fixed point 462 Font, J.M. 452, 462, 505 formula 11

meaning of 11 open 286 prenex 285 true 11

four valued logic 148 Fraenkel, A.A. 207 Frechet, M. 201, 245, 249, 505 Frege, G. 99, 110, 119, 207, 211,

505 Fresison 134 function 185

characteristic of a set 218 continuous 233 contracting 224 fuzzy membership 428 injective 185 isotone 188 propositional 275 pseudo-inverse 467 quotient 202 lower semi-continuous 4 7 4 upper semi-continuous 474

functional dependence 302 functor 96 fuzzy logic 427

completeness of 441

Galois connection 209 Gao, J.M. 498, 499, 510

INDEX

Gentzen, G. 107, 111, 119, 291, 292, 293, 295, 390, 407, 505

Gentzen implication 107 Glazek, K. 323, 327, 505 Georgescu, G. 157, 502 Goguen, J.A. 428, 451, 462, 505 Goldberg, D.E. 37, 45, 505 Goldberg, H. 139, 142, 153, 157,

505 Godel, K. 293, 295, 505 granule of knowledge 4

elementary 4, 6 Greco, S. 22, 45, 505, 506 Grzymala-Busse, J. 323, 327, 506

Hahn, H. 245, 249, 506 Hajek, P. 420, 452, 462, 506 Hasenjaeger, G. 203, 295, 506 Hauptsatz 119 Hausdorff, F. 192, 237, 238, 240,

245,249,345,346,348,356,358,506 Hausdorff space 246 Hellendoorn, H. 498, 504 Henkin, L. 281, 293, 295, 506 Herbrand, J. 103, 119, 506 Heyting, A. 293, 295, 506 Hilbert cube 231 Hohle, U. 485, 486, 499, 506 Hughes, G.E. 169, 172, 506, 507 Hurewicz, W. 345, 359, 507 Hutchinson, J.E. 346, 359, 507

Ideal 200 maximal 200 principal 200

idempotency law 176 image 186

INDEX

inverse 186 implicant 19

prime 19 independence 303

linear 300 of meaningful expressions 288

indiscernibility logic 389 completeness of 394 relation 4

indiscernible elements 4 infinitely valued logic 414

completeness of 427 infinity axiom 190 information system 5

logic 15 many-valued 15

interior 225 operator 10, 225

interpretation frame 275 canonical 283 generalized 283

intersection of sets 175 of a family 182

interval, left 188 intuitionism 290 inventory 27 Iturrioz, L. 404, 407, 507 Iwinski, T. 9, 45, 369, 407, 507

Joscelin de Soissons 207

Kacprzyk, J. 48 Kalmar, L. 104, 111, 119, 507 Kanger, S. 107, 109, 110, 111, 119,

291, 292, 293, 295, 507 kernel, of necessity 496

of non-possibility 496 of possibility 496

McKinsey, J.C.C. 266, 267, 267, 270, 271, 507

Knaster, B. 188, 207, 212, 507 knowledge base 5 Kolmogorov, A.N. 497, 499, 507 Konig, D. 210, 212, 507

Konig lemma 210 Kripke, S. 169, 172, 507 Kripke semantics 162

525

Kuratowski, C. 192, 209, 212, 245, 249, 507

Langford, C.H. 169, 172, 508 lattice 189, 252

bounded 254 Brouwer-Zadeh 494 distributive 254 pseudo-complemented 257 residuated 431

Lukasiewicz 431 discrete 449

Stone 258 Zadeh 481

Leblanc, H. 139, 142, 153, 157, 505 Lebesgue, H. 245, 249, 508 Lemmon, E.J. 163, 169, 172, 508 Lesniewski, S. 25, 162 Lewis, C.I. 169, 172, 508 lifting 438 limit point 217

of a sequence of sets 240 lower 240 upper 240

Lin, T.Y. 45, 46, 48, 49, 50, 64, 65, 66

Lindenbaum, A. 16, 278, 279, 294, 470, 499, 508

Ling, C.-H. 468, 470, 476, 477, 499, 508

Luxenburger, M. 323, 327, 508

Lukasiewicz, J. 12, 13, 15, 16, 96, 99, 100, 110, 111, 118, 119, 120, 124, 125, 126, 129, 130, 131, 135, 137, 138, 139, 145, 148, 149, 150, 151, 153, 155, 157,158,187,188,289,293,295,322, 433, 434, 462, 463, 479, 480, 508, 509

Mac Neille, H.M. 188, 189, 208, 212, 509

526

Mandelbrot, B. 347, 358, 361, 511 de Mantaras, L. 484, 485, 499, 509 Marcus, S. 245, 249, 509 Marczewski, E. 323, 328, 509 Marek, W. 356, 359, 509 Matarazzo, B. 45, 56, 57, 58, 59,

505, 506 maximum principle 192 McNaughton, R. 413, 463, 509 meaningful expression 97

equiform 99 equivalence of 97 open 286 prenex form of 285

Menger, K. 208, 212, 509 Menu, J. 475, 478, 499, 509 Meredith, C.A. 415, 451, 452, 463,

509 metric 214

D on rough sets 336 D* on rough sets 336 discrete 214 Euclidean 214 fuzzy 485 Hausdorff-Pompeju 237 Manhattan 214 natural 214 product 218 space 214

Mitchell, T. 16, 18, 46, 509 modal expression 148 Moisil, G. 404, 407, 510 Mostert, P.S. 475, 476, 479, 499,

510 Mostowski, A. 209, 212, 507 Mundici, D. 451, 462, 503

Nakamura, A. 498, 499, 510 necessitation 160 necessity operator 15 negation 96 neighborhood 216

basis 216 Nelson, D. 362, 407, 510

INDEX

Nelson algebra 362 Nguyen, H.S. 22, 42, 46, 66, 70, 71,

510 Nguyen, S.H. 22, 39, 42, 44, 45, 46,

70, 71, 72, 510 Nistic6, G. 495, 498, 503 Novak, V. 420, 452, 463, 511 Novotny, M. 13, 46, 306, 308, 309,

310,321,323,324,325,328,329,511, 512

n-valued logic 145

Obtulowicz, A. 9, 46, 402, 407, 512 open ball 214

of radius r 214 open covering 220 ordered k-tuple 181 ordered pair 181 ordering 186

complete 187 linear 186

Orlowska, E. 11, 15, 46, 47, 55, 56, 327, 361, 395, 406, 408, 501, 502,512, 513

Pagliani, P. 9, 16, 47, 362, 363, 367, 371, 402, 408, 513

Pal, S.K. 498, 499, 513 Panti, G. 451, 463, 513 part 25 partition 203

fuzzy 487 Pavelka, J. 414, 428, 429, 430, 431,

433,436,438,439,440,441,442,444, 446,449,452,454,455,463,478,479, 499, 513

Pawlak, z. 3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16, 20, 21, 46, 47, 48, 72, 73, 74, 75, 76, 77,306,308,322,323,324, 326,328,329,356,360,361,395,408, 498, 500, 513, 514

Pedrycz, W. 498, 500, 514 Peirce C.S. 293 Poincare, H. 204, 208, 212, 514

INDEX

point 214 Polkowski, L. 10, 25, 31, 48, 245,

249, 354, 356, 357, 359, 514, 515 M. Polkowska-Semeniuk, 31, 48,

515 polynomial 422

formula 422 over lattice 269 over reals 422

Pompeju, D. 237, 238, 240, 245, 249, 516

Pomykala, J. 9, 48, 369, 404, 408, 516

Pomykala, J.A. 48, 369, 408, 516 Post, E. 111, 119, 138, 153, 158,

388, 408, 516 power set 183 Prade, H. 466, 481, 483, 486, 497,

498, 504 predicate 273

calculus of 273 premise 124

major 124 minor 124

principle of mathematical induc­tion 190

proof 100 proposition 95

forming functor 96 of necessity 159 of possibility 159

propositional logic 95 completeness of 106 fuzzy 427 intuitionistic 290 modal165

pseudo-complement 256 relative 256

pseudo-complementation 363 weak relative 363

quasi-complementation 363 quotient topology 236

527

Ramsey, F.P. 211, 212, 516 Rasiowa, H. 11, 49, 245, 249, 266,

267,269,271,272,279,281,283,293, 296,356,359,362,406,409,434,463, 509, 516

Rauszer, C. 15, 18, 19, 20, 21, 49, 323,324,326,329,330,389,390,394, 409, 516, 517

reduced R-rough set 9 reduct 18

(B, C,ry)- 18 6-decision 32 dynamic 35

generalized 36 s-discerning 33 local 34 m-decision 330 relative 20

region, positive 14 B-positive 17

Reinfrank, M. 462, 498, 504 relation 183

accessibility 162 informational inclusion 395 inverse 184 linear 186 a-indiscernibility 6 B-indiscernibility 6 containment 15 d-indiscernibility 16 equivalence 4, 202

finer 203 fuzzy similarity 483 identity 185 indiscernibility 4, 395 likeness 484 pre-order 252 probabilistic similarity 484 similarity 23 tolerance 23, 204

informational 395 reflexive 186 symmetric 186 transitive 186

528

weak anti-symmetric 186 relational system 401 residuation 432 residuated implication 465, 473

dual 485 restriction 185 Riemann, B. 472 Riesz, F. 245, 249, 517 Rissanen, J. 18, 49, 517 r-net 220 Rodriguez, A.J. 462, 505 Rose, A. 413, 414, 415, 416, 419,

421,423,424,427,451,455,456,457, 458, 459, 460, 461, 463, 517

Rosenbloom, P. 388, 409, 517 Rosser, J.B. 145, 146, 148, 153,

155,158,413,414,415,416,419,421, 423,424,427,451,455,456,457,458, 459,460,461,463,517

rough equality 9 bottom 318 top 316

rough inclusion 26 rough membership function 21

generalized 23 rough mereology 26 rough set 7 Rousseau, G. 404, 409, 517 Rudeanu, S. 157, 502 Ruspini, E.H. 484, 500, 517 Russell, B. 244 Russell antinomy 207

Schroder, E. 209, 212, 269, 272, 517 Schweizer, B. 471, 497, 500, 517 Scott, D.S. 163, 169, 172, 508 selection 29 selector 192 semantic consequence 141, 431 semi-lattice 306 sentence 95 sentential calculus 95 sequent 107 set 174

bounded 187 from above 187 from below 187

clopen 229 closed 226 countable 198 definable 320 empty 181 finite 194

INDEX

in Dedekind' sense 194 in Tarski' sense 195

fuzzy rough 483 infinite 195 interior of 225 meager 223 nowhere dense 223 of cuts 43

consistent 43 irreducible 43 optimal43

of 1st category 223 open 225 ITo-exact 332 ITo-rough 332

almost 341 positive 13 quotient 5, 202 regular closed 227 regular open 227 R-rough fuzzy 482 syntactically consistent 139

maximal139 syntactically inconsistent 139 well-ordered 192

sequence 217 converging 255 fundamental 261 limit of 256

sequent 127 calculus 127 true 129

Shafer, G. 483, 500, 518 Shields, A.L. 476, 477, 479, 499,

510

INDEX

Sierpinski, W. 470, 500, 518 Sikorski, R. 233, 245, 249, 266, 267,

269,272,279,281,293,296,434,463, 516

similarity relation 23 Sklar, A. 471, 497, 500, 517 Skowron, A. 9, 11, 18, 19, 21, 22,

25, 31, 42, 46, 48, 49, 50, 51, 52, 53, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 92, 322, 324, 329, 356, 360, 402, 498, 499, 514, 515, 516, 517

Slowinnski, R. 22, 45, 49, 56, 57, 58, 59, 61, 84, 85, 92, 498, 504, 515, 518

Slupecki, J. 130, 136, 153, 156, 158, 518

stability coefficient 36 standard object 27 Stepaniuk, J. 22, 49, 83, 84, 85, 86,

324, 330, 518 Stoics 110 Stone, M.H. 245, 250, 258, 264,

269, 272, 518 Stone duality theorem 230 Stone space 232

lattice 258 subset 179

s-discerning 33 m-defining 34

substitution 99 superposition 185 surjection 186 syllogistic 122 symmetric difference 175 syntactic consequence 430 Synak, P. 45, 502 system K 160 system S4 166 system S5 167 system T 166 system (W) 139 Sl~zak, D. 31, 34, 49, 71, 87, 88,

325, 330, 518

529

Stepanek, P. 192, 211, 501

Taitslin, M.A. 402, 406, 501 Tarski, A. 16, 188, 195, 196, 207,

266,267,269,270,271,272,278,279, 294, 296, 509, 518, 519

template 39 generalized 39 quality of 39

theorem 98

188

Baire category 223 Banach fixed point 224 Bayes logical 289 Cantor diagonal 197 Cantor completeness 222 Cantor-Bernstein 197 Dedekind- MacNeille 188 Herbrand deduction 103 Knaster-Tarski fixed point

Tikhonov 231 thesis 99 Tikhonov, A.N. 231, 245, 250, 519 tolerance relation 23, 204

class 205 pre-class 205

topological Boolean algebra 267 rough set 332 space 225

Cech 242 To, T1, T2 246

Torrens, A. 462, 505 Traczyk, T. 409, 519 transitive closure 206 triangular conorm (t-conorm) 472 triangular norm (t-norm) 466

archimedean 467 strict 467

truth value 12 falsity 95 fractional 287 of a proposition 95 truth 95

Tsumoto, S. 48, 50, 66, 89

530

Turquette, A.R. 145, 146, 148, 153, 155, 158, 419, 420, 463, 517

Ultrafilter 200 union of sets 175

axiom 182 unordered pair axiom 181 Urysohn, P.S. 245, 249, 501

Vakarelov, D. 11, 15, 49, 394, 397, 401, 409, 519

valuation 141 value set 6 Valverde, L. 484, 485, 487, 488,

499, 500, 509, 519 Vanderpooten, D. 49, 518 Varlet, J. 404, 409, 519 Vaught, R.L. 194, 208, 212, 519 Vaught principle 194 Vietoris, L. 245, 250, 519

VVajsberg, 11. 16, 138, 139, 153, 158, 414, 427, 451, 464, 519

VVallman, H. 345, 359, 507 VVeaver, G. 139, 142, 153, 157, 505 VViweger, A. 356, 360, 519 world, possible 162

accessible from w 162 von VVright, G.H. 169, 172, 519 VVr6blewski, J. 45, 502

Zadeh, L.A. 427, 451, 464, 466, 479, 480, 481, 483, 485, 500, 519, 520

Zeeman, E.C. 204, 208, 212, 520 Zeno of Kithion 110 Zermelo, E. 192, 207, 212, 520 Zermelo-Fraenkel system 207 Zorn, 11. 192, 208, 212, 520 Zorn lemma 192 Zytkow, J. 245, 249, 515

Yaeger, R.R. 48

INDEX

List of Symbols

Rc equivalence relation 4 [x]R equivalence class 4 (U, A) information system 6 IN DB B-indiscernibility relation 6 RY R-lower approximation 7 RY R-upper approximation 7 BY B-lower approximation 8 BY B-upper approximation 8 Y = Z rough equality 9 (I, D) disjoint representation 9 (a, v) elementary formula in DL-logic 11 [a] meaning of formula 11 f= semantic satisfaction 11 lXI cardinality of X 13 POSsC positive region (set) 14 (U, A, d) decision system 16 MA discernibility matrix 19 RED(A) set of reducts of information system 19 Ad decision system 20 J-t~(x) rough membership function 21 X elY mereological element 25 lab( ag) label of agent ag 27 N negation functor 96 0 R, V alternation functor 96 AND, 1\ conjunction functor 96 C, => implication functor 97 pfa substition of a for p 100 f- yields symbol 103 ---+ Gentzen implication 107 r ---+ .6. sequent 107 Tr( v, S) extended valuation on a sequent 109

532 LIST OF SYMBOLS

f= semantic consequence 142 Jk Rosser-Turquette functor 146

negation in n-logic 148 C implication in n-logic 148 -1 rejection symbol 149 L necessity functor 149 M possibility functor 149 K modal system K 160 (W, R, v) interpretation frame 162 [a]~ meaning of modal formula 162 T modal system T 166 84 modal system 84 166 85 modal system S5 166 Con(r) consistent set 164 Conmax(r) maximal consistent set 164 V general quantifier 174 3 existential quantifier 17 4 => implication functor 174

negation functor 17 4 X U Y union of sets 175 X n Y intersection of sets 175 X \ Y set difference 175 6 symmetric difference 176 0 empty set 177 C set containment 179 < a, b > ordered pair 181 < a1, ... , ak > ordered k-tuple 181 U X union of family of sets 182 n X intersection of family of sets 182 X x Y Cartesian product of sets 183 domR domain of relation 184 codomR codomain of relation 184 R-1 inverse relation 184 R o 8 composition (superposition) of relations 185 fA image of set by function 186 f- 1 A inverse image of set by function 186 inf A greatest lower bound 187 supA least upper bound 187 A+ set of upper bounds of A 187 A- set of lower bounds of A 187 x U y lattice join 189 x n y lattice meet 189 -x lattice complement 189 l x J integer part of x 199

LIST OF SYMBOLS 533

cof N Frechet filter 201 X I R quotient of X by R 202 R 1 set of reals 213 B(x,r) open ball at x of radius r 214 (X,p) metricspace 214 IntA interior of A 215 ClA closure of A 216 BdA boundary of A 216 limxn limit of sequence Xn 217 2 the set {0, 1} 218 2N the Cantor cube 218 B(i1, ... ,in) binary base in Cantor cube 219 Aj_ complement to closure of A 227 RO(X) Boolean algebra of regular open sets 228 CO(X) Boolean algebra of clopen (=closed-open) sets 229 RC(X) Boolean algebra of regular closed sets 229 S(B) Stone space of Boolean algebra B 231 TR quotient topology 236 6p Hausdorff metric on closed sets 238 LiAn lower limit of set sequence 240 LsAn upper limit of set sequence 240 LimAn limit of set sequence 240 Clc Cech closure 242 yl unit element 254 n zero element 254 xc pseudo-complement to x 256 -- x complement to x 258 C(S) center of Stone algebra 260 I±J join in C(S) 261 B IF quotient of Boolean algebra by filter 265 [[a]]~ valuation value on formula 276 II all equivalence class in Lindenbaum-Tarski algebra 279 w( a) fractional value of a 287 D.F fiber product of functions 304 * semi-lattice operation 306 (A, R) dependence space 308 RED(x, R) set of R-reducts of x 310 SU BRED(x, R) set of sub-reducts of x 311 1--t R dependence arrow 312 CR closure operator in approximation space 315 DR interior operator in approximation space 317

rough top equality 317 "' rough bottom equality 318 (R, II0 ) space of topological rough sets 334

534 LIST OF SYMBOLS

I ntrr0 interior operator with respect to topology Ilo 332 Clrr0 closure operator with respect to topology Ilo 332 dH Hausdorff metric 335 D((Qb T1), (Q2, T2)) metric on rough sets 336 D*((QI. T1), (Q2, T2)) metric on rough sets 336 D' ( ( Q, T), ( Q', T')) metric on almost rough sets 343 dim11.(T) Hausdorff dimension 346 dimM (T) Minkowski dimension 348 dimA fractal dimension in information system 351 r X l least upper integer bound 355 ,...., quasi-complementation in Nelson algebra 362 --, pseudo-complementation in Nelson algebra 362 ---+ weak relative pseudo-complementation in Nelson algebra 362 ::::} relative pseudo-complemantation in Heyting algebra of rough sets

367 tx pseudo-complement in Heyting algebra of rough sets 367 cPi(x) operator in Lukasiewicz algebra 372 '---+ implication in Wajsberg algebra 375 .l empty term 389 t(t) atomic formula of indiscernibility logic 389 xly (informational) indiscernibility relation 395 xCy informational inclusion relation 395 xTy informational tolerance relation 395 T unit in Lindenbaum-Tarski algebra of informational logic 397 P F(f) set of polynomial formulae 422 XA fuzzy membership function 428 ( ®,---+) pair of adjoints 431 x ® y Lukasiewicz multiplication 432 ---+ Lukasiewicz adjoint implication 432 (F D) fuzzy detachment 438 (La) lifting 438 T triangular norm 466 ST conorm associated with T 472 ---+T residuated implication 473 Jb: the lower .r-approximation 489 B :F the upper .r-approximation 489 x~ Brouwer complement 494 x' Kleene-Zadeh complement 494 A ex crisp set associated with fuzzy set A 497 8 A characteristic function of crisp set A 497