Gierz G., Hofmann K.H., Keimel K. — Continuous Lattices and Domains :: Электронная библиотека попечительского совета мехмата МГУ

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Gierz G., Hofmann K.H., Keimel K. — Continuous Lattices and Domains

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Название: Continuous Lattices and Domains

Авторы: Gierz G., Hofmann K.H., Keimel K.

Аннотация:

Information content and programming semantics are just two of the applications of the mathematical concepts of order, continuity and domains. This authoritative and comprehensive account of the subject will be an essential handbook for all those working in the area. An extensive index and bibliography make this an ideal sourcebook for all those working in domain theory.

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Рубрика: Computer science/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2003

Количество страниц: 591

Добавлена в каталог: 04.06.2008

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Предметный указатель

Continuous lattice is embeddable in a product of chains      299 IV—3.20
Continuous lattice is quotient of an arithmetic lattice      123 I—4.18
Continuous lattice is quotient of an arithmetic lattice, of same weight      245 III—4.7
Continuous lattice, closure operator on      87 I—2.12
Continuous lattice, closure properties      86 I—2.11
Continuous lattice, congruences on      88 I—2.14 303
Continuous lattice, Continuous poset      54 I—1.6
Continuous lattice, Dedekind cuts in      301 IV—3.24
Continuous lattice, distributive      see “Distributive continuous lattice”
Continuous lattice, equational characterization      83 I—2.7
Continuous lattice, free over a compact Hausdorff space      454 VI—3.8(ii)
Continuous lattice, free over a compact pospace      455 VI—3.10
Continuous lattice, free over a set      455 remarks following VI—3.9
Continuous lattice, homomorphic images      86 I—2.10
Continuous lattice, homomorphism of      86 I—2.10
Continuous lattice, interval topology is Hausdorff      510 VII—3.4
Continuous lattice, irreducible elements      400 V—2.1 401
Continuous lattice, is the upper topology      510 VII—3.4
Continuous lattice, kernel operator on      88 I—2.15 89
Continuous lattice, lattice of congruences      303 IV—3.29(i)
Continuous lattice, lattice of continuous closure operators      302 IV—3.26
Continuous lattice, lattice of continuous kernel operators      302 IV—3.26 303
Continuous lattice, lattice of ideals      402 V—2.6
Continuous lattice, Lawson closed subsemilattices      215 III—1.12
Continuous lattice, Lawson topology is the interval topology      510 VII—3.4 (see also “Lawson topology”)
Continuous lattice, order-generating sets in Id L      402 V—2.6
Continuous lattice, order-generating subsets      401 V—2.4(i)
Continuous lattice, primes order-generate      101 I—3.15
Continuous lattice, primes topologically generate      406 V—3.9
Continuous lattice, projection operator on      89 I—2.17
Continuous lattice, pseudoprimes are weak primes      405 V—3.5
Continuous lattice, quotient of      88 I—2.14
Continuous lattice, Scott topology is strongly sober      497 VII—1.10
Continuous lattice, smallest closed order-generating subset      400 V—2.1
Continuous lattice, smallest closed topologically generating subset      401 V—2.4(ii)
Continuous lattice, subalgebras      86 I—2.10
Continuous lattice, topologically generating sets in      400 V—2.3
Continuous lattice, topologically generating sets in Id L      402 V—2.6
Continuous lattice, weak irreducibles are closed      404 V—3.2
Continuous lattice, weak primes are closed      404 V—3.2
Continuous lattice, weak primes are closed, are weak irreducibles      405 V—3.6
Continuous lattice, weak primes are closed, equal weak irreducibles      407 V—3.11
Continuous lattice, weight of      326 IV—5.13
Continuous lattice, weight of projective limit      327 IV—5.14
Continuous lattice, when completely distributive      102 I—3.16
Continuous semilattice      54 I—1.6
Continuous semilattice, irreducible elements      97 I—3.7
Continuous semilattice, smallest approximating relation on      60 I—1.16
Continuous semilattice, with order-generating primes, but not distributive      109 I—3.33
Continuous valuation      379 IV—9.5
Converse relation      4 O—1.7
Countably based domain      242 III—4.4
Countably based domain is a Polish space      421 V—5.17
cube      15 O—2.7(9)
dcpo      9 O—2.1
dcpo is a quasicontinuous domain iff Scott topology is hypercontinuous      513 VII—3.9
dcpo of Scott-continuous functions      161 II—2.5 162
dcpo, Lawson topology on      211 III—1.5
dcpo, open filter determined      285 IV—2.11
dcpo, order consistent topology on      152 II—1.30
dcpo, Scott closed subsets      279 IV—1.25
dcpo, Scott topology is a continuous lattice      197 II—4.13
dcpo, Scott topology is a continuous lattice on      134 II—1.3
dcpo, when Lawson topology is productive      221 III—2.6
dcpo, with continuous Scott topology      221 III—2.6
dcpo, with nonsober Scott topology      155 II—1.36
dcpo-algebra      359 IV—8.1
dcpo-algebra, free over X      360 IV—8.2
dcpo-cone      388 IV—9.20
dcpo-semilattice      360 remarks following IV—8.2 364
Deflationary semilattice      363 IV—8.8
Dense element, in a lattice      113 I—3.5
Densely injective space      182 II—3.10
Densely injective space and bounded complete domains      182 II—3.11
Density, of a domain      248 III—4.13
Diagonal, of a space      88 I—2.13
Diagram, cone over      305 IV—4.1
Diagram, cone over, in a category      305 IV—4.1
Direct limit, in a category      308 remarks following IV—4.3
Direct system, in a category      308 remarks following IV—4.3
Directed complete poset      9 O—2.1 (see also “dcpo”)
Directed complete semilattice      9 O—2.1 40 40
Directed distributive law      83 I—2.7
Directed net      2 O—1.2
Directed set      1 O—1.1
Disjoint sum      73 I—1.31 321
Distributive algebraic lattice, compact open sets are a basis for the spectrum      423 V—5.21(i)
Distributive algebraic lattice, compact open sets are a basis for the spectrum, patch topology on primes is compact      420 V—5.13(ii)
Distributive arithmetic lattice, categorically equivalent to distributive lattices      423 V—5.22
Distributive arithmetic lattice, categorically equivalent to distributive lattices, Priestley duality      491 VI—7.10
Distributive arithmetic lattice, categorically equivalent to distributive lattices, primes are closed      406 iiV—3.7(ii)
Distributive arithmetic lattice, categorically equivalent to distributive lattices, spectrum is totally order disconnected      490 VI—7.9(i)
Distributive complete lattice is linked bicontinuous if bicontinuous      503 VII—2.10
Distributive complete lattice is linked bicontinuous if bicontinuous, way-below relation in      105 I—3.23
Distributive continuous lattice      see also “Continuous frame”
Distributive continuous lattice and prime preserving maps      278 IV—1.23
Distributive continuous lattice is a continuous frame      101 I—3.15
Distributive continuous lattice is topological      501 VII—2.4
Distributive continuous lattice, dual to locally compact sober spaces      423 V—5.20 426
Distributive continuous lattice, is a frame      101 I—3.15
Distributive continuous lattice, is the Lawson topology      419 V—5.12
Distributive continuous lattice, patch topology on primes is compact      420 V—5.13
Distributive continuous lattice, prime element in      99 I—3.12
Distributive continuous lattice, primes are closed      406 V—3.7(i)
Distributive continuous lattice, pseudoprimes equal weak primes      405 V—3.5
Distributive continuous lattice, spectrum is sober locally compact      417 V—5.5
Distributive continuous lattice, when stably continuous      420 V—5.13
Distributive continuous lattice, when way-below relation is multiplicative      406 V—3.7(i)
Distributive lattice      12 O—2.6
Distributive lattice, algebraic      see “Distributive algerabic lattice”
Distributive lattice, arithmetic      see “Distributive arithmetic lattice”
Distributive lattice, continuous      see “Distributive continuous lattice”
Distributive semilattice      98 I—3.11
Distributive semilattice, prime element in      99 I—3.12
Domain      54 I—1.6
Domain equation, construction of minimal solution      344 IV—7.1
Domain is a quotient of an algebraic domain      246 I—4.17
Domain is a quotient of an algebraic domain , of same weight      245 III—4.7
Domain is embeddable in a cube      291 IV—3.2
Domain not closed under quotients      262 III—5.22
Domain of formal balls      435 V—6.8 437
Domain, basis for      240 III—4.1
Domain, basis for the Scott topology      138 II—1.10
Domain, characterization in Scott topology      142 II—1.14 154
Domain, characterization in Scott topology, through S-convergence      138 II—1.9
Domain, countably based      242 III—4.4 244 250
Domain, density of      248 III—4.13
Domain, environment      433 V—6.1
Domain, is meet continuous      222 III—2.11
Domain, Lawson topology is productive      221 III—2.6
Domain, Lawson topology is productive, is separable metric      244 III—4.6
Domain, morphisms into the unit interval      291 IV—3.1
Domain, properties of weight on      247 III—4.12
Domain, quasicontinuous      226 III—3.2
Domain, relation between weight and density      248 III—4.14
Domain, Scott open subsets      136 II—1.6
Domain, Scott topology has basis of open filters      142 II—1.14
Domain, Scott topology has basis of open filters, is Baire      142 II—1.13
Domain, Scott topology has basis of open filters, is locally compact sober      142 II—1.13
Domain, topological characterization      184 II—3.16
Domain, way-below relation in      62 I—1.20
Domain, weight of      242 III—4.4 243
Dual of a dcpo      283 IV—2.7
Dual topology      479 VI—6.17
Duality of categories      266 IV—1
Duality of categories, $CL-CL^{op}$       271 IV—1.10(iv)
Duality of categories, -      271 IV—1.10(i)
Duality of categories,       271 IV—1.10(iii)
Duality of categories, $INF\uparrow$ -      271 IV—1.10(ii)
Duality of categories, -      267 IV—1.3

Duality of categories, AL-SEM      274 IV—1.16
Duality of categories, ALGDOM-POID      274 IV—1.15
Duality of categories, algebraic lattices and completely distributive algebraic lattices      521 VII—4.10
Duality of categories, compact semilattices and distributive continuous lattices      519 VII—4.6
Duality of categories, continuous lattices and completely distributive lattices      521 VII—4.10
Duality of categories, DAR-CCSOB      423 V—5.22
Duality of categories, distributive algebraic lattices and sober spaces having a basis of compact open sets      423 V—5.21(ii)
Duality of categories, distributive arithmetic lattices and totally order disconnected pspaces      490 VI—7.9
Duality of categories, distributive continuous lattices and locally compact sober spaces      423 V—5.20 426
Duality of categories, distributive continuous resp., lattices with CL-maps preserving primes and distributive continuous resp., lattices with $CL^{op}$ -maps preserving finite infs      278 IV—1.24
Duality of categories, distributive lattices and totally order disconnected pospaces      491 VI—7.10
Duality of categories, DLat-CCSOB      423 V—5.22
Duality of categories, domains and completely distributive lattices      398 V—1.7
Duality of categories, frames with enough points and sober spaces      426 V—5.27
Duality of categories, INF-SUP      267 IV—1.3
Duality of categories, Lawson duality      286 IV—2.14 287
Duality of categories, stably continuous frames and stably locally compact spaces      489 VI—7.4
Duality open-compact      288 IV—2.18
Environment      433 V—6.1
Equivalence of categories, $ArL^{op}$ -LAT      274 IV—1.18
Equivalence of categories, $ArL^{op}$ -LAT, stably compact spaces and compact pospaces      482 VI—6.23
Evaluation map is Scott-continuous      163 II—2.9
Evaluation map is Scott-continuous, on $[L \rightarrow A]$       340 IV—6.14
Exponentiable space      196 II—4.11
Exponentiable space, characterization      196 II—4.12
Extremally disconnected space      17 O—2.9
F-algebra      333 IV—6.3 346
F-algebra isomorphisms, construction by a pro-continuous functor F      338 IV—6.9
F-algebra, endomorphism      346 IV—7.2
F-algebra, morphism      333 IV—6.3 346
F-algebra, quotient of      338 remarks preceding IV—6.9
F-coalgebra      346 IV—7.2
F-coalgebra, endomorphism      346 IV—7.2
F-coalgebra, morphism      346 IV—7.2
Family of finite character, in a power set      136 II—1.5(4)
Filter      3 O—1.3
Filter of sets      103 remarks precedingI—3.18
Filter, open      see “Open filter”
Filter, prime      see “Prime filter”
Filter, principal      3 O—1.3
Filter, Scott open      135 remarks following II—1.3
Filtered net      2 O—1.2
Filtered set      1 O—1.1
Final F-coalgebra      348 IV—7.5 349
Finite element      128 remarks preceding I—4.29
Finitely additive      375 remarks following IV—9.1
Finitely separating function      166 II—2.15
Fixed point theorem, for monotone self-maps      20 O—2.20
Fixed point theorem, for monotone self-maps, for Scott-continuous self-maps      160 II—2.4
Fixed point theorem, for monotone self-maps, Pataraia’s      20 O—2.21
Fixed point theorem, for monotone self-maps, Tarski’s      10 O—2.3
Formal ball      435 V—6.8
Formal union      360 remarks following IV—8.2
Frame      12 O—2.6 101
Frame as a function space      200 II—4.19
Frame is meet continuous      38 O—4.3
Frame, closure properties      34 O—3.25
Frame, dual to sober spaces      426 V—5.27
Frame, homomorphism      34 O—3.24
Frame, subalgebra      34 O—3.24
Free continuous lattice      123 I—4.19 455 460
Free continuous lattice, over a compact Hausdorff space      454 VI—3.8(ii)
Free continuous lattice, over a compact pospace      455 VI—3.10
Free deflationary semilattice, over a domain      363 IV—8.10
Free inflationary semilattice, over a dcpo      362 IV—8.6
Free semilattice, over a domain      367 IV—8.12
FS-domain      166 II—2.15
FS-domain is Lawson-compact      258 III—5.14
FS-domain, preservation properties      167 II—2.17
Function space      162 II—2.6
Function space functor      162 II—2.7 321
Function space functor, Funct      324 IV—5.10
Function space functor, Funct, preserves injective (surjective) maps      325 IV—5.10
Function space functor, Funct, preserves projective limits      325 IV—5.10
Function space is a continuous lattice      192 II—4.6 193
Function space is a domain      190 II—4.4
Function space, Isbell and Scott topology agree      192 II—4.6 260 261
Function, idempotent      25 remarks preceding O—3.6
Function, idempotent, lower semicontinuous      see “Lower semicontinuous function”
Function, idempotent, monotone      5 O—1.9
Function, idempotent, open      269 remarks preceding VI—1.5
Function, idempotent, order preserving      5 O—1.9
Function, idempotent, partial      15 O—2.7(10)
Function, idempotent, preserving arbitrary infs      5 O—1.9
Function, idempotent, preserving arbitrary sups      5 O—1.9
Function, idempotent, preserving directed sups      5 O—1.9
Function, idempotent, preserving filtered infs      5 O—1.9
Function, idempotent, preserving finite infs      5 O—1.9
Function, idempotent, preserving finite sups      5 O—1.9
Function, idempotent, Scott-continuous      see “Scott-continuous function”
Function, idempotent, semicontinuous      17 O—2.10
Function, idempotent, upper semicontinuous      17 O—2.10
Galois adjunction      see “Galois connection”
Galois connection      22 O—3 22
Greatest lower bound      1 O—1.1
Hausdorff space is locally compact iff O(X) is continuous      417 V—5.7
Heyting algebra      30 O—3.16 (see also “Frame”)
Heyting algebra, complete      12 O—2.6
Heyting algebra, Continuous      see “Distributive continuous lattice”
Hilbert space      15 O—2.7(8)
Hoare powerdomain      361 IV—8.3 362
Hoare powerdomain of an algebraic domain      372 IV—8.22
Hofmann — Mislove Theorem      146 II—1.20 288 417
Hofmann — Mislove Theorem, holds only for sober spaces      147 II—1.21
Homomorphism of L-domains      92 I—2.23
Homomorphism of L-domains, of continuous lattices and bounded complete domains      86 I—2.10
Homomorphism of L-domains, of frames      34 O—3.24
Homomorphism of L-domains, of semilattices      5 remarks following O—1.9
Homomorphism of L-domains, onto chains, separation of points in complete lattices      299 IV—3.19
Hull-kernel topology, on the spectrum      409 V—4.3
Hypercontinuous lattice      509 VII—3.2
Hypercontinuous lattice is continuous      509 VII—3.3
Hypercontinuous lattice, characterizations      510 VII—3.4
Hypercontinuous lattice, Scott topology is the upper topology      510 VII—3.4
Ideal      3 O—1.3
Ideal functor Id      325 IV—5.12
Ideal functor Id is locally order preserving      325 IV—5.12
Ideal functor Id is not locally continuous      325 IV—5.12
Ideal, prime      see “Prime ideal”
Ideal, principal      3 O—1.3
Idempotent function      25 remarks preceding O—3.6
Infimum      1 O—1.1
Inflationary semilattice      361 IV—8.3
Initial F-algebra      348 IV—7.5
Initial F-algebra - final F-coalgebra coincidence      349 IV—7.6 350
Injective space      176 II—3 177
Injective space and algebraic lattices      186 II—3.18
Injective space is a continuous lattice      180 II—3.7
Injective space, characterization of      178 II—3.4
Injective space, closure properties of      177 II—3.2
Injective space, equivalent conditions for      185 II—3.17
Interpolation property      56 I—1.9 60
Interval domain      70 I—1.26.1
Interval topology      43 O—5.4 217 501
Interval topology is the Lawson topology      510 VII—3.4
Interval topology, when Hausdorff      239 III—3.31 506 510
Irreducible closed set      43 O—5.5 101 141
Irreducible element      95 I—3 97
Irreducible element in a continuous semilattice      97 I—3.7
Irreducible element in a function space      202 II—4.23 203
Irreducible element in a modular lattice      108 I—3.29
Irreducible element, order generate a continuous semilattice      98 I—3.10
Irreducible subset of a space      43 O—5.5 46
Isbell topology      188 II—4.1
Isolated element      49 I—1.1
Isomorphism      5 O—1.9
Join      1 O—1.1
Join continuous lattice      36 O—4.1
Join continuous semilattice      36 O—4.1
Join-compact space      485 VI—6.31

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