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









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: Continuous Lattices and Domains

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

:

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.


: en

: Computer science/

:

ed2k: ed2k stats

: 2003

: 591

: 04.06.2008

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$Spec : SUP^{^} \rightarrow TOP^{op}$ is left adjoint to $\mathcal{O}: TOP^{op} > SUP^{^}$      412 V4.7
$T_0$ space, all irreducible subspaces are Baire      427 V5.30
$T_0$ space, all irreducible subspaces are Baire is sober      427 V5.30
$T_0$ space, all irreducible subspaces are Baire, all subspaces are Baire      427 V5.30
$T_0$ space, all irreducible subspaces are Baire, order generates a continuous lattice      418 V5.10
$T_0$ space, all irreducible subspaces are Baire, sobrification is locally compact      418 V5.10
$T_0$ space, all irreducible subspaces are Baire, strict embedding in a locally compact space      418 V5.10
$T_0$ space, all irreducible subspaces are Baire, when $\mathcal{O}(X)$ is a continuous lattice      418 V5.10
$T_0$ space, all irreducible subspaces are Baire, with $\mathcal{O}(X)$ continuous but not locally compact      425 V5.25(i)
$\omega$ continuous function      328 IV5.16
$\omega$ continuous function is Scott-continuous on countably based domains      250 III4.20
$\omega$-complete posets      328 IV5.16
Adjoint, lower      see Lower adjoint
Adjoint, upper      see Upper adjoint
Alexanders Lemma, generalization of      105 I3.22
Algebraic domain      115 I4.2
Algebraic domain is Lawson-compact      259 III5.15
Algebraic domain, as a domain of ideals of a poset      118 I4.10
Algebraic domain, characterization in Scott topology      143 II1.15
Algebraic domain, domain of open filters      128 I4.31
Algebraic domain, products are algebraic      119 I4.12
Algebraic lattice      115 I4 115
Algebraic lattice and injective spaces      186 II3.18
Algebraic lattice and Scott-continuous functions      175 II2.36
Algebraic lattice is a projective limit of finite lattices      317 IV4.14
Algebraic lattice is a subalgebra of a power set lattice      121 I4.16
Algebraic lattice, closed order generating subsets      402 V2.5(i)
Algebraic lattice, closed topologically generating subsets      402 V2.5(i)
Algebraic lattice, completely irreducible elements      126 I4.25 402
Algebraic lattice, distributive      see Distributive algebraic lattice
Algebraic lattice, irreducibles order-generate      126 I4.26
Algebraic lattice, smallest closed order generating subset      402 V2.5(i)
Algebraic lattice, subalgebra of      120 I4.14
Algebraic lattice, topologically generating subsets      402 V2.5(ii)
Algebraic lattice, when arithmetic      117 I4.8
Algebraic lattice, with multiplicative way-below relation      117 I4.8
Algebraic poset      115 I4.2
Algebraic semilattice      115 I4.2
Algebraic semilattice, domain of open filters      128 I4.31
Antichain      4 O1.6
Antitone net      2 O1.2
Approximate identity      165 II2.13
Arc-chain, in a compact pospace      470 VI5.9
Arc-chain, in a pospace      469 VI5.5
Arc-chain, limit of in a compact pospace      469 VI5.7
Arithmetic lattice      117 I4.7
Arithmetic lattice, distributive      see Distributive arithmetic lattice
Arithmetic lattice, pseudo-prime elements      118 I4.9
Ascending chain condition      52 I1.3(4)
Ascending chain condition and domains      55 I1.7
Asymmetric space      485 VI6.31
atom      13 O2.7(1)
Atomic lattice      13 O2.7(1)
Atomless Boolean algebra      14 O2.7(3)
Atomless Boolean algebra, Auxiliary order      see Auxiliary relation
Auxiliary relation      57 I1.11
Auxiliary relation, approximating      59 I1.13 293
Auxiliary relation, auxiliary relation with strong interpolation property derived from      72 I1.28 301
Auxiliary relation, interpolation property for      61 I1.17
Auxiliary relation, multiplicative      107 I3.27
Auxiliary relation, on a complete lattice      293 IV3.4
Auxiliary relation, strong interpolation property for      60 I1.17 301 301
Auxiliary relation, sup closure of      72 I1.29
Axiom of approximation      54 I1.6
Baire category theorem      112 I3.40
Baire Category Theorem for continuous lattices      113 I3.40.7
Baire Category Theorem for locally compact spaces      113 I3.40.8
Baire space      45 O5.13
Basis of a domain      240 III4.1
Basis of a topology      44 O5.8
Basis, abstract basis      249 III4.15
Bi-Scott topology      501 Remark following VII2.3
Bi-Scott topology, when Hausdorff      505 VII2.12
Bicontinuous function      218 III1.21
Bicontinuous lattice      501 VII2.5 (see also Linked bicontinuous lattice)
Bifinite domain      169 II2.21
Bifinite domain is a projective limit of finite domains      316 IV4.12
Bifinite domain is Lawson-compact      258 III5.14
Bitopological space      218 III1.21
Boolean algebra      12 O2.6
Boolean algebra is a continuous lattice      124 I4.20
Boolean algebra is algebraic      124 I4.20
Boolean algebra is arithmetic      124 I4.20
Boolean algebra is atomic      124 I4.20
Boolean algebra is completely distributive      124 I4.20
Boolean algebra, complete      12 O2.6
Boolean algebra, prime element in      99 I3.12
Boolean algebra, way-below relation in      52 I1.3(3)
Boolean lattice      see Boolean algebra
Bottom, of a poset      5 O1.8
Bound, lower      1 O1.1
Bound, upper      1 O1.1
Bounded complete domain      54 I1.6
Bounded complete domain and densely injective spaces      182 II3.11
Bounded complete domain is an FS-domain      202 II4.21
Bounded complete domain, closure properties      86 I2.11
Bounded complete poset      9 O2.1
C*-algebra      62 I1.21
C*-algebra, closed prime ideals      109 I3.34
C*-algebra, primitive ideal of      109 remarks following I3.34
Cantor tree      434 V6.4
Cartesian closed category      163 remarks preceding II2.10
Cartesian closed category, algebraic bounded complete domains and Scott-continuous maps      173 II2.31
Cartesian closed category, algebraic L-domains and Scott-continuous maps      173 II2.32
Cartesian closed category, algebraic lattices and Scott-continuous maps      165 II2.12 173
Cartesian closed category, bifinite domains and Scott-continuous maps      170 II2.23
Cartesian closed category, bounded complete domains and Scott-continuous maps      173 II2.31
Cartesian closed category, complete lattices and Scott-continuous maps      164 II2.10
Cartesian closed category, continuous lattices and Scott-continuous maps      165 II2.12 173
Cartesian closed category, countably based bifinite domains      251 III4.22
Cartesian closed category, countably based continuous lattices      247 remarks following III4.12
Cartesian closed category, countably based FS-domains      251 III4.21
Cartesian closed category, counterexample: countably based L-domains      251 III4.23
Cartesian closed category, dcpos and Scott-continuous maps      164 II2.10
Cartesian closed category, FS-domains and Scott-continuous maps      168 II2.19
Cartesian closed category, L-domains and Scott-continuous maps      173 II2.32 202
Category of algebraic lattices and sup and way-below preserving maps      272 IV1.13
Category of algebraic lattices and sup, and way-below preserving maps      272 IV1.13
Category of algebraic lattices is cartesian closed      165 II2.12
Category of algebraic lattices, and inf and directed sup preserving maps      272 IV1.13
Category of algebraic lattices, and Scott-continuous functions      158 II2.2
Category of arithmetic lattices, and inf and directed sup preserving maps      272 IV1.13
Category of bifinte domains, and Scott-continuous functions is cartesian closed      170 II2.23
Category of complete lattices, and inf and directed sup preserving maps      270 IV1.9
Category of complete lattices, and inf and directed sup preserving maps, and inf preserving maps      266 IV1.1
Category of complete lattices, and inf and directed sup preserving maps, and Scott-continuous functions      158 II2.2
Category of complete lattices, and inf and directed sup preserving maps, and Scott-continuous functions is cartesian closed      164 II2.10
Category of complete lattices, and inf and directed sup preserving maps, and sup and Scott open set preserving maps      270 IV1.9
Category of complete lattices, and inf and directed sup preserving maps, and sup preserving maps      266 IV1.1
Category of continuous lattices and inf and directed sup preserving maps      270 IV1.9
Category of continuous lattices and Scott-continuous functions      158 II2.2
Category of continuous lattices and Scott-continuous functions is cartesian closed      165 II2.12
Category of continuous lattices and sup and way-below preserving maps      270 IV1.9
Category of continuous lattices, having weight less than a fixed cardinal      330 IV5.18
Category of continuous semilattices      281 IV2.2
Category of domains with open filter morphisms      281 IV2.2
Category of FS-domains, and Scott-continuous functions is cartesian closed      168 II2.19
Category of posets and lower adjoints      266 IV1.1
Category of posets and upper adjoints      266 IV1.1
Category of sup semilattices with 0, and maps preserving sup and 0      272 IV1.13
Category of sup semilattices, and monotone maps      175 II2.36
Category, duality of      see Duality of categories
Chain      4 O1.6
Chain Modification Lemma, for strict chains      295 IV3.11
Chain, Complete      see Complete chain
Chain, gap in      128 I4.30
Chain, gap in is embeddable in a cube      300 IV3.21
Chain, way-below relation      51 I1.3(1)
Character of a dcpo      283 IV2.7
Character poset of a dcpo      283 IV2.7
Clopen set      17 O2.9
Closed sets in a compact Hausdorff space, form a continuous lattice      454 VI3.8
Closed sets in a compact Hausdorff space, form a continuous lattice, Vietoris topology on      454 VI3.8
Closure operator      26 O3.8
Closure operator is Scott open      270 IV1.8
Closure operator, image is closed under directed sups      270 IV1.8
Closure operator, lattice of, on a complete lattice      301 IV3.25
Closure operator, lattice of, on a complete lattice, on a continuous lattice      301 IV3.25
Closure operator, on a continuous lattice      87 I2.12
Closure operator, on an algebraic domain      119 I4.13
Closure operator, on an algebraic lattice      120 I4.14
Closure operator, preserves sups      29 O3.12
Closure system      29 remarks preceding O3.13 29
Closure system, closed under directed sups      29 O3.14 82
Co-compact topology      44 O5.10
Co-compact topology, on a domain      482 VI6.24
Co-cone, in a category      308 remarks following IV4.3
Co-prime element      98 I3.11
Co-prime element, form a dcpo      111 I3.39
Co-retraction      179 remarks preceding II3.5
Coalesced sum      73 I1.31 327
Cofinal map      24 remarks preceding O3.4
Coherent space      474 VI6.2
Colimit, in a category      308 remarks following IV4.3
Compact convex set      110 I3.36
Compact convex set, closed convex subsets form a continuous lattice      66 I1.23
Compact convex set, converse of Krein-Milman Theorem      399 V1.11
Compact convex set, primes in $Con (K)^{op}$ topologically generate      407 V3.10
Compact convex set, where closed convex subsets do not form a continuous lattice      467 VI4.6
Compact element      49 I1.1 115 126
Compact element in the lattice of open sets of a space      127 I4.28(i)
Compact lattice, characterization of connectivity      472 VI5.15
Compact lattice, characterization of connectivity has bi-Scott topology      501 VII2.3
Compact lattice, characterization of connectivity has Scott and dual Scott topology      501 VII2.3
Compact metric semilattice, with small semilattices      458 VI3.17
Compact metrizable pospace admits a radially convex metric      445 VI1.18
Compact open topology      187 remarks preceding II4.1
Compact pospace      479 VI6.18
Compact pospace is stably compact      477 VI6.11
Compact pospace, embedding in a continuous lattice      459 VI3.21
Compact pospace, metrizable, admits a radially convex metric      445 VI1.18
Compact pospace, totally order-disconnected      490 VI7.8
Compact saturated sets      66 I1.24
Compact saturated sets and open filters in O(X)      146 II1.20
Compact semilattice      443 VI1.11
Compact semilattice has enough subinvariant pseudometrics      446 VI2.3
Compact semilattice has small semilattices      458 VI3.19
Compact semilattice is complete      443 VI1.13
Compact semilattice is embeddable in a compact lattice      500 VII2.1
Compact semilattice is Hausdorff by convention      443 VI1.11
Compact semilattice is meet continuous      443 VI1.13
Compact semilattice is stably compact      483 VI6.25
Compact semilattice with small semilattices      450 VI3.1
Compact semilattice with small semilattices is a complete continuous semilattice      451 VI3.4(ii)
Compact semilattice with small semilattices is embeddable in a cube      453 VI3.7
Compact semilattice with small semilattices, at a point      450 VI3.1 456
Compact semilattice with small semilattices, characterization      453 VI3.7
Compact semilattice with small semilattices, continuous morphisms between      452 VI3.4(iii)
Compact semilattice with small semilattices, quotient has small semilattices      453 VI3.5
Compact semilattice, characterization of connectivity      470 VI5.11
Compact semilattice, characterization of connectivity, of continuous homomorphisms      448 VI2.7
Compact semilattice, characterization of connectivity, of convergence in      447 VI2.6
Compact semilattice, characterization of connectivity, of order connectivity      471 VI5.14
Compact semilattice, closed lower sets      449 VI2.10 456
Compact semilattice, closed lower sets, form a compact lattice      500 remarks preceding VII2.1
Compact semilattice, closed subsemilattices      448 VI2.8(i) 449
Compact semilattice, compact elements      449 VI2.12
Compact semilattice, Fundamental Theorem      451 VI3.4
Compact semilattice, local minimum is compact      470 VI5.10(ii)
Compact semilattice, metric, with small semilattices      458 VI3.17
Compact semilattice, points joined by arc-chains      471 VI5.12(ii)
Compact semilattice, topology is compatible      443 VI1.13
Compact semilattice, universal continuous lattice quotient      461 VI3.24
Compact semilattice, when a continuous lattice      451 VI3.4 455
Compact semilattice, when a topological lattice      500 VII2.2
Compact semilattice, which is not a continuous lattice      466 VI4.5
Compact semitopological semilattice has closed graph      520 VII4.7
Compact semitopological semilattice has closed graph is topological      521 VII4.8 521
Compact space      43 O5.7
Compact totally disconnected semilattice, Fundamental Theorem      457 VI3.13
Compatible topology, for a poset      440 VI1.2
Complemented lattice      12 O2.6
Complete Boolean Algebra      see Boolean algebra
Complete category      307 remarks preceding IV4.2
Complete chain      9 O2.1
Complete chain is a continuous lattice      55 I1.7
Complete chain, when algebraic      128 I4.30
Complete continuous semilattice      54 I1.6
Complete distributive lattice      see Distributive complete lattice
Complete distributive law      85 I2.8
Complete Heyting algebra      see Heyting algebra and
Complete lattice      9 O2.1
Complete lattice is sober      198 II4.16
Complete lattice with continuous and join continuous Scott topology      518 VII4.4
Complete lattice with continuous Scott topology      516 VII4.1 517
Complete lattice, $INF\uparrow$-maps preserve irreducibles      402 V2.8
Complete lattice, characterization when linked bicontinuous      502 VII2.9
Complete lattice, distributive      see Distributive complete lattice
Complete lattice, function space is a frame      200 II4.19
Complete lattice, function space is meet continuous      200 II4.19
Complete lattice, irreducible elements order generate      402 V2.7
Complete lattice, lattice of congruences      302 IV3.27
Complete lattice, lattice of continuous kernel operators      302 IV3.27
Complete lattice, Lawson topology on      see Lawson topology
Complete lattice, lower topology on      see Lower topology
Complete lattice, patch topology on primes is functorial      489 VI7.6
Complete lattice, Scott topology is a continuous lattice      198 II4.16 200
Complete lattice, smallest closed order generating subset      402 V2.7
Complete lattice, sober subspaces in the lower topology      414 V4.8
Complete lattice, spectrum      408 V4.1
Complete lattice, when bi-Scott topology is Hausdorff      505 VII2.12
Complete lattice, when interval topology is Hausdorff      502 VII2.9
Complete lattice, when Scott topology is locally compact sober      516 VII4.1(iii)
Complete lattice, when Scott topology is productive      498 VII1.13
Complete lattice, when topology contains Scott topology      496 VII1.9
Complete lattice, when topology is Scott topology      496 VII1.9
Complete semilattice      9 O2.1
Complete semilattice, closed lower sets      448 VI1.8(ii)
Complete semilattice, Lawson closed subsemilattices      237 III3.26
Completely distributive algebraic lattice      521 VII4.10
Completely distributive lattice      85 I2.8 85 521
Completely distributive lattice and injective spaces      185 II3.17
Completely distributive lattice is continuous and dually continuous      102 I3.16
Completely distributive lattice is embeddable in a cube      303 IV3.32
Completely distributive lattice is hypercontinuous      515 VII3.12
Completely distributive lattice is linked bicontinuous      503 VII2.10
Completely distributive lattice, co-primes form a domain      398 V1.7
Completely distributive lattice, co-primes form a domain, in      397 V1.6
Completely distributive lattice, way-way-below relation      303 IV3.31
Completely irreducible element      125 I4.21 125
Completely prime filter      414 V4.10
Composition is Scott-continuous      163 II2.9 206
Condition, ($\dagger$)      433 V6.1
Condition, ($\ddagger$)      433 V6.1
Conditional sup semilattice      117 I4.5
Cone, over a diagram      305 IV4.1
Congruence relation      14 O2.7(4)
Congruence relation, on a continuous lattice      88 I2.14
Construction of function space algebras, on DCPO      339 IV6.11
Construction of Scott topology algebras, on $INF\uparrow$      339 IV6.12
Continuous (semi)lattice is meet-continuous      56 I1.8
Continuous frame      101 I3.15 (see also Distributive continuous lattice)
Continuous Heyting algebra      see Distributive continuous lattice
Continuous lattice      54 I1.6 (see also Complete lattice)
Continuous lattice has small semilattices      451 VI3.4
Continuous lattice is a compact semilattice      224 III2.15 303
Continuous lattice is a retract of a power set lattice      123 I4.18
Continuous lattice is an FS-domain      202 II4.21
Continuous lattice is an injective space in the Scott topology      179 II3.5
Continuous lattice is embeddable in a cube      292 IV3.3
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