Category:Continuous Lattices
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This category contains results about Continuous Lattices in the context of Lattice Theory.
Let $\struct {S, \preceq}$ be an ordered set.
Then $\struct {S, \preceq}$ is continuous if and only if
- (for all elements $x$ of $S$: the way below closure $x^\ll$ of $x$ is directed) and:
- $\struct {S, \preceq}$ is up-complete and satisfies the Axiom of Approximation.
Pages in category "Continuous Lattices"
The following 42 pages are in this category, out of 42 total.
A
C
- Continuous iff For Every Element There Exists Ideal Element Precedes Supremum
- Continuous iff Mapping at Element is Supremum
- Continuous iff Mapping at Element is Supremum of Compact Elements
- Continuous iff Meet-Continuous and There Exists Smallest Auxiliary Approximating Relation
- Continuous iff Way Below Closure is Ideal and Element Precedes Supremum
- Continuous iff Way Below iff There Exists Element that Way Below and Way Below
- Continuous Lattice and Way Below implies Preceding implies Preceding
- Continuous Lattice iff Auxiliary Approximating Relation is Superset of Way Below Relation
- Continuous Lattice is Meet-Continuous
- Continuous Lattice Subframe of Algebraic Lattice is Algebraic Lattice
E
I
- Ideals form Algebraic Lattice
- Ideals form Arithmetic Lattice
- Image of Compact Subset under Directed Suprema Preserving Closure Operator
- Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset
- Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice
- Interior is Union of Way Above Closures
M
S
W
- Way Above Closure is Open
- Way Above Closures Form Basis
- Way Above Closures that Way Below Form Local Basis
- Way Below has Interpolation Property
- Way Below has Strong Interpolation Property
- Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element
- Way Below implies There Exists Way Below Open Filter Subset of Way Above Closure
- Way Below is Approximating Relation