User:Leigh.Samphier/OrderTheory/Completed

Ordered Sets

Successor is Supremum

Predecessor is Infimum

Ordered Set has Upper Bound iff has Greatest Element

Ordered Set has Lower Bound iff has Smallest Element

Definition:Ideal (Order Theory)/Proper Ideal

Definition:Proper Ideal (Order Theory)

Definition:Completely Prime Filter

Definition:Completely Prime Ideal

Filter Contains Greatest Element

Semilattices

Join Semilattice is Dual to Meet Semilattice

Equivalence of Definitions of Join Semilattice

Definition:Meet Semilattice

Equivalence of Definitions of Meet Semilattice

Join Semilattice has Smallest Element iff has Identity

Meet Semilattice has Greatest Element iff has Identity

Definition:Meet-Irreducible Element

Definition:Join-Irreducible Element

Lattice Homomorphisms

Axiom:Lattice Homomorphism Axioms

Definition:Lattice Homomorphism

Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism

Lattice Homomorphism is Order-Preserving

Bounded Lattices

Definition:Bounded Lattice

Definition:Bounded Lattice/Notation

Bounded Lattice has Both Greatest Element and Smallest Element

Equivalence of Definitions of Bounded Lattice

Bounded Lattice Homomorphisms

Axiom:Bounded Lattice Homomorphism Axioms

Definition:Bounded Lattice Homomorphism

Distributive Lattices

Definition:Well Inside Relation

Element Well Inside Itself Iff Has Complement

Well Inside Implies Predecessor

Well Inside Relation Extends to Predecessor and Successor

Well Inside Elements Form Filter

Elements Well Inside Form Ideal

Properties of Well Inside Relation

Galois Connections

Definition:Galois Connection

Definition:Galois Connection/Upper Adjoint

Definition:Upper Adjoint

Definition:Galois Connection/Lower Adjoint

Definition:Lower Adjoint

Definition:Galois Connection/Also known as

Definition:Galois Connection/Notation

Complete Lattices

Definition:Complete Lattice

Definition:Complete Lattice/Notation

Complete Lattice has Both Greatest Element and Smallest Element

Definition:Complete Join Semilattice

Definition:Complete Meet Semilattice

Complete Join Semilattice is Dual to Complete Meet Semilattice

Equivalence of Complete Semilattice and Complete Lattice

Frames

Axiom:Infinite Join Distributive Law

Definition:Frame (Lattice Theory)

Definition:Finite Meet Preserving Mapping

Definition:Finite Join Preserving Mapping

Definition:Arbitrary Meet Preserving Mapping

Definition:Arbitrary Join Preserving Mapping

Definition:Frame Homomorphism

Composite Frame Homomorphism is Frame Homomorphism

Identity Mapping is Frame Homomorphism

Definition:Category of Frames

Category of Frames is Category

Characterization of Compact Element in Complete Lattice

Characterization of Compact Element in Complete Lattice/Statement 1 implies Statement 3

Characterization of Compact Element in Complete Lattice/Statement 3 implies Statement 2

Characterization of Compact Element in Complete Lattice/Statement 2 implies Statement 1

Locales

Definition:Category of Locales

Definition:Locale (Lattice Theory)

Definition:Continuous Map (Locale)

Category of Locales is Category

Relative Pseudocomplement Preserves Order

Characterization of Locale

Characterization of Locale/Statement 3 Implies Statement 4

Characterization of Locale/Statement 5 Implies Statement 3

Characterization of Compact Element in Frame or Locale

Definition:Compact Locale

Definition:Regular Locale

Definition:Compact Regular Locale