Category:Order Theory

F

• [×] Filter Theory‎ (12 P)

L

• [×] Lattice Theory‎ (11 P)

M

• [×] Max and Min Operations‎ (19 P)

N

• [+] Nets (Topology)‎ (1 C)

O

• [×] Order Isomorphisms‎ (18 P)

O cont.

• [+] Order Monomorphisms‎ (1 C, 8 P)
• [×] Order Topology‎ (2 P)
• [×] Ordered Geometry‎ (1 P)
• [×] Ordered Integral Domains‎ (16 P)
• [×] Ordered Product‎ (1 P)

O cont.

• [×] Ordered Sum‎ (1 P)
• [×] Ordinals‎ (20 P)

T

• [+] Total Orderings‎ (3 C, 22 P)
• [×] Totally Ordered Fields‎ (3 P)

W

• [×] Well-Orderings‎ (18 P)

B

• Boolean Ring Trivial Ordering

C

• Cancellability in Ordered Semigroup
• Cancellability in Totally Ordered Semigroup
• Cardinals are Totally Ordered
• Coarseness Relation on Topologies is Partial Ordering
• Cofinal Sets of Natural Numbers
• Constant Mapping both Increasing and Decreasing

D

• Diagonal Relation is Ordering and Equivalence
• Divides is Partial Ordering on Positive Integers
• Dominates is an Ordering

E

• Equality of Ordered Pairs
• Equality of Ordered Tuples
• Equivalence Relation is Preordering

F

• Finite Lexicographic Order on Well-Ordered Sets is Well-Ordering
• Finite Subsets form Directed Set

G

• Greatest Element is Maximal
• Greatest Element is Unique
• Greatest Element is Upper Bound

H

• Hausdorff Maximal Principle
• Hausdorff Maximal Principle/Proof 1
• Hausdorff Maximal Principle/Proof 2
• Hausdorff Maximal Principle/Proof 3

I

• Identity Mapping is Ordered Ring Automorphism
• Increasing Sequence in Ordered Set Terminates iff Maximal Element
• Infimum of Empty Set is Greatest Element
• Infimum of Power Set
• Infimum of Subset
• Infinite Lexicographic Order on Well-Ordered Sets is Not Well-Ordering
• Injection Induces Total Ordering

I cont.

• Injection Induces Well-Ordering
• Intersection of Strict Lower Closures in Toset
• Intersection of Strict Upper Closures in Toset
• Intersection of Subset with Lower Bounds
• Intersection of Subset with Upper Bounds
• Intersection of Weak Lower Closures in Toset
• Intersection of Weak Upper Closures in Toset
• Inverse of Ordered Ring Isomorphism is Ordered Ring Isomorphism
• Inverse of Ordering is Ordering
• Invertibility in Ordered Semigroup
• Isomorphism to Closed Interval

L

• Lexicographic Order on Cartesian Product
• Lower Bound for Subset

M

• Mappings Partially Ordered by Extension
• Maximal Element Not Necessarily Greatest Element

N

• Natural Numbers are Comparable

O

• Order Completion Unique up to Isomorphism
• Ordered Group Equivalences
• Ordered Semigroup Isomorphism is Surjective Monomorphism
• Ordering Equivalent to a Subset Relation
• Ordering Induced by Injection is Ordering
• Ordering is Strict Ordering Union Diagonal Relation
• Ordering of Inverses
• Ordering on an Ordinal is Subset Relation
• Ordering on Extended Real Numbers is Ordering
• Ordering on Extended Real Numbers is Total Ordering

P

• Poset has Maximal Chain
• Positive Elements of Ordered Ring
• Preorder Compatible with Operation is a Congruence

P cont.

• Properties of Ordered Ring

R

• Relation Compatibility in Totally Ordered Semigroup
• Restriction of Monotone Function is Monotone

S

• Smallest Element is Lower Bound
• Smallest Element is Minimal
• Smallest Element is Unique
• Strict Lower Closure in Restricted Ordering
• Strict Ordering is Asymmetric
• Strict Upper Closure in Restricted Ordering
• Strict Weak Ordering Induces Partition
• Strictly Decreasing Mapping is Decreasing
• Strictly Increasing Mapping is Increasing
• Strictly Monotone Mapping is Injective
• Strictly Monotone Mapping is Monotone
• Strictly Precedes is a Strict Ordering
• Subset Relation is Ordering
• Subset Relation on Power Set is Partial Ordering
• Supremum and Infimum Unique
• Supremum Not Less than Infimum
• Supremum of Empty Set is Smallest Element
• Supremum of Power Set
• Supremum of Subset

T

• Trivial Ordering

U

• Unions of Chains of Chains
• Upper Bound for Subset
• Upper Bound is Lower Bound for Inverse Ordering
• Upper Semilattice on Classical Set is Semilattice

W

• Weak Lower Closure in Restricted Ordering
• Weak Upper Closure in Restricted Ordering