# Category:Order Theory

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This category contains results about Order Theory.

Definitions specific to this category can be found in Definitions/Order Theory.

**Order theory** is the branch of relation theory which studies orderings.

## Subcategories

This category has the following 74 subcategories, out of 74 total.

### A

### B

### C

### D

### E

### F

### G

### H

### I

### J

### L

### M

### N

### O

### P

### R

### S

### T

### U

### W

## Pages in category "Order Theory"

The following 198 pages are in this category, out of 198 total.

### A

### C

- Canonical Order Well-Orders Ordered Pairs of Ordinals
- Cartesian Product of Ordered Sets is Ordered Set
- Chain is Directed
- Closure Operator from Closed Elements
- Coarseness Relation on Topologies is Partial Ordering
- Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema
- Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema
- Complement of Subset with Property (S) is Closed under Directed Suprema
- Composition of Inflationary Mappings is Inflationary
- Condition for Power Set to be Totally Ordered
- Condition for Well-Foundedness
- Copi's Identity
- Correctness of Definition of Increasing Mappings Satisfying Inclusion in Lower Closure
- Countable Subset of Minimal Uncountable Well-Ordered Set Has Upper Bound

### D

- Dedekind Completeness is Self-Dual
- Dedekind Completion is Unique up to Isomorphism
- Diagonal Relation is Ordering and Equivalence
- Directed Set has Strict Successors iff Unbounded Above
- Divisor Relation on Positive Integers is Partial Ordering
- Dominance Relation is Ordering
- Dual of Dual Ordering
- Dual of Dual Statement (Order Theory)
- Dual Ordering is Ordering
- Dual Pairs (Order Theory)
- Duality Principle (Order Theory)
- Duality Principle (Order Theory)/Global Duality
- Duality Principle (Order Theory)/Local Duality

### E

- Element of Finite Ordered Set is Between Maximal and Minimal Elements
- Equivalence Induced on Preordering defines Ordering
- Equivalence of Definitions of Maximal Element
- Equivalence of Definitions of Minimal Element
- Equivalence of Definitions of Order Complete Set
- Equivalence of Definitions of Ordering
- Equivalence of Definitions of Strict Ordering
- Equivalence of Definitions of Strict Well-Ordering
- Every Element is Directed and Every Two Elements are Included in Third Element implies Union is Directed
- Existence of Dedekind Completion
- Existence of Minimal Uncountable Well-Ordered Set
- Extension of Directed Suprema Preserving Mapping to Complete Lattice Preserves Directed Suprema
- Extension of Infima Preserving Mapping to Complete Lattice Preserves Infima

### F

- Filter is Ideal in Dual Ordered Set
- Finite Infima Set and Upper Closure is Filter
- Finite Infima Set and Upper Closure is Smallest Filter
- Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set
- Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements
- Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary
- Finite Suprema Set and Lower Closure is Ideal
- Finite Suprema Set and Lower Closure is Smallest Ideal
- Fixed Point of Composition of Inflationary Mappings

### G

### I

- Ideal is Filter in Dual Ordered Set
- Ideals form Complete Lattice
- Image of Directed Subset under Increasing Mapping is Directed
- Infima Preserving Mapping on Filters is Increasing
- Infima Preserving Mapping on Filters Preserves Filtered Infima
- Infimum in Ordered Subset
- Infimum is not necessarily Smallest Element
- Infimum of Infima
- Infimum of Power Set
- Infimum of Subset
- Initial Segment of Canonical Order is Set
- Intersection of Subset with Lower Bounds
- Intersection of Subset with Upper Bounds
- Interval of Ordered Set is Convex
- Inverse Image of Convex Set under Monotone Mapping is Convex
- Isomorphism to Closed Interval

### L

### M

- Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema
- Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset
- Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence
- Mapping Preserves Finite and Directed Suprema
- Mapping Preserves Finite and Filtered Infima
- Mappings Partially Ordered by Extension
- Maximal Element is Dual to Minimal Element
- Maximal Element need not be Greatest Element
- Maximal Element of Chain is Greatest Element
- Maximal Element of Complement of Filter is Meet Irreducible
- Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
- Meet Irreducible iff Finite Infimum equals Element
- Minimal Element of Chain is Smallest Element
- Minimal Uncountable Well-Ordered Set Unique up to Isomorphism
- Multiplicative Auxiliary Relation iff Congruent
- Multiplicative Auxiliary Relation iff Images are Filtered

### N

### O

- Order Completion Unique up to Isomorphism
- Order Isomorphism between Ordinals and Proper Class/Lemma
- Order-Extension Principle
- Order-Extension Principle/Strict
- Order-Extension Principle/Strict/Finite Set
- Order-Extension Principle/Strict/Proof 1
- Order-Extension Principle/Strict/Proof 2
- Ordered Set is Convex in Itself
- Ordered Set of All Mappings is Ordered Set
- Ordered Set of Closure Systems is Ordered Set
- Ordered Subset of Ordered Set is Ordered Set
- Ordering can be Expanded to compare Additional Pair
- Ordering Cycle implies Equality
- Ordering Cycle implies Equality/General Case
- Ordering Induced by Injection is Ordering
- Ordering Induced by Join Semilattice
- Ordering is Equivalent to Subset Relation
- Ordering is Equivalent to Subset Relation/Lemma
- Ordering of Inverses in Ordered Monoid
- Ordering on Ordinal is Subset Relation
- Ordering on Partition Determines Preordering
- Ordering Principle

### P

### R

- Reflexive Closure of Strict Ordering is Ordering
- Reflexive Closure of Strict Total Ordering is Total Ordering
- Reflexive Closure of Transitive Antisymmetric Relation is Ordering
- Reflexive Reduction of Ordering is Strict Ordering
- Reflexive Reduction of Well-Founded Ordering is Foundational Relation
- Restriction of Monotone Function is Monotone
- Restriction of Well-Founded Ordering
- Restriction to Subset of Strict Total Ordering is Strict Total Ordering

### S

- Set is Subset of Finite Infima Set
- Set is Subset of Lower Closure
- Set is Subset of Upper Closure
- Set of Chains is Chain Complete for Inclusion
- Set of Infima for Sequence is Directed
- Set of Mappings can be Ordered by Inclusion
- Set of Relations can be Ordered by Inclusion
- Singleton is Chain
- Singleton of Bottom is Ideal
- Smallest Element is Infimum
- Smallest Element is Lower Bound
- Smallest Element is Minimal
- Smallest Element is Unique
- Smallest Element WRT Restricted Ordering
- Smallest Set is Unique
- Smallest Set may not Exist
- Strict Ordering can be Expanded to Compare Additional Pair
- Strict Ordering can be Expanded to Compare Additional Pair/Proof 1
- Strict Weak Ordering Induces Partition
- Strictly Monotone Mapping is Monotone
- Strictly Monotone Mapping with Totally Ordered Domain is Injective
- Strictly Precedes is Strict Ordering
- Strictly Succeed is Dual to Strictly Precede
- Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum
- Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum
- Subset of Natural Numbers is Cofinal iff Infinite
- Subset Relation is Ordering
- Subset Relation is Ordering/General Result
- Subset Relation on Power Set is Partial Ordering
- Succeed is Dual to Precede
- Suprema Preserving Mapping on Ideals is Increasing
- Suprema Preserving Mapping on Ideals Preserves Directed Suprema
- Supremum in Ordered Subset
- Supremum is Dual to Infimum
- Supremum is Increasing relative to Product Ordering
- Supremum is not necessarily Greatest Element
- Supremum is Unique
- Supremum of Cartesian Product
- Supremum of Power Set
- Supremum of Subset
- Supremum of Suprema
- Szpilrajn Extension Theorem

### T

### U

- Union of Chain of Chains is Chain
- Union of Initial Segments is Initial Segment or All of Woset
- Union of Nest of Orderings is Ordering
- Up-Complete Product
- Up-Complete Product/Lemma 1
- Up-Complete Product/Lemma 2
- Upper Bound for Subset
- Upper Bound is Dual to Lower Bound
- Upper Bound is Lower Bound for Inverse Ordering
- Upper Closure of Element without Element is Filter implies Element is Meet Irreducible
- Upper Closure of Subset is Subset of Upper Closure