# Category:Order Isomorphisms

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This category contains results about **Order Isomorphisms**.

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

Let $\phi: S \to T$ be a bijection such that:

- $\phi: S \to T$ is order-preserving
- $\phi^{-1}: T \to S$ is order-preserving.

Then $\phi$ is an **order isomorphism**.

## Subcategories

This category has the following 7 subcategories, out of 7 total.

### E

### F

### I

### O

## Pages in category "Order Isomorphisms"

The following 56 pages are in this category, out of 56 total.

### C

### D

### I

- Identity Mapping is Automorphism/Ordered Semigroups
- Identity Mapping is Ordered Semigroup Automorphism
- Induced Relation Generates Order Isomorphism
- Inverse of Order Isomorphism is Order Isomorphism
- Inversion Mapping is Isomorphism from Ordered Abelian Group to its Dual
- Isomorphic Ordinals are Equal
- Isomorphisms between Lower Sections of Well-Ordered Classes are Nested

### M

### O

- Order Automorphism on Well-Ordered Class is Forward Moving
- Order Automorphism on Well-Ordered Class is Identity Mapping
- Order Embedding into Image is Isomorphism
- Order Isomorphic Sets are Equivalent
- Order Isomorphism between Ordinals and Proper Class/Corollary
- Order Isomorphism between Tosets is not necessarily Unique
- Order Isomorphism between Wosets is Unique
- Order Isomorphism forms Galois Connection
- Order Isomorphism from Woset onto Subset
- Order Isomorphism iff Strictly Increasing Surjection
- Order Isomorphism is Equivalence Relation
- Order Isomorphism is Preserved by Antilexicographic Order
- Order Isomorphism is Preserved by Order Sum
- Order Isomorphism is Reflexive
- Order Isomorphism is Surjective Order Embedding
- Order Isomorphism is Symmetric
- Order Isomorphism is Transitive
- Order Isomorphism on Lattice preserves Lattice Structure
- Order Isomorphism on Strictly Well-Founded Relation preserves Strictly Well-Founded Structure
- Order Isomorphism on Totally Ordered Set preserves Total Ordering
- Order Isomorphism on Well-Ordered Set preserves Well-Ordering
- Order Isomorphism Preserves Infima and Suprema
- Order Isomorphism Preserves Initial Segments
- Order Isomorphism Preserves Lower Bounds
- Order Isomorphism Preserves Strictly Minimal Elements
- Order Isomorphism Preserves Upper Bounds
- Order-Preserving Bijection on Wosets is Order Isomorphism
- Order-Preserving Identity Mapping between Ordered Structures not necessarily Isomorphism
- Ordered Semigroup Monomorphism into Image is Isomorphism
- Orderings on Set with 3 Elements
- Orderings on Set with 4 Elements
- Ordinals Isomorphic to the Same Well-Ordered Set