# 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 $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

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 2 subcategories, out of 2 total.

### I

### O

## Pages in category "Order Isomorphisms"

The following 34 pages are in this category, out of 34 total.

### C

### I

### M

### O

- Order Embedding into Image is Isomorphism
- Order Isomorphic Sets are Equivalent
- 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 Surjective Order Embedding
- Order Isomorphism on Foundational Relation preserves Foundational Structure
- Order Isomorphism on Lattice preserves Lattice 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 Minimal Elements
- Order Isomorphism Preserves Upper Bounds
- Order-Preserving Bijection on Wosets is Order Isomorphism
- Ordered Semigroup Monomorphism into Image is Isomorphism
- Ordinals Isomorphic to the Same Well-Ordered Set