# Category:Homomorphisms

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This category contains results about **Homomorphisms** in the context of **Abstract Algebra**.

Definitions specific to this category can be found in Definitions/Homomorphisms.

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from $\struct {S, \circ}$ to $\struct {T, *}$.

Let $\circ$ have the morphism property under $\phi$, that is:

- $\forall x, y \in S: \map \phi {x \circ y} = \map \phi x * \map \phi y$

Then $\phi$ is a **homomorphism**.

## Subcategories

This category has the following 12 subcategories, out of 12 total.

## Pages in category "Homomorphisms"

The following 33 pages are in this category, out of 33 total.

### C

- Composite of Homomorphisms is Homomorphism
- Composite of Homomorphisms is Homomorphism/Algebraic Structure
- Composite of Homomorphisms is Homomorphism/R-Algebraic Structure
- Composite of Homomorphisms on Algebraic Structure is Homomorphism
- Composition of Mappings is Left Distributive over Homomorphism of Pointwise Operation
- Condition for Mapping between Structures to be Homomorphism
- Constant Mapping to Identity is Homomorphism

### E

- Equivalent Conditions for Entropic Structure/Mapping from External Direct Product is Homomorphism
- Equivalent Conditions for Entropic Structure/Pointwise Operation is Homomorphism
- Equivalent Conditions for Entropic Structure/Pointwise Operation of Homomorphisms from External Direct Product is Homomorphism
- Extension Theorem for Homomorphisms

### H

- Homomorphic Image of Vector Space
- Homomorphism of External Direct Products
- Homomorphism of External Direct Products/General Result
- Homomorphism of Powers
- Homomorphism of Powers/Integers
- Homomorphism of Powers/Natural Numbers
- Homomorphism of Powers/Naturally Ordered Semigroup
- Homomorphism on Induced Structure to Commutative Semigroup
- Homomorphism Preserves Subsemigroups
- Homomorphism to Group Preserves Identity
- Homomorphism to Group Preserves Inverses
- Homomorphism with Cancellable Codomain Preserves Identity
- Homomorphism with Identity Preserves Inverses