# Homomorphism

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Let and be algebraic structures of the same species, for example two groups or fields. A **homomorphism** is a function that preserves the structure of the species.

For example, if is a substructure (subgroup, subfield, etc.) of , the *inclusion map* such that for all is a homomorphism.

A homomorphism from a structure to itself is called an endomorphism. A homomorphism that is bijective is called an isomorphism. A bijective endomorphism is called an automorphism.

## Examples

If and are partially ordered sets, a homomorphism from to is a mapping such that for all , if , then .

If and are groups, with group law of , then a homomorphism is a mapping such that for all , Similarly, if and are fields or rings, a homomorphism from to is a mapping such that for all , In other words, distributes over addition and multiplication.