# Difference between revisions of "Homomorphism"

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== Examples == | == Examples == | ||

− | If <math>A</math> and <math>B</math> are [[partially ordered set]]s, a homomorphism from <math>A</math> to <math>B</math> is a | + | If <math>A</math> and <math>B</math> are [[partially ordered set]]s, a homomorphism from <math>A</math> to <math>B</math> is a mapping <math>\phi : A \to B</math> such that for all <math>a, b \in A</math>, if <math>a \le b</math>, then <math>\phi(a) \le \phi(b)</math>. |

If <math>A</math> and <math>B</math> are [[group]]s, with group law of <math>*</math>, then a homomorphism <math>\phi : A \to B</math> is a mapping such that for all <math>a,b \in A</math>, | If <math>A</math> and <math>B</math> are [[group]]s, with group law of <math>*</math>, then a homomorphism <math>\phi : A \to B</math> is a mapping such that for all <math>a,b \in A</math>, |

## Revision as of 13:23, 19 February 2008

*This article is a stub. Help us out by expanding it.*

Let and be algebraic structures of the same species. A **homomorphism** is a mapping that preserves the structure of the species.

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.