# Difference between revisions of "Homomorphism"

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− | Let <math>A</math> and <math>B</math> be algebraic structures of the same species. A '''homomorphism''' is a [[function]] <math>\phi : A \to B</math> that preserves the structure of the species. | + | Let <math>A</math> and <math>B</math> be algebraic structures of the same species, for example two [[group]]s or [[field]]s. A '''homomorphism''' is a [[function]] <math>\phi : A \to B</math> 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]]. | + | For example, if <math>A</math> is a [[substructure]] ([[subgroup]], [[subfield]], etc.) of <math>B</math>, the ''inclusion map'' <math>i: A \to B</math> such that <math>i(a) = a</math> for all <math>a \in A</math> is a homomorphism. |

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+ | 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 == | == 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 | + | 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>, |

<cmath> \phi( a*b) = \phi(a)* \phi(b) . </cmath> | <cmath> \phi( a*b) = \phi(a)* \phi(b) . </cmath> | ||

− | Similarly, if <math>A</math> and <math>B</math> are [[field]]s or [[ring]]s, a homomorphism from <math>A</math> to <math>B</math> is a | + | Similarly, if <math>A</math> and <math>B</math> are [[field]]s or [[ring]]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>, |

<cmath> \begin{align*} | <cmath> \begin{align*} | ||

\phi(a+b) &= \phi(a) + \phi(b) \\ | \phi(a+b) &= \phi(a) + \phi(b) \\ | ||

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* [[Isomorphism]] | * [[Isomorphism]] | ||

− | + | * [[Automorphism]] | |

+ | * [[Endomorphism]] | ||

+ | * [[Exact Sequence]] | ||

[[Category:Abstract algebra]] | [[Category:Abstract algebra]] |

## Latest revision as of 11:31, 21 February 2008

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

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.