Homomorphism: Difference between revisions
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* [[Isomorphism]] | * [[Isomorphism]] | ||
* [[Automorphism]] | |||
* [[Endomorphism]] | |||
* [[Exact Sequence]] | |||
[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
Revision as of 13:24, 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 
,
![\[\phi( a*b) = \phi(a)* \phi(b) .\]](http://latex.artofproblemsolving.com/b/6/4/b64dae3d0d9713d80138389bbed3caa176ac8aaa.png)
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.
