Difference between revisions of "Homomorphism"
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* [[Endomorphism]] | * [[Endomorphism]] | ||
* [[Exact Sequence]] | * [[Exact Sequence]] | ||
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[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
Latest revision as of 15:33, 11 February 2024
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 
,
![\[\phi( a*b) = \phi(a)* \phi(b) .\]](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.
