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 
,
![\[\phi( a*b) = \phi(a)* \phi(b) .\]](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.
