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. |
| + | |||
| + | 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]] | ||
| + | * [[Diffeomorphism]] | ||
[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
Latest revision as of 14:33, 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.
