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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 [[mapping]] <math>\phi : A \to B</math> that preserves the structure of the species.
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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.
  
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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]].
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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 ==
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* [[Isomorphism]]
 
* [[Isomorphism]]
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* [[Automorphism]]
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* [[Exact Sequence]]
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* [[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 $A$ and $B$ be algebraic structures of the same species, for example two groups or fields. A homomorphism is a function $\phi : A \to B$ that preserves the structure of the species.

For example, if $A$ is a substructure (subgroup, subfield, etc.) of $B$, the inclusion map $i: A \to B$ such that $i(a) = a$ for all $a \in A$ 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 $A$ and $B$ are partially ordered sets, a homomorphism from $A$ to $B$ is a mapping $\phi : A \to B$ such that for all $a, b \in A$, if $a \le b$, then $\phi(a) \le \phi(b)$.

If $A$ and $B$ are groups, with group law of $*$, then a homomorphism $\phi : A \to B$ is a mapping such that for all $a,b \in A$, \[\phi( a*b) = \phi(a)* \phi(b) .\] Similarly, if $A$ and $B$ are fields or rings, a homomorphism from $A$ to $B$ is a mapping $\phi : A \to B$ such that for all $a,b \in A$, \begin{align*} \phi(a+b) &= \phi(a) + \phi(b) \\ \phi(ab) &= \phi(a)\phi(b) . \end{align*} In other words, $\phi$ distributes over addition and multiplication.

See Also

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