Difference between revisions of "Homomorphism"

(stub)
 
m (mapping is a more generalized term for function)
Line 1: Line 1:
 
{{stub}}
 
{{stub}}
  
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.  A '''homomorphism''' is a [[mapping]] <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]].
 
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]].
Line 9: Line 9:
 
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 function <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 [[partially ordered set]]s, a homomorphism from <math>A</math> to <math>B</math> is a function <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 function 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>,
 
<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 function <math>\phi : A \to B</math> such that for all <math>a,b \in A</math>,
+
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) \\

Revision as of 14:22, 19 February 2008

This article is a stub. Help us out by expanding it.

Let $A$ and $B$ be algebraic structures of the same species. A homomorphism is a mapping $\phi : A \to B$ 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 $A$ and $B$ are partially ordered sets, a homomorphism from $A$ to $B$ is a function $\phi : A \to B$ 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

Invalid username
Login to AoPS