Difference between revisions of "Homomorphism"

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[[Category:Abstract algebra]]
 
[[Category:Abstract algebra]]

Revision as of 14:24, 19 February 2008

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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 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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