Difference between revisions of "Homomorphism"
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* [[Isomorphism]] | * [[Isomorphism]] | ||
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[[Category:Abstract algebra]] | [[Category:Abstract algebra]] |
Revision as of 14:24, 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
,
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.