Difference between revisions of "Homomorphism"
10000th User (talk | contribs) m (mapping is a more generalized term for function) |
10000th User (talk | contribs) m (→Examples: minor typos) |
||
Line 7: | Line 7: | ||
== 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 mapping 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>, |
Revision as of 14:23, 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.