Difference between revisions of "Homomorphism"
10000th User (talk | contribs) m (→See Also) |
m |
||
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 [[mapping]] <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, for example two [[group]]s or [[field]]s. 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]]. | + | 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. |
+ | |||
+ | 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 == |
Revision as of 15:04, 20 February 2008
This article is a stub. Help us out by expanding it.
Let and
be algebraic structures of the same species, for example two groups or fields. A homomorphism is a mapping
that preserves the structure of the species.
For example, if is a substructure (subgroup, subfield, etc.) of
, the inclusion map
such that
for all
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 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.