Difference between revisions of "Homomorphism"
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− | 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 [[ | + | 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 [[function]] <math>\phi : A \to B</math> that preserves the structure of the species. |
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. | 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. |
Revision as of 11:31, 21 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 function
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.