Homomorphism
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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.