Difference between revisions of "Homomorphism"
m |
Riemann1826 (talk | contribs) (→See Also) |
||
Line 25: | Line 25: | ||
* [[Endomorphism]] | * [[Endomorphism]] | ||
* [[Exact Sequence]] | * [[Exact Sequence]] | ||
+ | * [[Diffeomorphism]] | ||
[[Category:Abstract algebra]] | [[Category:Abstract algebra]] |
Latest revision as of 15:33, 11 February 2024
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.