Editing Homomorphism (section)

== Definition ==
A homomorphism is a map between two [[algebraic structure]]s of the same type (e.g. two groups, two fields, two vector spaces), that preserves the [[operation (mathematics)|operations]] of the structures. This means a [[map (mathematics)|map]] <math>f: A \to B</math> between two [[set (mathematics)|sets]] <math>A</math>, <math>B</math> equipped with the same structure such that, if <math>\cdot</math> is an operation of the structure (supposed here, for simplification, to be a [[binary operation]]), then

<math display="block">f(x\cdot y)=f(x)\cdot f(y)</math>

for every pair <math>x</math>, <math>y</math> of elements of <math>A</math>.<ref group="note">As it is often the case, but not always, the same symbol for the operation of both <math>A</math> and <math>B</math> was used here.</ref> One says often that <math>f</math> preserves the operation or is compatible with the operation.

Formally, a map <math>f: A\to B</math> preserves an operation <math>\mu</math> of [[arity]] <math>k</math>, defined on both <math>A</math> and <math>B</math> if 

<math display="block">f(\mu_A(a_1, \ldots, a_k)) = \mu_B(f(a_1), \ldots, f(a_k)),</math> 

for all elements <math>a_1, ..., a_k</math> in <math>A</math>.

The operations that must be preserved by a homomorphism include [[0-ary function|0-ary operations]], that is the constants. In particular, when an [[identity element]] is required by the type of structure, the identity element of the first structure must be mapped to the corresponding identity element of the second structure.

For example:

* A [[semigroup homomorphism]] is a map between [[semigroup]]s that preserves the semigroup operation.
* A [[monoid homomorphism]] is a map between [[monoid]]s that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid (the identity element is a [[0-ary function|0-ary operation]]).
* A [[group homomorphism]] is a map between [[group (mathematics)|groups]] that preserves the group operation. This implies that the group homomorphism maps the identity element of the first group to the identity element of the second group, and maps the [[inverse element|inverse]] of an element of the first group  to the inverse of the image of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism.
* A [[ring homomorphism]] is a map between [[ring (mathematics)|rings]] that preserves the ring addition, the ring multiplication, and the [[multiplicative identity]]. Whether the multiplicative identity is to be preserved depends upon the definition of ''ring'' in use. If the multiplicative identity is not preserved, one has a [[rng (algebra)|rng]] homomorphism.
* A [[linear map]] is a homomorphism of [[vector space]]s; that is, a group homomorphism between vector spaces that preserves the abelian group structure and [[scalar multiplication]].
* A [[module homomorphism]], also called a linear map between [[module (mathematics)|modules]], is defined similarly.
* An [[algebra homomorphism]] is a map that preserves the [[algebra over a field|algebra]] operations.

An algebraic structure may have more than one operation, and a homomorphism is required to preserve each operation. Thus a map that preserves only some of the operations is not a homomorphism of the structure, but only a homomorphism of the substructure obtained by considering only the preserved operations. For example, a map between monoids that preserves the monoid operation and not the identity element, is not a monoid homomorphism, but only a semigroup homomorphism.

The notation for the operations does not need to be the same in the source and the target of a homomorphism. For example, the [[real number]]s form a group for addition, and the positive real numbers form a group for multiplication. The [[exponential function]] 

<math display="block">x\mapsto e^x</math>

satisfies

<math display="block">e^{x+y} = e^xe^y,</math>

and is thus a homomorphism between these two groups. It is even an isomorphism (see below), as its [[inverse function]], the [[natural logarithm]], satisfies 

<math display="block">\ln(xy)=\ln(x)+\ln(y), </math>
and is also a group homomorphism.