Definition:Homomorphism (Abstract Algebra)

Definition
Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be a mapping from one algebraic structure $$\left({S, \circ}\right)$$ to another $$\left({T, *}\right)$$.

If $$\circ$$ has the morphism property under $$\phi$$, that is:


 * $$\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$$

then $$\phi$$ is a homomorphism.

This can be generalised to algebraic structures with more than one operation:

Let: be algebraic structures.
 * $$\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$$
 * $$\left({T, *_1, *_2, \ldots, *_n}\right)$$

Let $$\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({T, *_1, *_2, \ldots, *_n}\right)$$ be a mapping from $$\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$$ to $$\left({T, *_1, *_2, \ldots, *_n}\right)$$.

If, $$\forall k \in \left[{1 \,. \, . \, n}\right]$$, $$\circ_k$$ has the morphism property under $$\phi$$, that is:


 * $$\forall x, y \in S: \phi \left({x \circ_k y}\right) = \phi \left({x}\right) *_k \phi \left({y}\right)$$

then $$\phi$$ is a homomorphism.

Group Homomorphism
If both $$\left({S, \circ}\right)$$ and $$\left({T, *}\right)$$ are groups, then a homomorphism $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ is called a group homomorphism.

Ring Homomorphism
If both $$\left({R, +, \circ}\right)$$ and $$\left({S, \oplus, *}\right)$$ are rings, then a homomorphism $$\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$$ is called a ring homomorphism.

Field Homomorphism
If both $$\left({R, +, \circ}\right)$$ and $$\left({S, \oplus, *}\right)$$ are fields, then a homomorphism $$\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$$ is called a field homomorphism.

F-Homomorphism
Let $$R$$, $$R^\prime$$ be rings with unity.

Let $$F$$ be a subfield of both $$R$$ and $$R^\prime$$.

Then a ring homomorphism $$\varphi: R \to R^\prime$$ is called an $$F$$-homomorphism if:
 * $$\forall a \in F: \varphi\left({a}\right) = a$$.

That is, $$\varphi \restriction_F = 1_F$$ where:
 * $$\varphi \restriction_F$$ is the restriction of $$\varphi$$ to $$F$$;
 * $$1_F$$ is the identity mapping on $$F$$.

Furthermore, if $$\varphi$$ is an isomorphism, we call it an $$F$$-isomorphism and write $$R \cong_F R^\prime$$.

R-Algebraic Structure Homomorphism
Let $$\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$$ and $$\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$$ be $R$-algebraic structures.

Then $$\phi$$ is an $$R$$-algebraic structure homomorphism iff:


 * 1) $$\forall k: k \in \left[{1 \, . \, . \, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$$;
 * 2) $$\forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$$.

Note that this definition also applies to modules and vector spaces.

Image
As a homomorphism is a mapping, and therefore a relation, we define the image of a homomorphism in the same way:


 * $$\operatorname{Im} \left({\phi}\right) = \left\{{t \in T: \exists s \in S: t = \phi \left({s}\right)}\right\}$$

Homomorphism as Cartesian Product
Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be a mapping from one algebraic structure $$\left({S, \circ}\right)$$ to another $$\left({T, *}\right)$$.

We define the cartesian product $$\phi \times \phi: S \times S \to T \times T$$ as:
 * $$\forall \left({x, y}\right) \in S \times S: \left({\phi \times \phi}\right) \left({x, y}\right) = \left({\phi \left({x}\right), \phi \left({y}\right)}\right)$$

Hence we can state that $$\phi$$ is a homomorphism iff:
 * $$\ast \left({\left({\phi \times \phi}\right) \left({x, y}\right)}\right) = \phi \left({\circ \left({x, y}\right)}\right)$$

using the notation $$\circ \left({x, y}\right)$$ to denote the operation $$x \circ y$$.

The point of doing this is so we can illustrate what is going on in a commutative diagram:


 * Homomorphism.png

Thus we see that $$\phi$$ is a homomorphism iff both of the composite mappings from $$S \times S$$ to $$T$$ have the same effect on all elements of $$S \times S$$.

Also see

 * Epimorphism: a surjective homomorphism;


 * Monomorphism: an injective homomorphism;


 * Isomorphism: a bijective homomorphism;


 * Endomorphism: a homomorphism from an algebraic structure to itself;


 * Automorphism: an isomorphism from an algebraic structure to itself.

Group definition

 * : $$\S 7.1$$
 * : $$\S 1.10$$
 * : Chapter $$\text{II}$$
 * : $$\S 47$$
 * : $$\S 8$$: Definition $$8.1$$

Ring definition

 * : $$\S 2.2$$: Definition $$2.4$$
 * : $$\S 57$$

R-Algebraic Structure definition

 * : $$\S 28$$