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.

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.

Linguistic Note
The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.