Definition:Homomorphism (Abstract Algebra)

Definition
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

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

Let $\circ$ have 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 $\left({T, *_1, *_2, \ldots, *_n}\right)$.

Let, $\forall k \in \left[{1 \,.\,.\, n}\right]$, $\circ_k$ have 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.

Also known as
Some sources refer to a homomorphism as a morphism, but this term is best reserved for its use in category theory.

Also see

 * Definition:Epimorphism (Abstract Algebra): a surjective homomorphism


 * Definition:Monomorphism (Abstract Algebra): an injective homomorphism


 * Definition:Isomorphism (Abstract Algebra): a bijective homomorphism


 * Endomorphism: a homomorphism from an algebraic structure to itself


 * Definition:Automorphism (Abstract Algebra): 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.