Definition:Homomorphism (Abstract Algebra)

Definition
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from $\struct {S, \circ}$ to $\struct {T, *}$.

Let $\circ$ have the morphism property under $\phi$, that is:


 * $\forall x, y \in S: \map \phi {x \circ y} = \map \phi x * \map \phi y$

Then $\phi$ is a homomorphism.

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

Let:
 * $\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
 * $\struct {T, *_1, *_2, \ldots, *_n}$

be algebraic structures.

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

Let, $\forall k \in \closedint 1 n$, $\circ_k$ have the morphism property under $\phi$, that is:


 * $\forall x, y \in S: \map \phi {x \circ_k y} = \map \phi x *_k \map \phi y$

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


 * Definition:Endomorphism: a homomorphism from a magma to itself


 * Definition:Automorphism (Abstract Algebra): an isomorphism from a magma to itself.