# Homomorphism

Let $A$ and $B$ be algebraic structures of the same species, for example two groups or fields. A homomorphism is a function $\phi : A \to B$ that preserves the structure of the species.

For example, if $A$ is a substructure (subgroup, subfield, etc.) of $B$, the inclusion map $i: A \to B$ such that $i(a) = a$ for all $a \in A$ is a homomorphism.

A homomorphism from a structure to itself is called an endomorphism. A homomorphism that is bijective is called an isomorphism. A bijective endomorphism is called an automorphism.

## Examples

If $A$ and $B$ are partially ordered sets, a homomorphism from $A$ to $B$ is a mapping $\phi : A \to B$ such that for all $a, b \in A$, if $a \le b$, then $\phi(a) \le \phi(b)$.

If $A$ and $B$ are groups, with group law of $*$, then a homomorphism $\phi : A \to B$ is a mapping such that for all $a,b \in A$, $$\phi( a*b) = \phi(a)* \phi(b) .$$ Similarly, if $A$ and $B$ are fields or rings, a homomorphism from $A$ to $B$ is a mapping $\phi : A \to B$ such that for all $a,b \in A$, \begin{align*} \phi(a+b) &= \phi(a) + \phi(b) \\ \phi(ab) &= \phi(a)\phi(b) . \end{align*} In other words, $\phi$ distributes over addition and multiplication.