Homomorphism

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

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