Definition 3

A mapping f of E into E' such that relation (1) holds for every ordered pair $(x, y) \in E \times E$ is called a homomorphism, or morphism, of E into E'; if $E = E'$, f is called an endomorphism of E.

The identity mapping of a magma E is a homomorphism, the composition of two homomorphism is a homomorphism.

† The following is an example where this principle would lead to confusion and should therefore not be used. Suppose that the law of composition in question is the law $(A, B) \mapsto A \cup B$ between subsets of a set E; a law of composition

is derived between subsets of $\mathfrak{P}(E)$, $F(\mathscr{A}, \mathscr{B})$ being the set of $A \cup B$ with $A \in \mathscr{A}, B \in \mathscr{B}$; but $F(\mathscr{A}, \mathscr{B})$ should not be denoted by $\mathscr{A} \cup \mathscr{B}$, as this notation already has a different meaning (the union of $\mathscr{A}$ and $\mathscr{B}$ considered as subsets of $\mathfrak{P}(E)$).

For a mapping $f$ of E into E' to be an isomorphism, it is necessary and sufficient that it be a bijective homomorphism and $f^{-1}$ is then an isomorphism of E' onto E.