Definition 1

Let E and E' be two sets each with an internal law of composition denoted by $(x, y) \mapsto x * y$, not necessarily everywhere defined. A mapping $f : E \to E'$ is a morphism if, for all $x, y$ in E such that $f(x) * f(y)$ is defined, the composition $x * y$ is also defined and:

(1)

More briefly, we may say that formula (1) must hold every time the right hand side has a meaning.

The notion of morphism is distinct from that of representation (Algebra, Chapter I, § 1, no. 1), where it is demanded that equation (1) hold whenever the left hand side has a meaning. Of course, the two notions coincide for laws of composition everywhere defined.