What is law of composition?

A law of composition on a set $E$ is a map $E \times E \to E$, written $(x, y) \mapsto x * y$. It is associative when $(x*y)*z = x*(y*z)$ always, commutative when $x * y = y * x$ always. An element $e$ is an identity when $e * x = x * e = x$ for all $x$; then $x'$ is an inverse of $x$ when $x * x' = x' * x = e$.