An D21: Algebraic structure $G = (X, \times)$ is a monoid if and only if

(1)	\begin{equation} \forall \, x, y \in X : x y \in X \end{equation}
(2)	\begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}
(3)	\begin{equation} \exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x \end{equation}