A D263: Groupoid $G = (X, \times)$ is a semigroup if and only if \begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}

An D21: Algebraic structure $G = (X, +)$ is a semigroup 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}

An D21: Algebraic structure $G = (X, f)$ is a semigroup if and only if

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

An D21: Algebraic structure $G = (X, \times)$ is a semigroup 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}