Let $S = (X, +)$ be an D21: Algebraic structure such that

(i)	\begin{equation} X \neq \emptyset \end{equation}
(ii)	$0_S$ is an D39: Identity element in $S$

A D2218: Set element $y \in X$ is an inverse element of $x \in X$ in $S$ if and only if

(1)	\begin{equation} y + x = 0_S \end{equation}	(D535: Left inverse element)
(2)	\begin{equation} x + y = 0_S \end{equation}	(D536: Right inverse element)

Let $S = (X, f)$ be an D21: Algebraic structure such that

(i)	\begin{equation} X \neq \emptyset \end{equation}
(ii)	$I_S$ is an D39: Identity element in $S$

A D2218: Set element $y \in X$ is an inverse element of $x \in X$ in $S$ if and only if

(1)	\begin{equation} f(y, x) = I_S \end{equation}	(D535: Left inverse element)
(2)	\begin{equation} f(x, y) = I_S \end{equation}	(D536: Right inverse element)

Let $S = (X, \times)$ be an D21: Algebraic structure such that

(i)	\begin{equation} X \neq \emptyset \end{equation}
(ii)	$1_S$ is an D39: Identity element in $S$

A D2218: Set element $y \in X$ is an inverse element of $x \in X$ in $S$ if and only if

(1)	\begin{equation} y x = 1_S \end{equation}	(D535: Left inverse element)
(2)	\begin{equation} x y = 1_S \end{equation}	(D536: Right inverse element)