ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D40
Inverse element

Let $S = (X, +)$ be an D21: Algebraic structure such that
 (i) $$X \neq \emptyset$$ (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) $$y + x = 0_S$$ (D535: Left inverse element) (2) $$x + y = 0_S$$ (D536: Right inverse element)

Let $S = (X, f)$ be an D21: Algebraic structure such that
 (i) $$X \neq \emptyset$$ (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) $$f(y, x) = I_S$$ (D535: Left inverse element) (2) $$f(x, y) = I_S$$ (D536: Right inverse element)

Let $S = (X, \times)$ be an D21: Algebraic structure such that
 (i) $$X \neq \emptyset$$ (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) $$y x = 1_S$$ (D535: Left inverse element) (2) $$x y = 1_S$$ (D536: Right inverse element)