Definition:Inverse (Abstract Algebra)

Let $$\left({S, \circ}\right)$$ be a monoid whose identity is $$e_S$$.

Left Inverse
An element $$x_L \in S$$ is called a left inverse of $$x$$ iff $$x_L \circ x = e_S$$.

Right Inverse
An element $$x_R \in S$$ is called a right inverse of $$x$$ iff $$x \circ x_R = e_S$$.

Inverse
An element $$y \in S$$ such that $$y \circ x = e_S = x \circ y$$, that is, $$y$$ is both a left inverse and a right inverse of $$x$$, then $$y$$ is a two-sided inverse (or simply inverse) of $$x$$.

The notation used to represent an inverse of an element depends on the set and binary operation under consideration. In the general case, the multiplicative notation is used. That is, if $$s \in S$$ has an inverse, it is denoted $$s^{-1}$$.