Definition:Inverse (Abstract Algebra)

Definition
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.

Various symbols are seen for a general inverse, for example $$\hat x$$ and $$x^*$$.

In multiplicative notation:
 * If $$s \in S$$ has an inverse, it is denoted $$s^{-1}$$.

If the operation concerned is commutative, then additive notation is often used:
 * If $$s \in S$$ has an inverse, it is denoted $$-s$$.

Also see

 * Left Inverse for All is Right Inverse