Definition:Inverse (Abstract Algebra)

Given a set $$S$$ together with a binary operation $$\circ$$ and identity element $$e \in S$$, if $$a,b \in S$$ where $$a \circ b=e$$, then $$a$$ is called a left inverse of $$b$$ and $$b$$ is called a right inverse of $$a$$.

If $$c \in S$$ is both a right and left inverse of some element $$d \in S$$, then $$c$$ is called a two-sided inverse (or simply inverse) of $$d$$. The notation used to represent an inverse of an element depends on the set and binary operation in consideration. In the general case, the multiplicative notation is used. That is, if $$s \in S$$ has an inverse, it is denoted $$s^{-1}$$.