Definition:Inverse (Abstract Algebra)/Inverse

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

Let an element $y \in S$ be such that:
 * $y \circ x = e_S = x \circ y$

that is, $y$ is both:
 * a left inverse of $x$

and:
 * a right inverse of $x$.

Then $y$ is an inverse of $x$.

Also known as
An inverse of $x$ can also be referred to as a two-sided 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

 * Definition:Left Inverse Element
 * Definition:Right Inverse Element