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 $-s$.