# Definition:Self-Inverse Element

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## Definition

Let $\struct {S, \circ}$ be a monoid whose identity element is $e_S$.

Let $x \in S$ be an element of $S$.

### Definition 1

$x$ is a **self-inverse element of $\struct {S, \circ}$** if and only if $x \circ x = e_S$.

### Definition 2

$x$ is a **self-inverse element of $\struct {S, \circ}$** if and only if:

- $x$ is invertible

and:

- $x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.

## Also known as

The definition of a **self-inverse element** is usually made in the context of a group.

Some sources refer to such an element as an **involution**.

## Also see

- Results about
**self-inverse elements**can be found**here**.