Equivalence of Definitions of Self-Inverse

Theorem

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

Let $x \in S$.

The following definitions of the concept of Self-Inverse Element in the context of Abstract Algebra are equivalent:

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 $x$.

Proof

Let $x \in S$.

\(\ds x \circ x\)

\(=\)

\(\ds e_S\)

by hypothesis

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {x \circ x} \circ x^{-1}\)

\(=\)

\(\ds e_S \circ x^{-1}\)

Monoid Axiom $\text S 0$: Closure

\(\ds \leadstoandfrom \ \ \)

\(\ds x \circ \paren {x \circ x^{-1} }\)

\(=\)

\(\ds e_S \circ x^{-1}\)

Monoid Axiom $\text S 1$: Associativity

\(\ds \leadstoandfrom \ \ \)

\(\ds x \circ \paren {x \circ x^{-1} }\)

\(=\)

\(\ds x^{-1}\)

Monoid Axiom $\text S 2$: Identity

\(\ds \leadstoandfrom \ \ \)

\(\ds x \circ e_S\)

\(=\)

\(\ds x^{-1}\)

Definition of Inverse Element

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds x^{-1}\)

Monoid Axiom $\text S 2$: Identity

$\blacksquare$