# Group is Inverse Semigroup with Identity

## Theorem

A group is an inverse semigroup with an identity.

## Proof

Let $\struct {S, \circ}$ be a group.

Let $a \in S$.

Then:

 $\ds e$ $=$ $\ds a \circ a^{-1}$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds \leadsto \ \$ $\ds e \circ a$ $=$ $\ds a \circ a^{-1} \circ a$ $\ds \leadsto \ \$ $\ds a$ $=$ $\ds a \circ a^{-1} \circ a$ Definition of Identity Element

and

 $\ds e$ $=$ $\ds a \circ a^{-1}$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds \leadsto \ \$ $\ds a^{-1} \circ e$ $=$ $\ds a^{-1} \circ a \circ a^{-1}$ $\ds \leadsto \ \$ $\ds a^{-1}$ $=$ $\ds a^{-1} \circ a \circ a^{-1}$ Definition of Identity Element

Thus the criteria of an inverse semigroup are fulfilled.

$\blacksquare$