Group is Inverse Semigroup with Identity

Theorem

A group is an inverse semigroup with an identity.

Proof

Let $\left({S, \circ}\right)$ be a group. Then for all $a \in S$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle e\)	\(=\)	\(\displaystyle a \circ a^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Every $a \in G$ is Invertible
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle e \circ a\)	\(=\)	\(\displaystyle a \circ a^{-1} \circ a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a\)	\(=\)	\(\displaystyle a \circ a^{-1} \circ a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Identity

and

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle e\)	\(=\)	\(\displaystyle a \circ a^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Every $a \in G$ is Invertible
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a^{-1} \circ e\)	\(=\)	\(\displaystyle a^{-1} \circ a \circ a^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a^{-1}\)	\(=\)	\(\displaystyle a^{-1} \circ a \circ a^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Identity

Thus the criteria of an inverse semigroup are fulfilled.

$\blacksquare$