Group is Inverse Semigroup with Identity

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Theorem

A group is an inverse semigroup with an identity.


Proof

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

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

and

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


Thus the criteria of an inverse semigroup are fulfilled.

$\blacksquare$