Group is Inverse Semigroup with Identity

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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.

$\blacksquare$