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.
$\blacksquare$