Left Identity while exists Left Inverse for All is Identity

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Let $\struct {S, \circ}$ be a semigroup with a left identity $e_L$ such that:

$\forall x \in S: \exists x_L: x_L \circ x = e_L$

That is, every element of $S$ has a left inverse with respect to the left identity.

Then $e_L$ is also a right identity, that is, is an identity.


From Left Inverse for All is Right Inverse we have that:

$x \circ x_L = e_L$


\(\ds x \circ e_L\) \(=\) \(\ds x \circ \paren {x_L \circ x}\) Definition of Left Inverse Element
\(\ds \) \(=\) \(\ds \paren {x \circ x_L} \circ x\) Semigroup Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds e_L \circ x\) Left Inverse for All is Right Inverse
\(\ds \) \(=\) \(\ds x\) Definition of Left Identity

So $e_L$ behaves as a right identity as well as a left identity.

That is, by definition, $e_L$ is an identity element.


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