Right Identity while exists Right Inverse for All is Identity

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Theorem

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

$\forall x \in S: \exists x_R: x \circ x_R = e_R$

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


Then $e_R$ is also a left identity, that is, is an identity.


Proof

Let $x \in S$ be any element of $S$.

From Right Inverse for All is Left Inverse we have that $x_R \circ x = e_R$.

Then:

\(\displaystyle e_R \circ x\) \(=\) \(\displaystyle \paren {x \circ x_R} \circ x\) Definition of Right Inverse Element
\(\displaystyle \) \(=\) \(\displaystyle x \circ \paren {x_R \circ x}\) as $\circ$ is associative
\(\displaystyle \) \(=\) \(\displaystyle x \circ e_R\) $x_R$ is a Right Inverse
\(\displaystyle \) \(=\) \(\displaystyle x\) Definition of Right Identity


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

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

$\blacksquare$


Also see


Sources