Product of Semigroup Element with Left Inverse is Idempotent

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Theorem

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

Let $x \in S$ such that $\exists x_L: x_L \circ x = e_L$, i.e. $x$ has a left inverse with respect to the left identity.


Then:

$\paren {x \circ x_L} \circ \paren {x \circ x_L} = x \circ x_L$

That is, $x \circ x_L$ is idempotent.


Proof

\(\displaystyle \paren {x \circ x_L} \circ \paren {x \circ x_L}\) \(=\) \(\displaystyle x \circ \paren {x_L \circ x} \circ x_L\) as $\circ$ is associative
\(\displaystyle \) \(=\) \(\displaystyle x \circ e_L \circ x_L\) Definition of Left Inverse Element
\(\displaystyle \) \(=\) \(\displaystyle x \circ x_L\) Definition of Left Identity

$\blacksquare$


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