Let $\displaystyle \left({S, \circ}\right)$ be a semigroup.
Let $\displaystyle u \in S$ be such that:
$\displaystyle \forall a \in S: \exists x, y \in S: u \circ x = a = y \circ u$
Prove that $\displaystyle \left({S, \circ}\right)$ has an identity element.![]()