Identity Property in Semigroup

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $s \in S$ be such that:


 * $\forall a \in S: \exists x, y \in S: s \circ x = a = y \circ s$

Then $\struct {S, \circ}$ has an identity.

Proof
Suppose that:
 * $\forall a \in S: \exists x, y \in S: s \circ x = a = y \circ s$.

Since $s \in S$, it follows that:
 * $\exists v, w \in S: s \circ v = s = w \circ s$.

Let $a \in S$.

Then:
 * $\exists x, y \in S: s \circ x = a = y \circ s$.

Thus:

Hence:
 * $w \circ a = a$ and $a \circ v = a$

for any $a \in S$.

In particular:
 * Letting $a = v$ in the first of these gives $w \circ v = v$
 * Letting $a = w$ in the second gives $w \circ v = w$.

Thus $v = w \circ v = w$ is the identity element in $S$.