Identity Property in Semigroup

Theorem
Let $$\left({S, \circ}\right)$$ 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 $$\left({S, \circ}\right)$$ 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 $$S$$.