Conditions under which Commutative Semigroup is Group/Warning

Warning concerning Conditions under which Commutative Semigroup is Group
Consider an algebraic structure $\struct {S, \circ}$ fulfilling the following conditions:

While it is the case that such an algebraic structure $\struct {S, \circ}$ is a group, if $\struct {S, \circ}$ is a semigroup which is not commutative, this does not necessarily follow.

Proof
Let $S$ be a set with more than $1$ element.

Let $\struct {S, \circ}$ be a semigroup such that $\circ$ is the left operation.

From Structure under Left Operation is Semigroup, $\struct {S, \circ}$ is indeed a semigroup.

From Right Operation is Anticommutative we have that $\circ$ is specifically not commutative.

By definition of the left operation:


 * $\forall x \in S: x \circ x = x$

Thus $x$ fulfils the role of $y$ in condition $(1)$:
 * $\forall x \in S: \exists y \in S: y \circ x = x$

and so $(1)$ is satisfied.

By definition of the left operation:
 * $\forall x, y \in S: y \circ x = y$

That is:
 * $\forall x, y \in S: \exists z \in S: z \circ x = y$

where here $z = y$.

Hence $x$ fulfils the role of $y$ and $y$ fulfils the role of $z$ in:
 * $\forall x, y \in S: y \circ x = x \implies \exists z \in S: z \circ x = y$

and so $(2)$ is satisfied.

But $\struct {S, \circ}$ is not a group.