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.