Naturally Ordered Semigroup Axioms are Independent

Theorem
Consider the naturally ordered semigroup axioms:

Each of the naturally ordered semigroup axioms is independent of all the others.

That is, you cannot drop any one of them and still uniquely define a naturally ordered semigroup.

Proof
This will be proved by demonstrating that for each of the naturally ordered semigroup axioms, it is possible to create an algebraic structure which fulfils all the others, but is not a naturally ordered semigroup.