Naturally Ordered Semigroup is Unique

Theorem
Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be naturally ordered semigroups.

Let:
 * $0'$ be the smallest element of $S'$
 * $1'$ be the smallest element of $S' \setminus \set {0'} = S'^*$.

Then the mapping $g: S \to S'$ defined as:
 * $\forall a \in S: \map g a = \circ'^a 1'$

is an isomorphism from $\struct {S, \circ, \preceq}$ to $\struct {S', \circ', \preceq'}$.

This isomorphism is unique.

Thus, up to isomorphism, there is only one naturally ordered semigroup.