Naturally_Ordered_Semigroup_is_Unique

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.