Naturally Ordered Semigroup is Unique/Isomorphism is Unique

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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 isomorphism $g: \struct {S, \circ, \preceq} \to \struct {S', \circ', \preceq'}$ defined as:

$\forall a \in S: \map g a = \circ'^a 1'$

is unique.


Proof

Let $f: S \to S'$ be another isomorphism different from $g$.

Aiming for a contradiction, suppose $\map f 1 \ne 1'$.

We show by induction that $1' \notin \Cdm f$.



... Thus $1' \notin \Cdm f$ which is a contradiction.


Thus $\map f 1 = 1$ and it follows



that $f = g$.

Thus the isomorphism $g$ is unique.

$\blacksquare$


Sources