Natural Numbers with Extension fulfil Naturally Ordered Semigroup Axioms 1, 3 and 4

Construction
There exists a unique total ordering $\le$ on $M$ such that:
 * the restriction of $\le$ to $\N$ is the given total ordering $\le$ on $\N$
 * $0 < \beta < 1$

such that the algebraic structure:
 * $\struct {M, +, \le}$

is an ordered semigroup which fulfils the axioms:



but:
 * does not fulfil
 * $\struct {M, +}$ is not isomorphic to $\struct {\N, +}$.

Proof
Recall the axioms:

Some lemmata:

Let $S \subseteq M$ such that $\beta \notin S$.

Then:
 * $S \subseteq \N$

By the Well-Ordering Principle, $\struct {\N, \le}$ is a well-ordered set.

Hence $S$ is well-ordered.

Hence by definition, $S$ has a smallest element

Let $T \subseteq M$ such that $\beta \in T$.


 * Case $1$: $0 \in T$

Then as $0 < \beta$ we have that:
 * $\forall x \in T: 0 \le x$

and so $T$ has a smallest element, that is, $0$.


 * Case $2$: $0 \notin T$

From Lemma $2$ we have that $\le$ is a total ordering such that:
 * $\forall x \in T: \beta \le x$

and so $T$ has a smallest element, that is, $\beta$.

Hence it has been shown that every subset of $M$ has a smallest element.

That is, $\struct {M, \le}$ is a well-ordered set.

Hence holds.

By the construction of the natural numbers, $\struct {\N, +, \le}$ is a naturally ordered semigroup.

Hence holds for $\N$:
 * $\forall m, n \in \N: m \le n \implies \exists p \in \N: m + p = n$

Hence:
 * $\forall m, n \in M \setminus \set \beta: m \le n \implies \exists p \in \N: m + p = n$

Now consider $\beta \in M$.

Let $m \in M: m \le \beta$.
 * Case 1:

Then either:
 * $m = \beta$

in which case:
 * $\exists \beta \in M: m + \beta = \beta$

or:
 * $m = 0$

in which case also:
 * $\exists \beta \in M: m + \beta = \beta$

Let $n \in M: \beta \le n$.
 * Case 2:

Then:
 * $\exists n \in \N: n + \beta = n$

and it is seen that holds.

We have that:
 * $0 \in M$

and:
 * $1 \in M$

and trivially holds.

We have that:


 * $\forall n \in M \setminus \set 0: n + 0 = n + \beta = n$

but it is not the case that $0 = \beta$.

That is, $\struct {M, +, \le}$ does not fulfil.

Hence the result.