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
Some lemmata:

Hence holds.

Hence holds.

We have that:
 * $0 \in M$

and:
 * $1 \in M$

and trivially holds.

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