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

Construction
The algebraic structure:
 * $\struct {M, +}$

is not isomorphic to $\struct {\N, +}$.

Proof
there exists a (semigroup) isomorphism $\phi$ from $\struct {M, +}$ to $\struct {\N, +}$.

By definition of isomorphism:
 * $\phi$ is a homomorphism
 * $\phi$ is a bijection.

As $\phi$ is a fortiori a surjection, $\phi$ is also an epimorphism.

Hence from Epimorphism Preserves Identity:
 * $\map \phi 0 = 0$

Let $a \in M$ such that $a \ne \beta$ and $a \ne 0$.

Then:

But then:
 * $\map \phi \beta = \map \phi 0$

and so $\phi$ is not injective.

Hence, by definition, $\phi$ is not a bijection.

This contradicts our assertion that $\phi$ is an isomorphism.

Hence there can be no such semigroup isomorphism between $\struct {M, +}$ and $\struct {\N, +}$.