Singleton fulfils Naturally Ordered Semigroup Axioms 1 to 3

Theorem

Let $S$ be a singleton:

$S = \set s$

for an arbitrary object $s$.

Let $+$ be the operation on $S$ defined as:

$\forall s \in S: s + s = s$

Let $\le$ be the relation defined on $S$ as:

$s \le s$

Then the algebraic structure:

$\struct {S, +, \le}$

is an ordered semigroup which fulfils the axioms:

Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered

Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability

Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product

but:

does not fulfil Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements

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

Proof

Recall the axioms:

A naturally ordered semigroup is a (totally) ordered commutative semigroup $\struct {S, \circ, \preceq}$ satisfying:

\((\text {NO} 1)\)	$:$		$S$ is well-ordered by $\preceq$	\(\ds \forall T \subseteq S:\)	\(\ds T = \O \lor \exists m \in T: \forall n \in T: m \preceq n \)
\((\text {NO} 2)\)	$:$		$\circ$ is cancellable in $S$	\(\ds \forall m, n, p \in S:\)	\(\ds m \circ p = n \circ p \implies m = n \)
					\(\ds p \circ m = p \circ n \implies m = n \)
\((\text {NO} 3)\)	$:$		Existence of product	\(\ds \forall m, n \in S:\)	\(\ds m \preceq n \implies \exists p \in S: m \circ p = n \)
\((\text {NO} 4)\)	$:$		$S$ has at least two distinct elements	\(\ds \exists m, n \in S:\)	\(\ds m \ne n \)

Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered

First it is noted that $\le$ is a reflexive relation.

Hence from Reflexive Relation on Singleton is Well-Ordering:

$\le$ is a well-ordering.

Hence Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered holds.

$\Box$

Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability

Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability holds for $\struct {S, +}$ trivally.

$\Box$

Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product

We have that:

$0 \in M$

and:

$2 \in M$

and trivially Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product holds.

$\Box$

Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements

We have that $S$ is a singleton.

Hence:

$\forall a, b \in S: a = b$

and so Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements does not hold.

$\Box$

Lack of Isomorphism

It remains to demonstrate that $S$ and $\N$ are not isomorphic.

Aiming for a contradiction, suppose there exists a (semigroup) isomorphism $\phi$ from $\struct {\N, +}$ to $\struct {S, +}$.

By definition of isomorphism:

$\phi$ is a homomorphism

$\phi$ is a bijection.

From Mapping is Constant iff Image is Singleton:

$\forall n \in \N: \map \phi n = s$

Thus for example:

$\map \phi 0 = s$

and:

$\map \phi 1 = s$

and it is immediate that $\phi$ is not an injection.

Hence $\phi$ is not a bijection.

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

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

$\blacksquare$