Positive Rational Numbers under Addition fulfil Naturally Ordered Semigroup Axioms 2 to 4

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Theorem

Let $\Q_{\ge 0}$ denote the set of positive rational numbers.

Consider the naturally ordered semigroup 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      \(\ds \exists m, n \in S:\) \(\ds m \ne n \)      


The algebraic structure:

$\struct {\Q_{\ge 0}, +, \le}$

is an ordered semigroup which fulfils the axioms:

Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability
Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product
Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements

but:

does not fulfil Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered
$\struct {\Q_{\ge 0}, +}$is not isomorphic to $\struct {\N, +}$.


Proof

First we note that from Positive Rational Numbers under Addition form Ordered Semigroup:

$\struct {\Q_{\ge 0}, +, \le}$ is an ordered semigroup.


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

From Rational Numbers under Addition form Infinite Abelian Group, $\struct {\Q, +}$ is a group.

By the Cancellation Laws, all elements of $\struct {\Q, +}$ are cancellable.

From Cancellable Element is Cancellable in Subset, it follows that all elements of $\struct {\Q_{\ge 0}, +}$ are likewise cancellable.

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

$\Box$


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

Let $a, b \in \Q_{\ge 0}$ such that $a \le b$.

Then as $\struct {\Q, +}$ is a group it follows that:

$\exists c \in \Q: a + c = b$

and so:

$c = b + \paren {-a}$

But as $a \le b$ it follows that $c \ge 0$.

That is:

$\exists c \in \Q_{\ge 0}: a + c = b$

Hence 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:

$0 \in \Q_{\ge 0}$

and:

$1 \in \Q_{\ge 0}$

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

$\Box$


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

We consider the subset $S$ of $\struct {\Q_{\ge 0}, +, \le}$ defined as:

$S = \set {x \in \Q_{\ge 0}: x > 0}$

From Smallest Strictly Positive Rational Number does not Exist:

there exists no smallest element of $S$.

Hence by definition $\struct {\Q_{\ge 0}, +, \le}$ is not well-ordered by $\le$.

That is, $\struct {\Q_{\ge 0}, +, \le}$ does not fulfil Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered.

$\Box$


From Positive Rational Numbers under Addition not Isomorphic to Natural Numbers:

$\struct {\Q_{\ge 0}, +}$is not isomorphic to $\struct {\N, +}$.

$\blacksquare$


Sources

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