Positive Rational Numbers under Addition form Ordered Semigroup/Proof 1

Theorem

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

The algebraic structure:

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

forms an ordered semigroup.

Proof

It is necessary to ascertain that $\struct {\Q_{\ge 0}, +, \le}$ fulfils the ordered semigroup axioms:

An ordered semigroup is an algebraic system $\struct {S, \circ, \preceq}$ which satisfies the following properties:

\((\text {OS} 0)\)	$:$		Closure	\(\ds \forall a, b \in S:\)	\(\ds a \circ b \in S \)
\((\text {OS} 1)\)	$:$		Associativity	\(\ds \forall a, b, c \in S:\)	\(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)
\((\text {OS} 2)\)	$:$		Compatibility of $\preceq$ with $\circ$	\(\ds \forall a, b, c \in S:\)	\(\ds a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} \)
			where $\preceq$ is an ordering		\(\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} \)

From Rational Numbers form Totally Ordered Field, $\struct {\Q, +, \times, \le}$ is a totally ordered field.

Hence $\struct {\Q, +, \le}$ is an ordered group, and so an ordered semigroup.

From Positive Rational Numbers are Closed under Addition we have that $\text {OS} 0$ holds.

From Restriction of Associative Operation is Associative we have that $\text {OS} 1$ holds.

From Restriction of Ordering is Ordering we have that $\text {OS} 2$ holds.

The result follows.

$\blacksquare$