Positive Rational Numbers under Addition form Commutative Monoid
Theorem
Let $\Q_{\ge 0}$ denote the set of positive rational numbers.
The algebraic structure:
- $\struct {\Q_{\ge 0}, +}$
forms a commutative monoid.
Proof
From Rational Numbers form Field, $\struct {\Q, +, \times}$ is a field.
Hence $\struct {\Q, +}$ is an abelian group.
From Positive Rational Numbers are Closed under Addition we have that $\struct {\Q_{\ge 0}, +}$ is closed.
Hence from Subsemigroup Closure Test, $\struct {\Q_{\ge 0}, +}$ is a subsemigroup of $\struct {\Q, +}$.
From Restriction of Commutative Operation is Commutative, $\struct {\Q_{\ge 0}, +}$ is a commutative semigroup.
We have that $0$ is the identity element of $\struct {\Q, +}$.
Hence from Identity of Subsemigroup of Group, $0$ is also the identity element of $\struct {\Q_{\ge 0}, +}$.
So $\struct {\Q_{\ge 0}, +}$ is a commutative monoid.
$\blacksquare$