Definition:Multiplicative Group of Positive Rational Numbers

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The multiplicative group of positive rational numbers $\left({\Q_{> 0}, \times}\right)$ is the set of (strictly) positive rational numbers under the operation of multiplication.

Also see

Thus rational multiplication is:

Well-defined on $\Q_{> 0}$
Closed on $\Q_{> 0}$
Associative on $\Q_{> 0}$
Commutative on $\Q_{> 0}$
The identity of $\left({\Q_{> 0}, \times}\right)$ is $1$
Each element of $\left({\Q_{> 0}, \times}\right)$ has an inverse.