Definition:Multiplicative Group of Positive Rational Numbers

Definition

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

Also see

• Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group

Thus rational multiplication is:

Well-defined on $\Q_{> 0}$

Closed on $\Q_{> 0}$

Associative on $\Q_{> 0}$

Commutative on $\Q_{> 0}$

The identity of $\struct {\Q_{> 0}, \times}$ is $1$

Each element of $\struct {\Q_{> 0}, \times}$ has an inverse.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse: 2.