The multiplicative group of positive rational numbers
Content created by malarbol and Fredrik Bakke.
Created on 2024-04-25.
Last modified on 2025-04-24.
{-# OPTIONS --lossy-unification #-} module elementary-number-theory.multiplicative-group-of-positive-rational-numbers where
Imports
open import elementary-number-theory.inequality-integers open import elementary-number-theory.multiplication-integers open import elementary-number-theory.multiplication-rational-numbers open import elementary-number-theory.multiplicative-monoid-of-rational-numbers open import elementary-number-theory.positive-rational-numbers open import elementary-number-theory.rational-numbers open import elementary-number-theory.strict-inequality-integers open import foundation.binary-transport open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.universe-levels open import group-theory.abelian-groups open import group-theory.groups open import group-theory.monoids open import group-theory.submonoids
Idea
The submonoid of positive rational numbers in the multiplicative monoid of rational numbers is a commutative group.
Definitions
The positive inverse of a positive rational number
inv-ℚ⁺ : ℚ⁺ → ℚ⁺ pr1 (inv-ℚ⁺ (x , P)) = inv-is-positive-ℚ x P pr2 (inv-ℚ⁺ (x , P)) = is-positive-denominator-ℚ x
The multiplicative group of positive rational numbers
group-mul-ℚ⁺ : Group lzero pr1 group-mul-ℚ⁺ = semigroup-mul-ℚ⁺ pr1 (pr2 group-mul-ℚ⁺) = is-unital-Monoid monoid-mul-ℚ⁺ pr1 (pr2 (pr2 group-mul-ℚ⁺)) = inv-ℚ⁺ pr1 (pr2 (pr2 (pr2 group-mul-ℚ⁺))) x = eq-ℚ⁺ ( left-inverse-law-mul-is-positive-ℚ ( rational-ℚ⁺ x) ( is-positive-rational-ℚ⁺ x)) pr2 (pr2 (pr2 (pr2 group-mul-ℚ⁺))) x = eq-ℚ⁺ ( right-inverse-law-mul-is-positive-ℚ ( rational-ℚ⁺ x) ( is-positive-rational-ℚ⁺ x))
Inverse laws in the multiplicative group of positive rational numbers
left-inverse-law-mul-ℚ⁺ : (x : ℚ⁺) → (inv-ℚ⁺ x) *ℚ⁺ x = one-ℚ⁺ left-inverse-law-mul-ℚ⁺ = left-inverse-law-mul-Group group-mul-ℚ⁺ right-inverse-law-mul-ℚ⁺ : (x : ℚ⁺) → x *ℚ⁺ (inv-ℚ⁺ x) = one-ℚ⁺ right-inverse-law-mul-ℚ⁺ = right-inverse-law-mul-Group group-mul-ℚ⁺
Properties
The multiplicative group of positive rational numbers is commutative
abelian-group-mul-ℚ⁺ : Ab lzero pr1 abelian-group-mul-ℚ⁺ = group-mul-ℚ⁺ pr2 abelian-group-mul-ℚ⁺ = commutative-mul-ℚ⁺
Inversion reverses inequality on the positive rational numbers
abstract inv-leq-ℚ⁺ : (x y : ℚ⁺) → leq-ℚ⁺ (inv-ℚ⁺ x) (inv-ℚ⁺ y) → leq-ℚ⁺ y x inv-leq-ℚ⁺ x y = binary-tr ( leq-ℤ) ( commutative-mul-ℤ ( denominator-ℚ⁺ x) ( numerator-ℚ⁺ y)) ( commutative-mul-ℤ ( denominator-ℚ⁺ y) ( numerator-ℚ⁺ x))
Inversion reverses strict inequality on the positive rational numbers
abstract inv-le-ℚ⁺ : (x y : ℚ⁺) → le-ℚ⁺ (inv-ℚ⁺ x) (inv-ℚ⁺ y) → le-ℚ⁺ y x inv-le-ℚ⁺ x y = binary-tr ( le-ℤ) ( commutative-mul-ℤ ( denominator-ℚ⁺ x) ( numerator-ℚ⁺ y)) ( commutative-mul-ℤ ( denominator-ℚ⁺ y) ( numerator-ℚ⁺ x)) inv-le-ℚ⁺' : (x y : ℚ⁺) → le-ℚ⁺ x y → le-ℚ⁺ (inv-ℚ⁺ y) (inv-ℚ⁺ x) inv-le-ℚ⁺' x y = binary-tr ( le-ℤ) ( inv ( commutative-mul-ℤ ( denominator-ℚ⁺ y) ( numerator-ℚ⁺ x))) ( inv ( commutative-mul-ℤ ( denominator-ℚ⁺ x) ( numerator-ℚ⁺ y)))
Inversion of positive rational numbers distributes over multiplication
abstract distributive-inv-mul-ℚ⁺ : (x y : ℚ⁺) → inv-ℚ⁺ (x *ℚ⁺ y) = inv-ℚ⁺ x *ℚ⁺ inv-ℚ⁺ y distributive-inv-mul-ℚ⁺ x y = distributive-inv-mul-Group' ( group-mul-ℚ⁺) ( x) ( y) ( commutative-mul-ℚ⁺ x y)
Recent changes
- 2025-04-24. malarbol. Bernoulli’s inequality on the positive rational numbers (#1371).
- 2024-04-25. malarbol and Fredrik Bakke. The discrete field of rational numbers (#1111).