# The multiplicative monoid of rational numbers

Content created by Fredrik Bakke and malarbol.

Created on 2024-04-25.

{-# OPTIONS --lossy-unification #-}

module elementary-number-theory.multiplicative-monoid-of-rational-numbers where

Imports
open import elementary-number-theory.multiplication-rational-numbers
open import elementary-number-theory.rational-numbers

open import foundation.dependent-pair-types
open import foundation.unital-binary-operations
open import foundation.universe-levels

open import group-theory.commutative-monoids
open import group-theory.monoids
open import group-theory.semigroups


## Idea

The type of rational numbers equipped with multiplication is a commutative monoid with unit one-ℚ.

## Definitions

### The multiplicative monoid of rational numbers

semigroup-mul-ℚ : Semigroup lzero
pr1 semigroup-mul-ℚ = ℚ-Set
pr1 (pr2 semigroup-mul-ℚ) = mul-ℚ
pr2 (pr2 semigroup-mul-ℚ) = associative-mul-ℚ

is-unital-mul-ℚ : is-unital mul-ℚ
pr1 is-unital-mul-ℚ = one-ℚ
pr1 (pr2 is-unital-mul-ℚ) = left-unit-law-mul-ℚ
pr2 (pr2 is-unital-mul-ℚ) = right-unit-law-mul-ℚ

monoid-mul-ℚ : Monoid lzero
pr1 monoid-mul-ℚ = semigroup-mul-ℚ
pr2 monoid-mul-ℚ = is-unital-mul-ℚ


## Properties

### The multiplicative monoid of rational numbers is commutative

commutative-monoid-mul-ℚ : Commutative-Monoid lzero
pr1 commutative-monoid-mul-ℚ = monoid-mul-ℚ
pr2 commutative-monoid-mul-ℚ = commutative-mul-ℚ