# Function monoids

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-03-13.

module group-theory.function-monoids where

Imports
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

open import group-theory.dependent-products-monoids
open import group-theory.monoids
open import group-theory.semigroups


## Idea

Given a monoid M and a type X, the function monoid M^X consists of functions from X to the underlying type of M. The multiplicative operation and the unit are given pointwise.

## Definition

module _
{l1 l2 : Level} (M : Monoid l1) (X : UU l2)
where

function-Monoid : Monoid (l1 ⊔ l2)
function-Monoid = Π-Monoid X (λ _ → M)

semigroup-function-Monoid : Semigroup (l1 ⊔ l2)
semigroup-function-Monoid = semigroup-Π-Monoid X (λ _ → M)

set-function-Monoid : Set (l1 ⊔ l2)
set-function-Monoid = set-Π-Monoid X (λ _ → M)

type-function-Monoid : UU (l1 ⊔ l2)
type-function-Monoid = type-Π-Monoid X (λ _ → M)

mul-function-Monoid :
(f g : type-function-Monoid) → type-function-Monoid
mul-function-Monoid = mul-Π-Monoid X (λ _ → M)

associative-mul-function-Monoid :
(f g h : type-function-Monoid) →
mul-function-Monoid (mul-function-Monoid f g) h ＝
mul-function-Monoid f (mul-function-Monoid g h)
associative-mul-function-Monoid =
associative-mul-Π-Monoid X (λ _ → M)

unit-function-Monoid : type-function-Monoid
unit-function-Monoid = unit-Π-Monoid X (λ _ → M)

left-unit-law-mul-function-Monoid :
(f : type-function-Monoid) →
mul-function-Monoid unit-function-Monoid f ＝ f
left-unit-law-mul-function-Monoid =
left-unit-law-mul-Π-Monoid X (λ _ → M)

right-unit-law-mul-function-Monoid :
(f : type-function-Monoid) →
mul-function-Monoid f unit-function-Monoid ＝ f
right-unit-law-mul-function-Monoid =
right-unit-law-mul-Π-Monoid X (λ _ → M)

is-unital-function-Monoid :
is-unital-Semigroup semigroup-function-Monoid
is-unital-function-Monoid = is-unital-Π-Monoid X (λ _ → M)