Function semigroups

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-03-13.
Last modified on 2023-03-21.

module group-theory.function-semigroups where
Imports 
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

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

Idea

Given a semigroup G and a type X, the function semigroup G^X consists of functions from X to the underlying type of G. The semigroup operation is given pointwise.

Definition

module _
  {l1 l2 : Level} (G : Semigroup l1) (X : UU l2)
  where

  function-Semigroup : Semigroup (l1 ⊔ l2)
  function-Semigroup = Π-Semigroup X (λ _ → G)

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

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

  mul-function-Semigroup :
    (f g : type-function-Semigroup) → type-function-Semigroup
  mul-function-Semigroup = mul-Π-Semigroup X (λ _ → G)

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

  has-associative-mul-function-Semigroup :
    has-associative-mul-Set set-function-Semigroup
  has-associative-mul-function-Semigroup =
    has-associative-mul-Π-Semigroup X (λ _ → G)

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