Powers of elements in monoids

Content created by Egbert Rijke, Fredrik Bakke, Louis Wasserman and Gregor Perčič.

Created on 2023-08-21.
Last modified on 2025-10-12.

module group-theory.powers-of-elements-monoids where
Imports 
open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.multiplication-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.existential-quantification
open import foundation.identity-types
open import foundation.propositions
open import foundation.universe-levels

open import group-theory.homomorphisms-monoids
open import group-theory.monoids

Idea

The power operation on a monoid is the map n x ↦ xⁿ, which is defined by iteratively multiplying x with itself n times.

Definitions

Powers of elements of monoids

module _
  {l : Level} (M : Monoid l)
  where

  power-Monoid : ℕ → type-Monoid M → type-Monoid M
  power-Monoid zero-ℕ x = unit-Monoid M
  power-Monoid (succ-ℕ zero-ℕ) x = x
  power-Monoid (succ-ℕ (succ-ℕ n)) x =
    mul-Monoid M (power-Monoid (succ-ℕ n) x) x

The predicate of being a power of an element of a monoid

We say that an element y is a power of an element x if there exists a number n such that xⁿ ＝ y.

module _
  {l : Level} (M : Monoid l)
  where

  is-power-of-element-prop-Monoid : (x y : type-Monoid M) → Prop l
  is-power-of-element-prop-Monoid x y =
    exists-structure-Prop ℕ (λ n → power-Monoid M n x ＝ y)

  is-power-of-element-Monoid : (x y : type-Monoid M) → UU l
  is-power-of-element-Monoid x y =
    type-Prop (is-power-of-element-prop-Monoid x y)

  is-prop-is-power-of-element-Monoid :
    (x y : type-Monoid M) → is-prop (is-power-of-element-Monoid x y)
  is-prop-is-power-of-element-Monoid x y =
    is-prop-type-Prop (is-power-of-element-prop-Monoid x y)

Properties

1ⁿ ＝ 1

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    power-unit-Monoid :
      (n : ℕ) →
      power-Monoid M n (unit-Monoid M) ＝ unit-Monoid M
    power-unit-Monoid zero-ℕ = refl
    power-unit-Monoid (succ-ℕ zero-ℕ) = refl
    power-unit-Monoid (succ-ℕ (succ-ℕ n)) =
      right-unit-law-mul-Monoid M _ ∙ power-unit-Monoid (succ-ℕ n)

xⁿ⁺¹ = xⁿx

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    power-succ-Monoid :
      (n : ℕ) (x : type-Monoid M) →
      power-Monoid M (succ-ℕ n) x ＝ mul-Monoid M (power-Monoid M n x) x
    power-succ-Monoid zero-ℕ x = inv (left-unit-law-mul-Monoid M x)
    power-succ-Monoid (succ-ℕ n) x = refl

xⁿ⁺¹ ＝ xxⁿ

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    power-succ-Monoid' :
      (n : ℕ) (x : type-Monoid M) →
      power-Monoid M (succ-ℕ n) x ＝ mul-Monoid M x (power-Monoid M n x)
    power-succ-Monoid' zero-ℕ x = inv (right-unit-law-mul-Monoid M x)
    power-succ-Monoid' (succ-ℕ zero-ℕ) x = refl
    power-succ-Monoid' (succ-ℕ (succ-ℕ n)) x =
      ( ap (mul-Monoid' M x) (power-succ-Monoid' (succ-ℕ n) x)) ∙
      ( associative-mul-Monoid M x (power-Monoid M (succ-ℕ n) x) x)

Powers by sums of natural numbers are products of powers

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    distributive-power-add-Monoid :
      (m n : ℕ) {x : type-Monoid M} →
      power-Monoid M (m +ℕ n) x ＝
      mul-Monoid M (power-Monoid M m x) (power-Monoid M n x)
    distributive-power-add-Monoid m zero-ℕ {x} =
      inv
        ( right-unit-law-mul-Monoid M
          ( power-Monoid M m x))
    distributive-power-add-Monoid m (succ-ℕ n) {x} =
      ( power-succ-Monoid M (m +ℕ n) x) ∙
      ( ap (mul-Monoid' M x) (distributive-power-add-Monoid m n)) ∙
      ( associative-mul-Monoid M
        ( power-Monoid M m x)
        ( power-Monoid M n x)
        ( x)) ∙
      ( ap
        ( mul-Monoid M (power-Monoid M m x))
        ( inv (power-succ-Monoid M n x)))

If x commutes with y then so do their powers

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    commute-powers-Monoid' :
      (n : ℕ) {x y : type-Monoid M} →
      ( mul-Monoid M x y ＝ mul-Monoid M y x) →
      ( mul-Monoid M (power-Monoid M n x) y) ＝
      ( mul-Monoid M y (power-Monoid M n x))
    commute-powers-Monoid' zero-ℕ H =
      left-unit-law-mul-Monoid M _ ∙ inv (right-unit-law-mul-Monoid M _)
    commute-powers-Monoid' (succ-ℕ zero-ℕ) {x} {y} H = H
    commute-powers-Monoid' (succ-ℕ (succ-ℕ n)) {x} {y} H =
      ( associative-mul-Monoid M (power-Monoid M (succ-ℕ n) x) x y) ∙
      ( ap (mul-Monoid M (power-Monoid M (succ-ℕ n) x)) H) ∙
      ( inv (associative-mul-Monoid M (power-Monoid M (succ-ℕ n) x) y x)) ∙
      ( ap (mul-Monoid' M x) (commute-powers-Monoid' (succ-ℕ n) H)) ∙
      ( associative-mul-Monoid M y (power-Monoid M (succ-ℕ n) x) x)

    commute-powers-Monoid :
      (m n : ℕ) {x y : type-Monoid M} →
      ( mul-Monoid M x y ＝ mul-Monoid M y x) →
      ( mul-Monoid M
        ( power-Monoid M m x)
        ( power-Monoid M n y)) ＝
      ( mul-Monoid M
        ( power-Monoid M n y)
        ( power-Monoid M m x))
    commute-powers-Monoid zero-ℕ zero-ℕ H = refl
    commute-powers-Monoid zero-ℕ (succ-ℕ n) H =
      ( left-unit-law-mul-Monoid M (power-Monoid M (succ-ℕ n) _)) ∙
      ( inv (right-unit-law-mul-Monoid M (power-Monoid M (succ-ℕ n) _)))
    commute-powers-Monoid (succ-ℕ m) zero-ℕ H =
      ( right-unit-law-mul-Monoid M (power-Monoid M (succ-ℕ m) _)) ∙
      ( inv (left-unit-law-mul-Monoid M (power-Monoid M (succ-ℕ m) _)))
    commute-powers-Monoid (succ-ℕ m) (succ-ℕ n) {x} {y} H =
      ( ap-mul-Monoid M (power-succ-Monoid M m x) (power-succ-Monoid M n y)) ∙
      ( associative-mul-Monoid M
        ( power-Monoid M m x)
        ( x)
        ( mul-Monoid M (power-Monoid M n y) y)) ∙
      ( ap
        ( mul-Monoid M (power-Monoid M m x))
        ( ( inv (associative-mul-Monoid M x (power-Monoid M n y) y)) ∙
          ( ap
            ( mul-Monoid' M y)
            ( inv (commute-powers-Monoid' n (inv H)))) ∙
          ( associative-mul-Monoid M (power-Monoid M n y) x y) ∙
          ( ap (mul-Monoid M (power-Monoid M n y)) H) ∙
          ( inv (associative-mul-Monoid M (power-Monoid M n y) y x)))) ∙
      ( inv
        ( associative-mul-Monoid M
          ( power-Monoid M m x)
          ( mul-Monoid M (power-Monoid M n y) y)
          ( x))) ∙
      ( ap
        ( mul-Monoid' M x)
        ( ( inv
            ( associative-mul-Monoid M
              ( power-Monoid M m x)
              ( power-Monoid M n y)
              ( y))) ∙
          ( ap
            ( mul-Monoid' M y)
            ( commute-powers-Monoid m n H)) ∙
          ( associative-mul-Monoid M
            ( power-Monoid M n y)
            ( power-Monoid M m x)
            ( y)) ∙
          ( ap
            ( mul-Monoid M (power-Monoid M n y))
            ( commute-powers-Monoid' m H)) ∙
          ( inv
            ( associative-mul-Monoid M
              ( power-Monoid M n y)
              ( y)
              ( power-Monoid M m x))) ∙
          ( ap
            ( mul-Monoid' M (power-Monoid M m x))
            ( inv (power-succ-Monoid M n y))))) ∙
        ( associative-mul-Monoid M
          ( power-Monoid M (succ-ℕ n) y)
          ( power-Monoid M m x)
          ( x)) ∙
        ( ap
          ( mul-Monoid M (power-Monoid M (succ-ℕ n) y))
          ( inv (power-succ-Monoid M m x)))

If x commutes with y, then powers distribute over the product of x and y

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    distributive-power-mul-Monoid :
      (n : ℕ) {x y : type-Monoid M} →
      (H : mul-Monoid M x y ＝ mul-Monoid M y x) →
      power-Monoid M n (mul-Monoid M x y) ＝
      mul-Monoid M (power-Monoid M n x) (power-Monoid M n y)
    distributive-power-mul-Monoid zero-ℕ H =
      inv (left-unit-law-mul-Monoid M (unit-Monoid M))
    distributive-power-mul-Monoid (succ-ℕ n) {x} {y} H =
      ( power-succ-Monoid M n (mul-Monoid M x y)) ∙
      ( ap
        ( mul-Monoid' M (mul-Monoid M x y))
        ( distributive-power-mul-Monoid n H)) ∙
      ( inv
        ( associative-mul-Monoid M
          ( mul-Monoid M (power-Monoid M n x) (power-Monoid M n y))
          ( x)
          ( y))) ∙
      ( ap
        ( mul-Monoid' M y)
        ( ( associative-mul-Monoid M
            ( power-Monoid M n x)
            ( power-Monoid M n y)
            ( x)) ∙
          ( ap
            ( mul-Monoid M (power-Monoid M n x))
            ( commute-powers-Monoid' M n (inv H))) ∙
          ( inv
            ( associative-mul-Monoid M
              ( power-Monoid M n x)
              ( x)
              ( power-Monoid M n y))))) ∙
      ( associative-mul-Monoid M
        ( mul-Monoid M (power-Monoid M n x) x)
        ( power-Monoid M n y)
        ( y)) ∙
      ( ap-mul-Monoid M
        ( inv (power-succ-Monoid M n x))
        ( inv (power-succ-Monoid M n y)))

Powers by products of natural numbers are iterated powers

module _
  {l : Level} (M : Monoid l)
  where

  abstract
    power-mul-Monoid :
      (m n : ℕ) {x : type-Monoid M} →
      power-Monoid M (m *ℕ n) x ＝
      power-Monoid M n (power-Monoid M m x)
    power-mul-Monoid zero-ℕ n {x} =
      inv (power-unit-Monoid M n)
    power-mul-Monoid (succ-ℕ zero-ℕ) n {x} =
      ap (λ t → power-Monoid M t x) (left-unit-law-add-ℕ n)
    power-mul-Monoid (succ-ℕ (succ-ℕ m)) n {x} =
      ( distributive-power-add-Monoid M (succ-ℕ m *ℕ n) n) ∙
      ( ap
        ( mul-Monoid' M (power-Monoid M n x))
        ( power-mul-Monoid (succ-ℕ m) n)) ∙
      ( inv
        ( distributive-power-mul-Monoid M n
          ( commute-powers-Monoid' M (succ-ℕ m) refl)))

Monoid homomorphisms preserve powers

module _
  {l1 l2 : Level} (M : Monoid l1) (N : Monoid l2) (f : hom-Monoid M N)
  where

  preserves-powers-hom-Monoid :
    (n : ℕ) (x : type-Monoid M) →
    map-hom-Monoid M N f (power-Monoid M n x) ＝
    power-Monoid N n (map-hom-Monoid M N f x)
  preserves-powers-hom-Monoid zero-ℕ x = preserves-unit-hom-Monoid M N f
  preserves-powers-hom-Monoid (succ-ℕ zero-ℕ) x = refl
  preserves-powers-hom-Monoid (succ-ℕ (succ-ℕ n)) x =
    ( preserves-mul-hom-Monoid M N f) ∙
    ( ap (mul-Monoid' N _) (preserves-powers-hom-Monoid (succ-ℕ n) x))

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