Powers of elements in commutative semirings

Content created by Fredrik Bakke, Jonathan Prieto-Cubides, Egbert Rijke and Maša Žaucer.

Created on 2023-02-20.
Last modified on 2023-08-21.

module commutative-algebra.powers-of-elements-commutative-semirings where
Imports 
open import commutative-algebra.commutative-semirings

open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.natural-numbers

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

open import ring-theory.powers-of-elements-semirings

Idea

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

Definition

power-Commutative-Semiring :
  {l : Level} (A : Commutative-Semiring l) →
  ℕ → type-Commutative-Semiring A → type-Commutative-Semiring A
power-Commutative-Semiring A = power-Semiring (semiring-Commutative-Semiring A)

Properties

xⁿ⁺¹ = xⁿx

module _
  {l : Level} (A : Commutative-Semiring l)
  where

  power-succ-Commutative-Semiring :
    (n : ℕ) (x : type-Commutative-Semiring A) →
    power-Commutative-Semiring A (succ-ℕ n) x ＝
    mul-Commutative-Semiring A (power-Commutative-Semiring A n x) x
  power-succ-Commutative-Semiring =
    power-succ-Semiring (semiring-Commutative-Semiring A)

Powers by sums of natural numbers are products of powers

module _
  {l : Level} (A : Commutative-Semiring l)
  where

  distributive-power-add-Commutative-Semiring :
    (m n : ℕ) {x : type-Commutative-Semiring A} →
    power-Commutative-Semiring A (m +ℕ n) x ＝
    mul-Commutative-Semiring A
      ( power-Commutative-Semiring A m x)
      ( power-Commutative-Semiring A n x)
  distributive-power-add-Commutative-Semiring =
    distributive-power-add-Semiring (semiring-Commutative-Semiring A)

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

module _
  {l : Level} (A : Commutative-Semiring l)
  where

  distributive-power-mul-Commutative-Semiring :
    (n : ℕ) (x y : type-Commutative-Semiring A) →
    power-Commutative-Semiring A n (mul-Commutative-Semiring A x y) ＝
    mul-Commutative-Semiring A
      ( power-Commutative-Semiring A n x)
      ( power-Commutative-Semiring A n y)
  distributive-power-mul-Commutative-Semiring n x y =
    distributive-power-mul-Semiring
      ( semiring-Commutative-Semiring A)
      ( n)
      ( commutative-mul-Commutative-Semiring A x y)

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