Classical.Structures.Group.AbelianGroup

Abelian Groups

This is the Classical.Structures.Group.AbelianGroup module of the Agda Universal Algebra Library.

An abelian group is an inhabitant of the type Σ[ 𝑨 ∈ Algebra α ρ ] 𝑨 ⊨ Th-AbelianGroup, where Algebra is parameterized by the signature type Sig-Group.

This is an equation-only extension of Group, structurally identical to the way CommutativeMonoid extends Monoid; that is, abelianGroup→group is a pure theory-reindex (proj₁ on the underlying algebra), and AbelianGroup-Op inherits _∙_, ε, _⁻¹, and all five group laws through it, adding comm-law.

{-# OPTIONS --without-K --exact-split --safe #-}

module Classical.Structures.Group.AbelianGroup where

open import Agda.Primitive                         using () renaming ( Set to Type )

-- Imports from the Agda Standard Library ---------------------------------------
open import Data.Fin.Base                          using ( Fin )
open import Data.Fin.Patterns                      using ( 0F ; 1F ; 2F )
open import Data.Product                           using ( Σ-syntax ; _×_ ; _,_ ; proj₁ ; proj₂ )
open import Level                                  using ( Level ; _⊔_ ; suc )
open import Relation.Binary                        using ( Setoid )
open import Relation.Binary.PropositionalEquality  using ( _≡_ )

-- Imports from the Agda Universal Algebra Library ------------------------------
open import Classical.Signatures.Group             using ( Sig-Group )
open import Classical.Structures.Group.Basic       using ( Group ; module Group-Op ; opsToBareGroup )
open import Classical.Theories.Group               using ( assoc ; idˡ ; idʳ ; invˡ ; invʳ )
open import Classical.Theories.AbelianGroup        using ( Eq-AbelianGroup ; Th-AbelianGroup ; comm )
                                                   renaming ( assoc to assocᵃ ; idˡ to idˡᵃ ; idʳ to idʳᵃ
                                                            ; invˡ to invˡᵃ ; invʳ to invʳᵃ )
open import Overture.Terms                         using ( Term ; ℊ )
open import Setoid.Algebras.Basic                  using ( Algebra ; 𝔻[_] ; 𝕌[_] )
open import Setoid.Varieties.EquationalLogic using ( _⊧_≈_ )

private variable α ρ : Level

Satisfaction predicate and the AbelianGroup type

infix  _⊨ᵃᵍ_
_⊨ᵃᵍ_ : (𝑨 : Algebra {𝑆 = Sig-Group} α ρ) (ℰ : Eq-AbelianGroup → Term (Fin ) × Term (Fin )) → Type (α ⊔ ρ)
𝑨 ⊨ᵃᵍ ℰ = ∀ i → 𝑨 ⊧ proj₁ (ℰ i) ≈ proj₂ (ℰ i)

AbelianGroup : (α ρ : Level) → Type (suc α ⊔ suc ρ)
AbelianGroup α ρ = Σ[ 𝑨 ∈ Algebra α ρ ] 𝑨 ⊨ᵃᵍ Th-AbelianGroup

The forgetful projection to groups

abelianGroup→group discards commutativity. Since an abelian group is a group with one extra equation and no extra operations, the signature is unchanged and no reduct is needed; the function keeps the algebra and re-indexes the satisfaction witness, mapping each constructor of Eq-Group to its counterpart in Eq-AbelianGroup. (This is the shape forgetful shape as that of commutativeMonoid→monoid, in contrast with the reduct-and-re-prove shape of monoid→semigroup.)

abelianGroup→group : AbelianGroup α ρ → Group α ρ
abelianGroup→group (𝑨 , mod) = 𝑨 , λ  { assoc → mod assocᵃ
                                      ; idˡ   → mod idˡᵃ
                                      ; idʳ   → mod idʳᵃ
                                      ; invˡ  → mod invˡᵃ
                                      ; invʳ  → mod invʳᵃ }

The AbelianGroup-Op module

AbelianGroup-Op𝑨 inherits the whole group interface through the forgetful projection (three operations, both congruences, all three containment lemmas and all five laws) and adds two names: equations, the new satisfaction witness, and comm-law, commutativity in curried form.

module AbelianGroup-Op {α ρ : Level} ((𝑨 , laws) : AbelianGroup α ρ) where
  open Setoid 𝔻[ 𝑨 ]

  open Group-Op (abelianGroup→group (𝑨 , laws)) public
    using  ( _∙_ ; ε ; _⁻¹ ; ∙-cong ; ⁻¹-cong ; interp-node-∙ ; interp-node-ε
           ; interp-node-⁻¹ ; assoc-law ; idˡ-law ; idʳ-law ; invˡ-law ; invʳ-law )

  comm-law : ∀ x y → x ∙ y ≈ y ∙ x
  comm-law x y = trans (sym (interp-node-∙ (ℊ 0F) (ℊ 1F) {η}))
                       (trans (laws comm η) (interp-node-∙ (ℊ 1F) (ℊ 0F) {η}))
               -- the same three-step shape every added law in the hierarchy has:
               -- `laws comm` between two applications of `interp-node-∙`, one
               -- for each side of the equation.
    where
    η : Fin  → 𝕌[ 𝑨 ]
    η = λ { 0F → x ; 1F → y ; 2F → x }

eqsToAbelianGroup

eqsToAbelianGroup is the constructor a downstream user calls: a carrier, a binary operation, an identity, an inverse, and the six laws as propositional equations. Every obligation discharges by definitional reduction, since under ≡.setoid A the setoid equality is propositional equality and each interpreted term reduces to the corresponding application of the supplied operations.

eqsToAbelianGroup : {A : Type α} (_·_ : A → A → A) (e : A) (i : A → A)
  → (·-assoc : ∀ a b c → (a · b) · c ≡ a · (b · c))
  → (·-idˡ : ∀ a → e · a ≡ a) (·-idʳ : ∀ a → a · e ≡ a)
  → (·-invˡ : ∀ a → (i a) · a ≡ e) (·-invʳ : ∀ a → a · (i a) ≡ e)
  → (·-comm : ∀ a b → a · b ≡ b · a)
  → AbelianGroup α α
eqsToAbelianGroup _·_ e i ·-assoc ·-idˡ ·-idʳ ·-invˡ ·-invʳ ·-comm = opsToBareGroup _·_ e i , proof
  where
  proof : opsToBareGroup _·_ e i ⊨ᵃᵍ Th-AbelianGroup
  proof assocᵃ ρ = ·-assoc (ρ 0F) (ρ 1F) (ρ 2F)
  proof idˡᵃ   ρ = ·-idˡ   (ρ 0F)
  proof idʳᵃ   ρ = ·-idʳ   (ρ 0F)
  proof invˡᵃ  ρ = ·-invˡ  (ρ 0F)
  proof invʳᵃ  ρ = ·-invʳ  (ρ 0F)
  proof comm   ρ = ·-comm  (ρ 0F) (ρ 1F)