[ draft ] Alternative design for sub- and quotient- (Abelian)Groups

CHANGELOG.md

@@ -79,6 +79,16 @@ Deprecated names
 New modules
 -----------
 
+* `Algebra.Construct.Quotient.{{Abelian}Group|Ring}` for the definition of quotient (Abelian) groups and rings.
+
+* `Algebra.Construct.Sub.{Abelian}Group` for the definition of sub-(Abelian)groups.
+
+* `Algebra.Construct.Sub.Group.Normal` for the definition of normal subgroups.
+
+* `Algebra.Construct.Sub.Ring.Ideal` for the definition of ideals of a ring.
+
+* `Algebra.Module.Construct.Sub.Bimodule` for the definition of sub-bimodules.
+
 * `Algebra.Properties.BooleanRing`.
 
 * `Algebra.Properties.BooleanSemiring`.

src/Algebra/Construct/Quotient/AbelianGroup.agda

@@ -0,0 +1,39 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Quotient Abelian groups
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+open import Algebra.Bundles using (Group; AbelianGroup)
+import Algebra.Construct.Sub.AbelianGroup as AbelianSubgroup
+import Algebra.Construct.Quotient.Group as Quotient
+
+module Algebra.Construct.Quotient.AbelianGroup
+  {c ℓ} (G : AbelianGroup c ℓ)
+  (open AbelianSubgroup G using (Subgroup; normalSubgroup))
+  {c′ ℓ′} (N : Subgroup c′ ℓ′)
+  where
+
+private
+  module G = AbelianGroup G
+
+-- Re-export the quotient group
+
+open Quotient G.group (normalSubgroup N) public
+  hiding (_/_)
+
+-- With its additional bundle
+
+abelianGroup : AbelianGroup c _
+abelianGroup = record
+  { isAbelianGroup = record
+    { isGroup = isGroup
+    ; comm = λ g h → ≈⇒≋ (G.comm g h)
+    }
+  } where open Group group
+
+-- Public re-exports
+
+_/_ = abelianGroup

src/Algebra/Construct/Quotient/Group.agda

@@ -0,0 +1,128 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Quotient groups
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+open import Algebra.Bundles using (Group)
+open import Algebra.Construct.Sub.Group.Normal using (NormalSubgroup)
+
+module Algebra.Construct.Quotient.Group
+  {c ℓ} (G : Group c ℓ) {c′ ℓ′} (N : NormalSubgroup G c′ ℓ′) where
+
+open import Algebra.Definitions using (Congruent₁; Congruent₂)
+open import Algebra.Morphism.Structures using (IsGroupHomomorphism)
+open import Data.Product.Base using (_,_)
+open import Function.Base using (_∘_)
+open import Function.Definitions using (Surjective)
+open import Level using (_⊔_)
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
+
+private
+  open module G = Group G
+    using (_≈_; _∙_; ε; _⁻¹)
+  open module N = NormalSubgroup N
+    using (ι; module ι; conjugate; normal)
+
+open import Algebra.Properties.Group G using (⁻¹-anti-homo-∙)
+open import Algebra.Properties.Monoid G.monoid
+open import Relation.Binary.Reasoning.Setoid G.setoid
+
+infix 0 _by_
+
+data _≋_ (x y : G.Carrier) : Set (c ⊔ ℓ ⊔ c′) where
+  _by_ : ∀ g → ι g ∙ x ≈ y → x ≋ y
+
+≈⇒≋ : _≈_ ⇒ _≋_
+≈⇒≋ x≈y = N.ε by G.trans (G.∙-cong ι.ε-homo x≈y) (G.identityˡ _)
+
+refl : Reflexive _≋_
+refl = ≈⇒≋ G.refl
+
+sym : Symmetric _≋_
+sym {x} {y} (g by ιg∙x≈y) = g N.⁻¹ by begin
+  ι (g N.⁻¹) ∙ y      ≈⟨ G.∙-cong (ι.⁻¹-homo g) (G.sym ιg∙x≈y) ⟩
+  ι g ⁻¹ ∙ (ι g ∙ x)  ≈⟨ cancelˡ (G.inverseˡ (ι g)) x ⟩
+  x                   ∎
+
+trans : Transitive _≋_
+trans {x} {y} {z} (g by ιg∙x≈y) (h by ιh∙y≈z) = h N.∙ g by begin
+  ι (h N.∙ g) ∙ x ≈⟨ G.∙-congʳ (ι.∙-homo h g) ⟩
+  (ι h ∙ ι g) ∙ x ≈⟨ uv≈w⇒xu∙v≈xw ιg∙x≈y (ι h) ⟩
+  ι h ∙ y         ≈⟨ ιh∙y≈z ⟩
+  z               ∎
+
+∙-cong : Congruent₂ _≋_ _∙_
+∙-cong {x} {y} {u} {v} (g by ιg∙x≈y) (h by ιh∙u≈v) = g N.∙ h′ by begin
+  ι (g N.∙ h′) ∙ (x ∙ u) ≈⟨ G.∙-congʳ (ι.∙-homo g h′) ⟩
+  (ι g ∙ ι h′) ∙ (x ∙ u) ≈⟨ uv≈wx⇒yu∙vz≈yw∙xz (normal h x) (ι g) u ⟩
+  (ι g ∙ x) ∙ (ι h ∙ u)  ≈⟨ G.∙-cong ιg∙x≈y ιh∙u≈v ⟩
+  y ∙ v                  ∎
+  where h′ = conjugate h x
+
+⁻¹-cong : Congruent₁ _≋_ _⁻¹
+⁻¹-cong {x} {y} (g by ιg∙x≈y) = h by begin
+  ι h ∙ x ⁻¹        ≈⟨ normal (g N.⁻¹) (x ⁻¹) ⟩
+  x ⁻¹ ∙ ι (g N.⁻¹) ≈⟨ G.∙-congˡ (ι.⁻¹-homo g) ⟩
+  x ⁻¹ ∙ ι g ⁻¹     ≈⟨ ⁻¹-anti-homo-∙ (ι g) x ⟨
+  (ι g ∙ x) ⁻¹      ≈⟨ G.⁻¹-cong ιg∙x≈y ⟩
+  y ⁻¹              ∎
+  where h = conjugate (g N.⁻¹) (x ⁻¹)
+
+group : Group c (c ⊔ ℓ ⊔ c′)
+group = record
+  { isGroup = record
+    { isMonoid = record
+      { isSemigroup = record
+        { isMagma = record
+          { isEquivalence = record
+            { refl = refl
+            ; sym = sym
+            ; trans = trans
+            }
+          ; ∙-cong = ∙-cong
+          }
+        ; assoc = λ x y z → ≈⇒≋ (G.assoc x y z)
+        }
+      ; identity = ≈⇒≋ ∘ G.identityˡ , ≈⇒≋ ∘ G.identityʳ
+      }
+    ; inverse = ≈⇒≋ ∘ G.inverseˡ , ≈⇒≋ ∘ G.inverseʳ
+    ; ⁻¹-cong = ⁻¹-cong
+    }
+  }
+
+private module Q = Group group
+
+-- Public re-exports
+
+_/_ : Group c (c ⊔ ℓ ⊔ c′)
+_/_ = group
+
+π : G.Carrier → Q.Carrier
+π x = x -- because we do all the work in the relation _≋_
+
+π-surjective : Surjective _≈_ _≋_ π
+π-surjective g = g , ≈⇒≋
+
+π-isGroupHomomorphism : IsGroupHomomorphism G.rawGroup Q.rawGroup π
+π-isGroupHomomorphism = record
+  { isMonoidHomomorphism = record
+    { isMagmaHomomorphism = record
+      { isRelHomomorphism = record
+        { cong = ≈⇒≋
+        }
+      ; homo = λ _ _ → Q.refl
+      }
+    ; ε-homo = Q.refl
+    }
+  ; ⁻¹-homo = λ _ → Q.refl
+  }
+
+open IsGroupHomomorphism π-isGroupHomomorphism public
+  using ()
+  renaming (isMonoidHomomorphism to π-isMonoidHomomorphism
+           ; isMagmaHomomorphism to π-isMagmaHomomorphism
+           )

src/Algebra/Construct/Quotient/Ring.agda

@@ -0,0 +1,77 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Quotient rings
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+open import Algebra.Bundles using (AbelianGroup; Ring)
+open import Algebra.Construct.Sub.Ring.Ideal using (Ideal)
+import Algebra.Construct.Quotient.AbelianGroup as Quotient
+
+module Algebra.Construct.Quotient.Ring
+  {c ℓ} (R : Ring c ℓ) {c′ ℓ′} (I : Ideal R c′ ℓ′)
+  where
+
+open import Algebra.Definitions using (Congruent₂)
+open import Algebra.Morphism.Structures using (IsRingHomomorphism)
+open import Data.Product.Base using (_,_)
+open import Function.Base using (_∘_)
+open import Level using (_⊔_)
+
+private
+  module R = Ring R
+  module I = Ideal I
+  module R/I = Quotient R.+-abelianGroup I.subgroup
+
+
+open R/I public
+  using (_≋_; _by_; ≈⇒≋; π; π-isMonoidHomomorphism; π-surjective)
+  renaming (abelianGroup to +-abelianGroup)
+
+*-cong : Congruent₂ _≋_ R._*_
+*-cong {x} {y} {u} {v} (j by ιj+x≈y) (k by ιk+u≈v) =
+  ι j *ₗ k +ᴹ j *ᵣ u +ᴹ x *ₗ k by begin
+  ι (ι j *ₗ k +ᴹ j *ᵣ u +ᴹ x *ₗ k) + x * u       ≈⟨ +-congʳ (ι.+ᴹ-homo (ι j *ₗ k +ᴹ j *ᵣ u) (x *ₗ k)) ⟩
+  ι (ι j *ₗ k +ᴹ j *ᵣ u) + ι (x *ₗ k) + x * u    ≈⟨ +-congʳ (+-congʳ (ι.+ᴹ-homo (ι j *ₗ k) (j *ᵣ u))) ⟩
+  ι (ι j *ₗ k) + ι (j *ᵣ u) + ι (x *ₗ k) + x * u ≈⟨ +-congʳ (+-cong (+-cong (ι.*ₗ-homo (ι j) k) (ι.*ᵣ-homo u j)) (ι.*ₗ-homo x k)) ⟩
+  ι j * ι k + ι j * u + x * ι k + x * u          ≈⟨ binomial-expansion (ι j) x (ι k) u ⟨
+  (ι j + x) * (ι k + u)                          ≈⟨ R.*-cong ιj+x≈y ιk+u≈v ⟩
+  y * v                                          ∎
+  where
+  open R using (_+_; _*_; +-congʳ ;+-cong)
+  open import Algebra.Properties.Semiring R.semiring using (binomial-expansion)
+  open import Relation.Binary.Reasoning.Setoid R.setoid
+  open I using (ι; _*ₗ_; _*ᵣ_; _+ᴹ_)
+
+ring : Ring c (c ⊔ ℓ ⊔ c′)
+ring = record
+  { isRing = record
+    { +-isAbelianGroup = isAbelianGroup
+    ; *-cong = *-cong
+    ; *-assoc = λ x y z → ≈⇒≋ (R.*-assoc x y z)
+    ; *-identity = ≈⇒≋ ∘ R.*-identityˡ , ≈⇒≋ ∘ R.*-identityʳ
+    ; distrib = (λ x y z → ≈⇒≋ (R.distribˡ x y z)) , (λ x y z → ≈⇒≋ (R.distribʳ x y z))
+    }
+  } where open AbelianGroup +-abelianGroup using (isAbelianGroup)
+
+private module Q = Ring ring
+
+-- Public re-exports
+
+_/_ : Ring c (c ⊔ ℓ ⊔ c′)
+_/_ = ring
+
+π-isRingHomomorphism : IsRingHomomorphism R.rawRing Q.rawRing π
+π-isRingHomomorphism = record
+  { isSemiringHomomorphism = record
+    { isNearSemiringHomomorphism = record
+      { +-isMonoidHomomorphism = π-isMonoidHomomorphism
+      ; *-homo = λ _ _ → Q.refl
+      }
+    ; 1#-homo = Q.refl
+    }
+  ; -‿homo = λ _ → Q.refl
+  }
+

src/Algebra/Construct/Sub/AbelianGroup.agda

@@ -0,0 +1,78 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Subgroups of Abelian groups: necessarily Normal
+--
+-- This is a strict addition/extension to Nathan van Doorn (Taneb)'s PR
+-- https://github.com/agda/agda-stdlib/pull/2852
+-- and avoids the lemma `Algebra.NormalSubgroup.abelian⇒subgroup-normal`
+-- introduced in PR https://github.com/agda/agda-stdlib/pull/2854
+-- in favour of the direct definition of the field `normal` below.
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+open import Algebra.Bundles using (AbelianGroup)
+
+module Algebra.Construct.Sub.AbelianGroup {c ℓ} (G : AbelianGroup c ℓ) where
+
+-- As with the corresponding appeal to `IsGroup` in the module
+-- `Algebra.Construct.Sub.AbelianGroup`, this import drives the export
+-- of the `X` bundle corresponding to the `subX` structure/bundle
+-- being defined here.
+
+open import Algebra.Morphism.GroupMonomorphism using (isAbelianGroup)
+
+private
+  module G = AbelianGroup G
+
+-- Here, we could chose simply to expose the `NormalSubgroup` definition
+-- from `Algebra.Construct.Sub.Group.Normal`, but as this module will use
+-- type inference to allow the creation of `NormalSubgroup`s based on their
+-- `isNormal` fields alone, it makes sense also to export the type `IsNormal`.
+
+open import Algebra.Construct.Sub.Group.Normal G.group
+  using (IsNormal; NormalSubgroup)
+
+-- Re-export the notion of subgroup of the underlying Group
+
+open import Algebra.Construct.Sub.Group G.group public
+  using (Subgroup)
+
+-- Then, for any such Subgroup:
+-- * it is, in fact, Normal
+-- * and defines an AbelianGroup, not just a Group
+
+module _ {c′ ℓ′} (subgroup : Subgroup c′ ℓ′) where
+
+-- Here, we do need both the underlying function `ι` of the mono, and
+-- the proof `ι-monomorphism`, in order to be able to construct an
+-- `IsAbelianGroup` structure on the way to the `AbelianGroup` bundle.
+
+  open Subgroup subgroup public
+    using (ι; ι-monomorphism)
+
+-- Here, we eta-contract the use of `G.comm` in defining the `normal` field.
+
+  isNormal : IsNormal subgroup
+  isNormal = record { normal = λ n → G.comm (ι n) }
+
+-- And the use of `record`s throughout permits this 'boilerplate' construction.
+
+  normalSubgroup : NormalSubgroup c′ ℓ′
+  normalSubgroup = record { isNormal = isNormal }
+
+-- And the `public` re-export of its substructure.
+
+  open NormalSubgroup normalSubgroup public
+    using (conjugate; normal)
+
+-- As with `Algebra.Construct.Sub.Group`, there seems no need to create
+-- an intermediate manifest field `isAbelianGroup`, when this may be, and
+-- is, obtained by opening the bundle for `public` export.
+
+  abelianGroup : AbelianGroup c′ ℓ′
+  abelianGroup = record
+    { isAbelianGroup = isAbelianGroup ι-monomorphism G.isAbelianGroup }
+
+  open AbelianGroup abelianGroup public