Add algebra morphism identity and composition constructions

CHANGELOG.md

@@ -122,6 +122,12 @@ Deprecated names
 New modules
 -----------
 
+* Identity morphisms and composition of morphisms between algebraic structures:
+  ```
+  Algebra.Morphism.Construct.Composition
+  Algebra.Morphism.Construct.Identity
+  ```
+
 * New module for making system calls during type checking:
   ```agda
   Reflection.External
@@ -167,6 +173,20 @@ New modules
 Other minor additions
 ---------------------
 
+* In `Algebra.Bundles`, `Lattice` now provides
+  ```agda
+  ∨-commutativeSemigroup : CommutativeSemigroup c ℓ
+  ∧-commutativeSemigroup : CommutativeSemigroup c ℓ
+  ```
+  and their corresponding algebraic subbundles.
+
+* In `Algebra.Structures`, `IsLattice` now provides
+  ```
+  ∨-isCommutativeSemigroup : IsCommutativeSemigroup ∨
+  ∧-isCommutativeSemigroup : IsCommutativeSemigroup ∧
+  ```
+  and their corresponding algebraic substructures.
+
 * Added new relations to `Data.Fin.Base`:
   ```agda
   _≥_ = ℕ._≥_ on toℕ

src/Algebra/Bundles.agda

@@ -386,7 +386,7 @@ record Lattice c ℓ : Set (suc (c ⊔ ℓ)) where
     ; _∨_  = _∨_
     }
 
-  open RawLattice rawLattice
+  open RawLattice rawLattice public
     using (∨-rawMagma; ∧-rawMagma)
 
   setoid : Setoid _ _

src/Algebra/Morphism/Construct/Composition.agda

@@ -0,0 +1,305 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The composition of morphisms between algebraic structures.
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --without-K #-}
+
+module Algebra.Morphism.Construct.Composition where
+
+open import Algebra.Bundles
+open import Algebra.Morphism.Structures
+open import Data.Product
+open import Function.Base using (_∘_)
+import Function.Construct.Composition as Func
+open import Level using (Level)
+open import Relation.Binary.Morphism.Construct.Composition
+open import Relation.Binary.Definitions using (Transitive)
+
+private
+  variable
+    a b c ℓ₁ ℓ₂ ℓ₃ : Level
+
+------------------------------------------------------------------------
+-- Magmas
+
+module _ {M₁ : RawMagma a ℓ₁}
+         {M₂ : RawMagma b ℓ₂}
+         {M₃ : RawMagma c ℓ₃}
+         (open RawMagma)
+         (≈₃-trans : Transitive (_≈_ M₃))
+         {f : Carrier M₁ → Carrier M₂}
+         {g : Carrier M₂ → Carrier M₃}
+         where
+
+  isMagmaHomomorphism
+    : IsMagmaHomomorphism M₁ M₂ f
+    → IsMagmaHomomorphism M₂ M₃ g
+    → IsMagmaHomomorphism M₁ M₃ (g ∘ f)
+  isMagmaHomomorphism f-homo g-homo = record
+    { isRelHomomorphism = isRelHomomorphism F.isRelHomomorphism G.isRelHomomorphism
+    ; homo              = λ x y → ≈₃-trans (G.⟦⟧-cong (F.homo x y)) (G.homo (f x) (f y))
+    } where module F = IsMagmaHomomorphism f-homo; module G = IsMagmaHomomorphism g-homo
+
+  isMagmaMonomorphism
+    : IsMagmaMonomorphism M₁ M₂ f
+    → IsMagmaMonomorphism M₂ M₃ g
+    → IsMagmaMonomorphism M₁ M₃ (g ∘ f)
+  isMagmaMonomorphism f-mono g-mono = record
+    { isMagmaHomomorphism = isMagmaHomomorphism F.isMagmaHomomorphism G.isMagmaHomomorphism
+    ; injective           = F.injective ∘ G.injective
+    } where module F = IsMagmaMonomorphism f-mono; module G = IsMagmaMonomorphism g-mono
+
+  isMagmaIsomorphism
+    : IsMagmaIsomorphism M₁ M₂ f
+    → IsMagmaIsomorphism M₂ M₃ g
+    → IsMagmaIsomorphism M₁ M₃ (g ∘ f)
+  isMagmaIsomorphism f-iso g-iso = record
+    { isMagmaMonomorphism = isMagmaMonomorphism F.isMagmaMonomorphism G.isMagmaMonomorphism
+    ; surjective          = Func.surjective (_≈_ M₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsMagmaIsomorphism f-iso; module G = IsMagmaIsomorphism g-iso
+
+
+------------------------------------------------------------------------
+-- Monoids
+
+module _ {M₁ : RawMonoid a ℓ₁}
+         {M₂ : RawMonoid b ℓ₂}
+         {M₃ : RawMonoid c ℓ₃}
+         (open RawMonoid)
+         (≈₃-trans : Transitive (_≈_ M₃))
+         {f : Carrier M₁ → Carrier M₂}
+         {g : Carrier M₂ → Carrier M₃}
+         where
+
+  isMonoidHomomorphism
+    : IsMonoidHomomorphism M₁ M₂ f
+    → IsMonoidHomomorphism M₂ M₃ g
+    → IsMonoidHomomorphism M₁ M₃ (g ∘ f)
+  isMonoidHomomorphism f-homo g-homo = record
+    { isMagmaHomomorphism = isMagmaHomomorphism ≈₃-trans F.isMagmaHomomorphism G.isMagmaHomomorphism
+    ; ε-homo              = ≈₃-trans (G.⟦⟧-cong F.ε-homo) G.ε-homo
+    } where module F = IsMonoidHomomorphism f-homo; module G = IsMonoidHomomorphism g-homo
+
+  isMonoidMonomorphism
+    : IsMonoidMonomorphism M₁ M₂ f
+    → IsMonoidMonomorphism M₂ M₃ g
+    → IsMonoidMonomorphism M₁ M₃ (g ∘ f)
+  isMonoidMonomorphism f-mono g-mono = record
+    { isMonoidHomomorphism = isMonoidHomomorphism F.isMonoidHomomorphism G.isMonoidHomomorphism
+    ; injective            = F.injective ∘ G.injective
+    } where module F = IsMonoidMonomorphism f-mono; module G = IsMonoidMonomorphism g-mono
+
+  isMonoidIsomorphism
+    : IsMonoidIsomorphism M₁ M₂ f
+    → IsMonoidIsomorphism M₂ M₃ g
+    → IsMonoidIsomorphism M₁ M₃ (g ∘ f)
+  isMonoidIsomorphism f-iso g-iso = record
+    { isMonoidMonomorphism = isMonoidMonomorphism F.isMonoidMonomorphism G.isMonoidMonomorphism
+    ; surjective           = Func.surjective (_≈_ M₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsMonoidIsomorphism f-iso; module G = IsMonoidIsomorphism g-iso
+
+
+------------------------------------------------------------------------
+-- Groups
+
+module _ {G₁ : RawGroup a ℓ₁}
+         {G₂ : RawGroup b ℓ₂}
+         {G₃ : RawGroup c ℓ₃}
+         (open RawGroup)
+         (≈₃-trans : Transitive (_≈_ G₃))
+         {f : Carrier G₁ → Carrier G₂}
+         {g : Carrier G₂ → Carrier G₃}
+         where
+
+
+  isGroupHomomorphism
+    : IsGroupHomomorphism G₁ G₂ f
+    → IsGroupHomomorphism G₂ G₃ g
+    → IsGroupHomomorphism G₁ G₃ (g ∘ f)
+  isGroupHomomorphism f-homo g-homo = record
+    { isMonoidHomomorphism = isMonoidHomomorphism ≈₃-trans F.isMonoidHomomorphism G.isMonoidHomomorphism
+    ; ⁻¹-homo              = λ x → ≈₃-trans (G.⟦⟧-cong (F.⁻¹-homo x)) (G.⁻¹-homo (f x))
+    } where module F = IsGroupHomomorphism f-homo; module G = IsGroupHomomorphism g-homo
+
+  isGroupMonomorphism
+    : IsGroupMonomorphism G₁ G₂ f
+    → IsGroupMonomorphism G₂ G₃ g
+    → IsGroupMonomorphism G₁ G₃ (g ∘ f)
+  isGroupMonomorphism f-mono g-mono = record
+    { isGroupHomomorphism = isGroupHomomorphism F.isGroupHomomorphism G.isGroupHomomorphism
+    ; injective           = F.injective ∘ G.injective
+    } where module F = IsGroupMonomorphism f-mono; module G = IsGroupMonomorphism g-mono
+
+  isGroupIsomorphism
+    : IsGroupIsomorphism G₁ G₂ f
+    → IsGroupIsomorphism G₂ G₃ g
+    → IsGroupIsomorphism G₁ G₃ (g ∘ f)
+  isGroupIsomorphism f-iso g-iso = record
+    { isGroupMonomorphism = isGroupMonomorphism F.isGroupMonomorphism G.isGroupMonomorphism
+    ; surjective          = Func.surjective (_≈_ G₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsGroupIsomorphism f-iso; module G = IsGroupIsomorphism g-iso
+
+
+------------------------------------------------------------------------
+-- Near semirings
+
+module _ {R₁ : RawNearSemiring a ℓ₁}
+         {R₂ : RawNearSemiring b ℓ₂}
+         {R₃ : RawNearSemiring c ℓ₃}
+         (open RawNearSemiring)
+         (≈₃-trans : Transitive (_≈_ R₃))
+         {f : Carrier R₁ → Carrier R₂}
+         {g : Carrier R₂ → Carrier R₃}
+         where
+
+  isNearSemiringHomomorphism
+    : IsNearSemiringHomomorphism R₁ R₂ f
+    → IsNearSemiringHomomorphism R₂ R₃ g
+    → IsNearSemiringHomomorphism R₁ R₃ (g ∘ f)
+  isNearSemiringHomomorphism f-homo g-homo = record
+    { +-isMonoidHomomorphism = isMonoidHomomorphism ≈₃-trans F.+-isMonoidHomomorphism G.+-isMonoidHomomorphism
+    ; *-isMagmaHomomorphism  = isMagmaHomomorphism  ≈₃-trans F.*-isMagmaHomomorphism  G.*-isMagmaHomomorphism
+    } where module F = IsNearSemiringHomomorphism f-homo; module G = IsNearSemiringHomomorphism g-homo
+
+  isNearSemiringMonomorphism
+    : IsNearSemiringMonomorphism R₁ R₂ f
+    → IsNearSemiringMonomorphism R₂ R₃ g
+    → IsNearSemiringMonomorphism R₁ R₃ (g ∘ f)
+  isNearSemiringMonomorphism f-mono g-mono = record
+    { isNearSemiringHomomorphism = isNearSemiringHomomorphism F.isNearSemiringHomomorphism G.isNearSemiringHomomorphism
+    ; injective                  = F.injective ∘ G.injective
+    } where module F = IsNearSemiringMonomorphism f-mono; module G = IsNearSemiringMonomorphism g-mono
+
+  isNearSemiringIsomorphism
+    : IsNearSemiringIsomorphism R₁ R₂ f
+    → IsNearSemiringIsomorphism R₂ R₃ g
+    → IsNearSemiringIsomorphism R₁ R₃ (g ∘ f)
+  isNearSemiringIsomorphism f-iso g-iso = record
+    { isNearSemiringMonomorphism = isNearSemiringMonomorphism F.isNearSemiringMonomorphism G.isNearSemiringMonomorphism
+    ; surjective                 = Func.surjective (_≈_ R₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsNearSemiringIsomorphism f-iso; module G = IsNearSemiringIsomorphism g-iso
+
+
+------------------------------------------------------------------------
+-- Semirings
+
+module _
+  {R₁ : RawSemiring a ℓ₁}
+  {R₂ : RawSemiring b ℓ₂}
+  {R₃ : RawSemiring c ℓ₃}
+  (open RawSemiring)
+  (≈₃-trans : Transitive (_≈_ R₃))
+  {f : Carrier R₁ → Carrier R₂}
+  {g : Carrier R₂ → Carrier R₃}
+  where
+
+
+  isSemiringHomomorphism
+    : IsSemiringHomomorphism R₁ R₂ f
+    → IsSemiringHomomorphism R₂ R₃ g
+    → IsSemiringHomomorphism R₁ R₃ (g ∘ f)
+  isSemiringHomomorphism f-homo g-homo = record
+    { +-isMonoidHomomorphism = isMonoidHomomorphism ≈₃-trans F.+-isMonoidHomomorphism G.+-isMonoidHomomorphism
+    ; *-isMonoidHomomorphism = isMonoidHomomorphism ≈₃-trans F.*-isMonoidHomomorphism G.*-isMonoidHomomorphism
+    } where module F = IsSemiringHomomorphism f-homo; module G = IsSemiringHomomorphism g-homo
+
+  isSemiringMonomorphism
+    : IsSemiringMonomorphism R₁ R₂ f
+    → IsSemiringMonomorphism R₂ R₃ g
+    → IsSemiringMonomorphism R₁ R₃ (g ∘ f)
+  isSemiringMonomorphism f-mono g-mono = record
+    { isSemiringHomomorphism = isSemiringHomomorphism F.isSemiringHomomorphism G.isSemiringHomomorphism
+    ; injective              = F.injective ∘ G.injective
+    } where module F = IsSemiringMonomorphism f-mono; module G = IsSemiringMonomorphism g-mono
+
+  isSemiringIsomorphism
+    : IsSemiringIsomorphism R₁ R₂ f
+    → IsSemiringIsomorphism R₂ R₃ g
+    → IsSemiringIsomorphism R₁ R₃ (g ∘ f)
+  isSemiringIsomorphism f-iso g-iso = record
+    { isSemiringMonomorphism = isSemiringMonomorphism F.isSemiringMonomorphism G.isSemiringMonomorphism
+    ; surjective             = Func.surjective (_≈_ R₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsSemiringIsomorphism f-iso; module G = IsSemiringIsomorphism g-iso
+
+
+------------------------------------------------------------------------
+-- Rings
+
+module _ {R₁ : RawRing a ℓ₁}
+         {R₂ : RawRing b ℓ₂}
+         {R₃ : RawRing c ℓ₃}
+         (open RawRing)
+         (≈₃-trans : Transitive (_≈_ R₃))
+         {f : Carrier R₁ → Carrier R₂}
+         {g : Carrier R₂ → Carrier R₃}
+         where
+
+
+  isRingHomomorphism
+    : IsRingHomomorphism R₁ R₂ f
+    → IsRingHomomorphism R₂ R₃ g
+    → IsRingHomomorphism R₁ R₃ (g ∘ f)
+  isRingHomomorphism f-homo g-homo = record
+    { +-isGroupHomomorphism = isGroupHomomorphism   ≈₃-trans F.+-isGroupHomomorphism  G.+-isGroupHomomorphism
+    ; *-isMonoidHomomorphism = isMonoidHomomorphism ≈₃-trans F.*-isMonoidHomomorphism G.*-isMonoidHomomorphism
+    } where module F = IsRingHomomorphism f-homo; module G = IsRingHomomorphism g-homo
+
+  isRingMonomorphism
+    : IsRingMonomorphism R₁ R₂ f
+    → IsRingMonomorphism R₂ R₃ g
+    → IsRingMonomorphism R₁ R₃ (g ∘ f)
+  isRingMonomorphism f-mono g-mono = record
+    { isRingHomomorphism = isRingHomomorphism F.isRingHomomorphism G.isRingHomomorphism
+    ; injective = F.injective ∘ G.injective
+    } where module F = IsRingMonomorphism f-mono;  module G = IsRingMonomorphism g-mono
+
+  isRingIsomorphism
+    : IsRingIsomorphism R₁ R₂ f
+    → IsRingIsomorphism R₂ R₃ g
+    → IsRingIsomorphism R₁ R₃ (g ∘ f)
+  isRingIsomorphism f-iso g-iso = record
+    { isRingMonomorphism = isRingMonomorphism F.isRingMonomorphism G.isRingMonomorphism
+    ; surjective         = Func.surjective (_≈_ R₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsRingIsomorphism f-iso; module G = IsRingIsomorphism g-iso
+
+
+------------------------------------------------------------------------
+-- Lattices
+
+module _ {L₁ : RawLattice a ℓ₁}
+         {L₂ : RawLattice b ℓ₂}
+         {L₃ : RawLattice c ℓ₃}
+         (open RawLattice)
+         (≈₃-trans : Transitive (_≈_ L₃))
+         {f : Carrier L₁ → Carrier L₂}
+         {g : Carrier L₂ → Carrier L₃}
+         where
+
+  isLatticeHomomorphism
+    : IsLatticeHomomorphism L₁ L₂ f
+    → IsLatticeHomomorphism L₂ L₃ g
+    → IsLatticeHomomorphism L₁ L₃ (g ∘ f)
+  isLatticeHomomorphism f-homo g-homo = record
+    { ∧-isMagmaHomomorphism = isMagmaHomomorphism ≈₃-trans F.∧-isMagmaHomomorphism G.∧-isMagmaHomomorphism
+    ; ∨-isMagmaHomomorphism = isMagmaHomomorphism ≈₃-trans F.∨-isMagmaHomomorphism G.∨-isMagmaHomomorphism
+    } where module F = IsLatticeHomomorphism f-homo; module G = IsLatticeHomomorphism g-homo
+
+  isLatticeMonomorphism
+    : IsLatticeMonomorphism L₁ L₂ f
+    → IsLatticeMonomorphism L₂ L₃ g
+    → IsLatticeMonomorphism L₁ L₃ (g ∘ f)
+  isLatticeMonomorphism f-mono g-mono = record
+    { isLatticeHomomorphism = isLatticeHomomorphism F.isLatticeHomomorphism G.isLatticeHomomorphism
+    ; injective             = F.injective ∘ G.injective
+    } where module F = IsLatticeMonomorphism f-mono; module G = IsLatticeMonomorphism g-mono
+
+  isLatticeIsomorphism
+    : IsLatticeIsomorphism L₁ L₂ f
+    → IsLatticeIsomorphism L₂ L₃ g
+    → IsLatticeIsomorphism L₁ L₃ (g ∘ f)
+  isLatticeIsomorphism f-iso g-iso = record
+    { isLatticeMonomorphism = isLatticeMonomorphism F.isLatticeMonomorphism G.isLatticeMonomorphism
+    ; surjective            = Func.surjective (_≈_ L₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    } where module F = IsLatticeIsomorphism f-iso; module G = IsLatticeIsomorphism g-iso