------------------------------------------------------------------------
-- The Agda standard library
--
-- Morphisms between algebraic structures
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Algebra.Morphism where

import Algebra.Morphism.Definitions as MorphismDefinitions
open import Algebra
import Algebra.Properties.Group as GroupP
open import Function hiding (Morphism)
open import Level
open import Relation.Binary
import Relation.Binary.Reasoning.Setoid as EqR

private
  variable
    a b ℓ₁ ℓ₂ : Level
    A : Set a
    B : Set b

------------------------------------------------------------------------
-- Re-export

module Definitions {a b ℓ₁} (A : Set a) (B : Set b) (_≈_ : Rel B ℓ₁) where
  open MorphismDefinitions A B _≈_ public

open import Algebra.Morphism.Structures public


------------------------------------------------------------------------
-- DEPRECATED
------------------------------------------------------------------------
-- Please use the new definitions re-exported from
-- `Algebra.Morphism.Structures` as continuing support for the below is
-- no guaranteed.

-- Version 1.5

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : Semigroup c₁ ℓ₁)
         (To   : Semigroup c₂ ℓ₂) where

  private
    module F = Semigroup From
    module T = Semigroup To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsSemigroupMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      ⟦⟧-cong : ⟦_⟧ Preserves F._≈_ ⟶ T._≈_
      ∙-homo  : Homomorphic₂ ⟦_⟧ F._∙_ T._∙_

  IsSemigroupMorphism-syntax = IsSemigroupMorphism
  syntax IsSemigroupMorphism-syntax From To F = F Is From -Semigroup⟶ To

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : Monoid c₁ ℓ₁)
         (To   : Monoid c₂ ℓ₂) where

  private
    module F = Monoid From
    module T = Monoid To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsMonoidMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      sm-homo : IsSemigroupMorphism F.semigroup T.semigroup ⟦_⟧
      ε-homo  : Homomorphic₀ ⟦_⟧ F.ε T.ε

    open IsSemigroupMorphism sm-homo public

  IsMonoidMorphism-syntax = IsMonoidMorphism
  syntax IsMonoidMorphism-syntax From To F = F Is From -Monoid⟶ To

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : CommutativeMonoid c₁ ℓ₁)
         (To   : CommutativeMonoid c₂ ℓ₂) where

  private
    module F = CommutativeMonoid From
    module T = CommutativeMonoid To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsCommutativeMonoidMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      mn-homo : IsMonoidMorphism F.monoid T.monoid ⟦_⟧

    open IsMonoidMorphism mn-homo public

  IsCommutativeMonoidMorphism-syntax = IsCommutativeMonoidMorphism
  syntax IsCommutativeMonoidMorphism-syntax From To F = F Is From -CommutativeMonoid⟶ To

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : IdempotentCommutativeMonoid c₁ ℓ₁)
         (To   : IdempotentCommutativeMonoid c₂ ℓ₂) where

  private
    module F = IdempotentCommutativeMonoid From
    module T = IdempotentCommutativeMonoid To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsIdempotentCommutativeMonoidMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      mn-homo : IsMonoidMorphism F.monoid T.monoid ⟦_⟧

    open IsMonoidMorphism mn-homo public

    isCommutativeMonoidMorphism :
      IsCommutativeMonoidMorphism F.commutativeMonoid T.commutativeMonoid ⟦_⟧
    isCommutativeMonoidMorphism = record { mn-homo = mn-homo }

  IsIdempotentCommutativeMonoidMorphism-syntax = IsIdempotentCommutativeMonoidMorphism
  syntax IsIdempotentCommutativeMonoidMorphism-syntax From To F = F Is From -IdempotentCommutativeMonoid⟶ To

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : Group c₁ ℓ₁)
         (To   : Group c₂ ℓ₂) where

  private
    module F = Group From
    module T = Group To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsGroupMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      mn-homo : IsMonoidMorphism F.monoid T.monoid ⟦_⟧

    open IsMonoidMorphism mn-homo public

    ⁻¹-homo : Homomorphic₁ ⟦_⟧ F._⁻¹ T._⁻¹
    ⁻¹-homo x = let open EqR T.setoid in T.uniqueˡ-⁻¹ ⟦ x F.⁻¹ ⟧ ⟦ x ⟧ $ begin
      ⟦ x F.⁻¹ ⟧ T.∙ ⟦ x ⟧ ≈⟨ T.sym (∙-homo (x F.⁻¹) x) ⟩
      ⟦ x F.⁻¹ F.∙ x ⟧     ≈⟨ ⟦⟧-cong (F.inverseˡ x) ⟩
      ⟦ F.ε ⟧              ≈⟨ ε-homo ⟩
      T.ε ∎

  IsGroupMorphism-syntax = IsGroupMorphism
  syntax IsGroupMorphism-syntax From To F = F Is From -Group⟶ To

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : AbelianGroup c₁ ℓ₁)
         (To   : AbelianGroup c₂ ℓ₂) where

  private
    module F = AbelianGroup From
    module T = AbelianGroup To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsAbelianGroupMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      gp-homo : IsGroupMorphism F.group T.group ⟦_⟧

    open IsGroupMorphism gp-homo public

  IsAbelianGroupMorphism-syntax = IsAbelianGroupMorphism
  syntax IsAbelianGroupMorphism-syntax From To F = F Is From -AbelianGroup⟶ To

module _ {c₁ ℓ₁ c₂ ℓ₂}
         (From : Ring c₁ ℓ₁)
         (To   : Ring c₂ ℓ₂) where

  private
    module F = Ring From
    module T = Ring To
  open Definitions F.Carrier T.Carrier T._≈_

  record IsRingMorphism (⟦_⟧ : Morphism) :
         Set (c₁ ⊔ ℓ₁ ⊔ c₂ ⊔ ℓ₂) where
    field
      +-abgp-homo : ⟦_⟧ Is F.+-abelianGroup -AbelianGroup⟶ T.+-abelianGroup
      *-mn-homo   : ⟦_⟧ Is F.*-monoid -Monoid⟶ T.*-monoid

  IsRingMorphism-syntax = IsRingMorphism
  syntax IsRingMorphism-syntax From To F = F Is From -Ring⟶ To

{-# WARNING_ON_USAGE IsSemigroupMorphism
"Warning: IsSemigroupMorphism was deprecated in v1.5.
Please use IsSemigroupHomomorphism instead."
#-}
{-# WARNING_ON_USAGE IsMonoidMorphism
"Warning: IsMonoidMorphism was deprecated in v1.5.
Please use IsMonoidHomomorphism instead."
#-}
{-# WARNING_ON_USAGE IsCommutativeMonoidMorphism
"Warning: IsCommutativeMonoidMorphism was deprecated in v1.5.
Please use IsMonoidHomomorphism instead."
#-}
{-# WARNING_ON_USAGE IsIdempotentCommutativeMonoidMorphism
"Warning: IsIdempotentCommutativeMonoidMorphism was deprecated in v1.5.
Please use IsMonoidHomomorphism instead."
#-}
{-# WARNING_ON_USAGE IsGroupMorphism
"Warning: IsGroupMorphism was deprecated in v1.5.
Please use IsGroupHomomorphism instead."
#-}
{-# WARNING_ON_USAGE IsAbelianGroupMorphism
"Warning: IsAbelianGroupMorphism was deprecated in v1.5.
Please use IsGroupHomomorphism instead."
#-}