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Homomorphic images of setoid algebras

This is the Setoid.Homomorphisms.HomomorphicImages module of the Agda Universal Algebra Library.

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

open import Base.Algebras.Basic using (𝓞 ; 𝓥 ; Signature )

module Setoid.Homomorphisms.HomomorphicImages {𝑆 : Signature 𝓞 𝓥} where

-- Imports from Agda and the Agda Standard Library ------------------------------------------
open import Agda.Primitive   using ( _⊔_ ; lsuc )      renaming ( Set to Type )
open import Data.Product     using ( _,_ ; Σ-syntax )  renaming ( _×_ to _∧_ ; proj₁ to fst ; proj₂ to snd )
open import Function         using ( Func ; _on_ ; _∘_ ; id )
open import Level            using ( Level )
open import Relation.Binary  using ( Setoid ; _Preserves_⟶_ )
open import Relation.Unary   using ( Pred ; _∈_ )
open import Relation.Binary.PropositionalEquality as ≡ using ()

-- Imports from the Agda Universal Algebra Library ---------------------------------------------
open import Base.Overture.Preliminaries                 using ( ∣_∣ ; ∥_∥ ; transport )
open import Setoid.Overture.Preliminaries               using ( lift∼lower )
open import Setoid.Overture.Inverses                    using ( Ran ; _range ; _preimage ; _image ; Inv )
                                                        using ( _preimage≈image ; InvIsInverseʳ ; Image_∋_ )
open import Setoid.Overture.Surjective                  using ( IsSurjective ; ∘-IsSurjective )
open import Setoid.Algebras.Basic              {𝑆 = 𝑆}  using ( Algebra ; ov ; _̂_ ; ⟦_⟧ ; Lift-Algˡ )
                                                        using ( Lift-Alg ; 𝕌[_] )
open import Setoid.Homomorphisms.Basic         {𝑆 = 𝑆}  using ( hom ; IsHom )
open import Setoid.Homomorphisms.Isomorphisms  {𝑆 = 𝑆}  using ( _≅_ ; Lift-≅ )
open import Setoid.Homomorphisms.Properties    {𝑆 = 𝑆}  using ( Lift-homˡ ; ToLiftˡ ; lift-hom-lemma )
                                                        using ( 𝒾𝒹 ; ∘-hom )
private variable α ρᵃ β ρᵇ : Level

open Algebra

We begin with what seems, for our purposes, the most useful way to represent the class of homomorphic images of an algebra in dependent type theory.

open IsHom


_IsHomImageOf_ : (𝑩 : Algebra β ρᵇ)(𝑨 : Algebra α ρᵃ) → Type (𝓞 ⊔ 𝓥 ⊔ α ⊔ β ⊔ ρᵃ ⊔ ρᵇ)
𝑩 IsHomImageOf 𝑨 = Σ[ φ ∈ hom 𝑨 𝑩 ] IsSurjective ∣ φ ∣

HomImages : Algebra α ρᵃ → Type (α ⊔ ρᵃ ⊔ ov (β ⊔ ρᵇ))
HomImages {β = β}{ρᵇ = ρᵇ} 𝑨 = Σ[ 𝑩 ∈ Algebra β ρᵇ ] 𝑩 IsHomImageOf 𝑨

IdHomImage : {𝑨 : Algebra α ρᵃ} → 𝑨 IsHomImageOf 𝑨
IdHomImage {α = α}{𝑨 = 𝑨} = 𝒾𝒹 , λ {y} → Image_∋_.eq y refl
 where
 open Setoid (Domain 𝑨) using ( refl )

These types should be self-explanatory, but just to be sure, let’s describe the Sigma type appearing in the second definition. Given an 𝑆-algebra 𝑨 : Algebra α ρ, the type HomImages 𝑨 denotes the class of algebras 𝑩 : Algebra β ρ with a map φ : ∣ 𝑨 ∣ → ∣ 𝑩 ∣ such that φ is a surjective homomorphism.

The image algebra of a hom

Here we show how to construct a Algebra (called ImageAlgebra below) that is the image of given hom.

module _ {𝑨 : Algebra α ρᵃ}{𝑩 : Algebra β ρᵇ} where
 open Algebra 𝑨  using (Interp)  renaming (Domain to A )
 open Algebra 𝑩  using ()        renaming (Domain to B ; Interp to InterpB )

 open Setoid A  using ( )            renaming ( _≈_ to _≈₁_ ; Carrier to ∣A∣)
 open Setoid B  using ( reflexive )  renaming ( _≈_ to _≈₂_ ; refl to refl₂ )
                                     renaming ( sym to sym₂ ; trans to trans₂ ; Carrier to ∣B∣)

 open Func using ( cong ) renaming (f to _⟨$⟩_ )
 open IsHom

 HomImageOf[_] : hom 𝑨 𝑩 → Algebra (α ⊔ β ⊔ ρᵇ) ρᵇ
 HomImageOf[ h ] =
  record { Domain = Ran ∣ h ∣ ; Interp = record { f = f' ; cong = cong' } }
   where
   open Setoid (⟦ 𝑆 ⟧ (Ran ∣ h ∣)) using () renaming (Carrier to SRanh ; _≈_ to _≈₃_ ; refl to refl₃ )

   hhom : ∀ {𝑓}(x : ∥ 𝑆 ∥ 𝑓 → ∣ h ∣ range )
    →     (∣ h ∣ ⟨$⟩ (𝑓 ̂ 𝑨) ((∣ h ∣ preimage) ∘ x)) ≈₂ (𝑓 ̂ 𝑩) ((∣ h ∣ image) ∘ x)

   hhom {𝑓} x = trans₂ (compatible ∥ h ∥) (cong InterpB (≡.refl , (∣ h ∣ preimage≈image) ∘ x))

   f' : SRanh → ∣ h ∣ range
   f' (𝑓 , x) = (𝑓 ̂ 𝑩)((∣ h ∣ image)∘ x)       -- b : the image in ∣B∣
                , (𝑓 ̂ 𝑨)((∣ h ∣ preimage) ∘ x) -- a : the preimage in ∣A∣
                , hhom x                        -- p : proof that `(∣ h ∣ ⟨$⟩ a) ≈₂ b`

   cong' : ∀ {x y} → x ≈₃ y → ((∣ h ∣ image) (f' x)) ≈₂ ((∣ h ∣ image) (f' y))
   cong' {(𝑓 , u)} {(.𝑓 , v)} (≡.refl , EqA) = Goal

    where

    -- Alternative formulation of the goal:
    goal : (𝑓 ̂ 𝑩)(λ i → ((∣ h ∣ image)(u i))) ≈₂ (𝑓 ̂ 𝑩)(λ i → ((∣ h ∣ image) (v i)))
    goal = cong InterpB (≡.refl , EqA )

    Goal : (∣ h ∣ image) (f' (𝑓 , u)) ≈₂ (∣ h ∣ image) (f' (𝑓 , v))
    Goal = goal

    -- Note: `EqA : ∀ i → (∣ h ∣ image) (u i) ≈₂ (∣ h ∣ image) (v i)`

Homomorphic images of classes of setoid algebras

Given a class 𝒦 of 𝑆-algebras, we need a type that expresses the assertion that a given algebra is a homomorphic image of some algebra in the class, as well as a type that represents all such homomorphic images.

IsHomImageOfClass : {𝒦 : Pred (Algebra α ρᵃ)(lsuc α)} → Algebra α ρᵃ → Type (ov (α ⊔ ρᵃ))
IsHomImageOfClass {𝒦 = 𝒦} 𝑩 = Σ[ 𝑨 ∈ Algebra _ _ ] ((𝑨 ∈ 𝒦) ∧ (𝑩 IsHomImageOf 𝑨))

HomImageOfClass : Pred (Algebra α ρᵃ) (lsuc α) → Type (ov (α ⊔ ρᵃ))
HomImageOfClass 𝒦 = Σ[ 𝑩 ∈ Algebra _ _ ] IsHomImageOfClass {𝒦 = 𝒦} 𝑩

Lifting tools

Here are some tools that have been useful (e.g., in the road to the proof of Birkhoff’s HSP theorem). The first states and proves the simple fact that the lift of an epimorphism is an epimorphism.

module _ {𝑨 : Algebra α ρᵃ}{𝑩 : Algebra β ρᵇ} where

 open Algebra 𝑨 using ()              renaming ( Domain to A )
 open Algebra 𝑩 using ()              renaming ( Domain to B )
 open Setoid B        using ( sym ; trans ) renaming ( _≈_ to _≈₂_ )

 open Func            using ( cong )        renaming ( f to _⟨$⟩_ )
 open Level

 Lift-epi-is-epiˡ : (h : hom 𝑨 𝑩)(ℓᵃ ℓᵇ : Level)
  →                IsSurjective ∣ h ∣ → IsSurjective ∣ Lift-homˡ {𝑨 = 𝑨}{𝑩} h ℓᵃ ℓᵇ ∣

 Lift-epi-is-epiˡ h ℓᵃ ℓᵇ hepi {b} = Goal
  where
  open Algebra (Lift-Algˡ 𝑩 ℓᵇ) using () renaming (Domain to lB )
  open Setoid lB using () renaming ( _≈_ to _≈ₗ₂_ )

  a : 𝕌[ 𝑨 ]
  a = Inv ∣ h ∣ hepi

  lem1 : b ≈ₗ₂ (lift (lower b))
  lem1 = lift∼lower {𝑨 = B} b

  lem2' : (lower b) ≈₂ (∣ h ∣ ⟨$⟩ a)
  lem2' = sym  (InvIsInverseʳ hepi)

  lem2 : (lift (lower b)) ≈ₗ₂ (lift (∣ h ∣ ⟨$⟩ a))
  lem2 = cong{From = B} ∣ ToLiftˡ{𝑨 = 𝑩}{ℓᵇ} ∣ lem2'

  lem3 : (lift (∣ h ∣ ⟨$⟩ a)) ≈ₗ₂ ((∣ Lift-homˡ h ℓᵃ ℓᵇ ∣ ⟨$⟩ lift a))
  lem3 = lift-hom-lemma h a ℓᵃ ℓᵇ

  η : b ≈ₗ₂ (∣ Lift-homˡ h ℓᵃ ℓᵇ ∣ ⟨$⟩ lift a)
  η = trans lem1 (trans lem2 lem3)

  Goal : Image ∣ Lift-homˡ h ℓᵃ ℓᵇ ∣ ∋ b
  Goal = Image_∋_.eq (lift a) η


 Lift-Alg-hom-imageˡ : (ℓᵃ ℓᵇ : Level) → 𝑩 IsHomImageOf 𝑨
  →                   (Lift-Algˡ 𝑩 ℓᵇ) IsHomImageOf (Lift-Algˡ 𝑨 ℓᵃ)

 Lift-Alg-hom-imageˡ ℓᵃ ℓᵇ ((φ , φhom) , φepic) = Goal
  where
  lφ : hom (Lift-Algˡ 𝑨 ℓᵃ) (Lift-Algˡ 𝑩 ℓᵇ)
  lφ = Lift-homˡ {𝑨 = 𝑨}{𝑩} (φ , φhom) ℓᵃ ℓᵇ

  lφepic : IsSurjective ∣ lφ ∣
  lφepic = Lift-epi-is-epiˡ (φ , φhom) ℓᵃ ℓᵇ φepic
  Goal : (Lift-Algˡ 𝑩 ℓᵇ) IsHomImageOf (Lift-Algˡ 𝑨 ℓᵃ)
  Goal = lφ , lφepic


module _ {𝑨 : Algebra α ρᵃ}{𝑩 : Algebra β ρᵇ} where
 open _≅_
 Lift-HomImage-lemma : ∀{γ} → (Lift-Alg 𝑨 γ γ) IsHomImageOf 𝑩 → 𝑨 IsHomImageOf 𝑩
 Lift-HomImage-lemma {γ} φ = ∘-hom ∣ φ ∣ (from Lift-≅) ,
                             ∘-IsSurjective ∥ φ ∥ (fromIsSurjective (Lift-≅{𝑨 = 𝑨}))

module _ {𝑨 𝑨' : Algebra α ρᵃ}{𝑩 : Algebra β ρᵇ} where
 open _≅_
 HomImage-≅ : 𝑨 IsHomImageOf 𝑨' → 𝑨 ≅ 𝑩 → 𝑩 IsHomImageOf 𝑨'
 HomImage-≅ φ A≅B = ∘-hom ∣ φ ∣ (to A≅B) , ∘-IsSurjective ∥ φ ∥ (toIsSurjective A≅B)

 HomImage-≅' : 𝑨 IsHomImageOf 𝑨' → 𝑨' ≅ 𝑩 → 𝑨 IsHomImageOf 𝑩
 HomImage-≅' φ A'≅B = (∘-hom (from A'≅B) ∣ φ ∣) , ∘-IsSurjective (fromIsSurjective A'≅B) ∥ φ ∥

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← Setoid.Homomorphisms.Isomorphisms Setoid.Terms →