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Homomorphic Images

This section describes the Homomorphisms.HomomorphicImages module of the Agda Universal Algebra Library.

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

open import Algebras.Signatures using (Signature; π“ž; π“₯)

module Homomorphisms.HomomorphicImages {𝑆 : Signature π“ž π“₯} where

open import Homomorphisms.Isomorphisms{𝑆 = 𝑆} public

Images of a single 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.

module _ {𝓀 𝓦 : Universe} where

 HomImage : {𝑨 : Algebra 𝓀 𝑆}(𝑩 : Algebra 𝓦 𝑆)(Ο• : hom 𝑨 𝑩) β†’ ∣ 𝑩 ∣ β†’ 𝓀 βŠ” 𝓦 Μ‡
 HomImage 𝑩 Ο• = Ξ» b β†’ Image ∣ Ο• ∣ βˆ‹ b

 HomImagesOf : Algebra 𝓀 𝑆 β†’ π“ž βŠ” π“₯ βŠ” 𝓀 βŠ” 𝓦 ⁺ Μ‡
 HomImagesOf 𝑨 = Ξ£ 𝑩 κž‰ (Algebra 𝓦 𝑆) , Ξ£ Ο• κž‰ (∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣) , is-homomorphism 𝑨 𝑩 Ο• Γ— Epic Ο•

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 𝓀 𝑆, the type HomImagesOf 𝑨 denotes the class of algebras 𝑩 : Algebra 𝓦 𝑆 with a map Ο† : ∣ 𝑨 ∣ β†’ ∣ 𝑩 ∣ such that Ο† is a surjective homomorphism.

Since we take the class of homomorphic images of an algebra to be closed under isomorphism, we consider 𝑩 to be a homomorphic image of 𝑨 if there exists an algebra π‘ͺ which is a homomorphic image of 𝑨 and isomorphic to 𝑩. The following type expresses this.

 _is-hom-image-of_ : (𝑩 : Algebra 𝓦 𝑆)(𝑨 : Algebra 𝓀 𝑆) β†’ ov 𝓦 βŠ” 𝓀 Μ‡
 𝑩 is-hom-image-of 𝑨 = Ξ£ π‘ͺΟ• κž‰ (HomImagesOf 𝑨) , ∣ π‘ͺΟ• ∣ β‰… 𝑩

Images of a class of 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.

module _ {𝓀 : Universe} where

 _is-hom-image-of-class_ : Algebra 𝓀 𝑆 β†’ Pred (Algebra 𝓀 𝑆)(𝓀 ⁺) β†’ ov 𝓀 Μ‡
 𝑩 is-hom-image-of-class π“š = Ξ£ 𝑨 κž‰ (Algebra 𝓀 𝑆) , (𝑨 ∈ π“š) Γ— (𝑩 is-hom-image-of 𝑨)

 HomImagesOfClass : Pred (Algebra 𝓀 𝑆) (𝓀 ⁺) β†’ ov 𝓀 Μ‡
 HomImagesOfClass π“š = Ξ£ 𝑩 κž‰ (Algebra 𝓀 𝑆) , (𝑩 is-hom-image-of-class π“š)

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.

open Lift

module _ {𝓧 𝓨 : Universe} where

 lift-of-alg-epic-is-epic : (𝓩 : Universe){𝓦 : Universe}
                            {𝑨 : Algebra 𝓧 𝑆}(𝑩 : Algebra 𝓨 𝑆)(h : hom 𝑨 𝑩)
  β†’                         Epic ∣ h ∣  β†’  Epic ∣ Lift-hom 𝓩 𝓦 𝑩 h ∣

 lift-of-alg-epic-is-epic 𝓩 {𝓦} {𝑨} 𝑩 h hepi y = eq y (lift a) Ξ·
  lh : hom (Lift-alg 𝑨 𝓩) (Lift-alg 𝑩 𝓦)
  lh = Lift-hom 𝓩 𝓦 𝑩 h

  ΞΆ : Image ∣ h ∣ βˆ‹ (lower y)
  ΞΆ = hepi (lower y)

  a : ∣ 𝑨 ∣
  a = Inv ∣ h ∣ ΢

  Ξ² : lift (∣ h ∣ a) ≑ (lift ∘ ∣ h ∣ ∘ lower{𝓦}) (lift a)
  Ξ² = ap (Ξ» - β†’ lift (∣ h ∣ ( - a))) (lower∼lift {𝓦} )

  Ξ· : y ≑ ∣ lh ∣ (lift a)
  Ξ· = y               β‰‘βŸ¨ (happly lift∼lower) y ⟩
      lift (lower y)  β‰‘βŸ¨ ap lift (InvIsInv ∣ h ∣ ΞΆ)⁻¹ ⟩
      lift (∣ h ∣ a)  β‰‘βŸ¨ Ξ² ⟩
      ∣ lh ∣ (lift a) ∎

 Lift-alg-hom-image : {𝓩 𝓦 : Universe}
                      {𝑨 : Algebra 𝓧 𝑆}{𝑩 : Algebra 𝓨 𝑆}
  β†’                   𝑩 is-hom-image-of 𝑨
  β†’                   (Lift-alg 𝑩 𝓦) is-hom-image-of (Lift-alg 𝑨 𝓩)

 Lift-alg-hom-image {𝓩}{𝓦}{𝑨}{𝑩} ((π‘ͺ , Ο• , Ο•hom , Ο•epic) , Cβ‰…B) =
  (Lift-alg π‘ͺ 𝓦 , ∣ lΟ• ∣ , βˆ₯ lΟ• βˆ₯ , lΟ•epic) , Lift-alg-iso Cβ‰…B
   lΟ• : hom (Lift-alg 𝑨 𝓩) (Lift-alg π‘ͺ 𝓦)
   lΟ• = (Lift-hom 𝓩 𝓦 π‘ͺ) (Ο• , Ο•hom)

   lΟ•epic : Epic ∣ lΟ• ∣
   lΟ•epic = lift-of-alg-epic-is-epic 𝓩 π‘ͺ (Ο• , Ο•hom) Ο•epic

← Homomorphisms.Isomorphisms Terms β†’