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Kernels of Homomorphisms

This is the Base.Homomorphisms.Kernels module of the Agda Universal Algebra Library.

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

open import Overture using ( Signature; 𝓞 ; 𝓥 )

module Base.Homomorphisms.Kernels {𝑆 : Signature 𝓞 𝓥} where

-- Imports from Agda and the Agda Standard Library --------------------------------
open import Data.Product   using ( _,_ )
open import Function.Base  using ( _∘_ )
open import Level          using ( Level ; _⊔_ ; suc )

open  import Relation.Binary.PropositionalEquality
      using ( _≡_ ; module ≡-Reasoning ; refl )

-- Imports from the Agda Universal Algebras Library --------------------------------
open import Overture        using ( ∣_∣ ; ∥_∥ ; _⁻¹ )
open import Base.Functions  using ( Image_∋_ ; IsSurjective )
open import Base.Equality   using ( swelldef )
open import Base.Relations  using ( ker ; ker-IsEquivalence ; ⟪_⟫ ; mkblk )

open  import Base.Algebras {𝑆 = 𝑆}
      using ( Algebra ; compatible ; _̂_ ; Con ; mkcon ; _╱_ ; IsCongruence ; /-≡ )

open import Base.Homomorphisms.Basic {𝑆 = 𝑆}  using ( hom ; epi ; epi→hom )

private variable α β : Level

Definition

The kernel of a homomorphism is a congruence relation and conversely for every congruence relation θ, there exists a homomorphism with kernel θ (namely, that canonical projection onto the quotient modulo θ).

module _ {𝑨 : Algebra α 𝑆} where
 open ≡-Reasoning
 homker-comp :  swelldef 𝓥 β → {𝑩 : Algebra β 𝑆}(h : hom 𝑨 𝑩)
  →             compatible 𝑨 (ker ∣ h ∣)

 homker-comp wd {𝑩} h f {u}{v} kuv =
  ∣ h ∣((f ̂ 𝑨) u)    ≡⟨ ∥ h ∥ f u ⟩
  (f ̂ 𝑩)(∣ h ∣ ∘ u)  ≡⟨ wd(f ̂ 𝑩)(∣ h ∣ ∘ u)(∣ h ∣ ∘ v)kuv ⟩
  (f ̂ 𝑩)(∣ h ∣ ∘ v)  ≡⟨ (∥ h ∥ f v)⁻¹ ⟩
  ∣ h ∣((f ̂ 𝑨) v)    ∎

(Notice, it is here that the swelldef postulate comes into play, and because it is needed to prove homker-comp, it is postulated by all the lemmas below that depend upon homker-comp.)

It is convenient to define a function that takes a homomorphism and constructs a congruence from its kernel. We call this function kercon.

 kercon : swelldef 𝓥 β → {𝑩 : Algebra β 𝑆} → hom 𝑨 𝑩 → Con{α}{β} 𝑨
 kercon wd {𝑩} h = ker ∣ h ∣ , mkcon (ker-IsEquivalence ∣ h ∣)(homker-comp wd {𝑩} h)

With this congruence we construct the corresponding quotient, along with some syntactic sugar to denote it.

 kerquo : swelldef 𝓥 β → {𝑩 : Algebra β 𝑆} → hom 𝑨 𝑩 → Algebra (α ⊔ suc β) 𝑆
 kerquo wd {𝑩} h = 𝑨 ╱ (kercon wd {𝑩} h)

ker[_⇒_]_↾_ :  (𝑨 : Algebra α 𝑆)(𝑩 : Algebra β 𝑆) → hom 𝑨 𝑩 → swelldef 𝓥 β
 →             Algebra (α ⊔ suc β) 𝑆

ker[ 𝑨 ⇒ 𝑩 ] h ↾ wd = kerquo wd {𝑩} h

Thus, given h : hom 𝑨 𝑩, we can construct the quotient of 𝑨 modulo the kernel of h, and the syntax for this quotient in the agda-algebras library is 𝑨 [ 𝑩 ]/ker h ↾ fe.

The canonical projection

Given an algebra 𝑨 and a congruence θ, the canonical projection is a map from 𝑨 onto 𝑨 ╱ θ that is constructed, and proved epimorphic, as follows.

module _ {α β : Level}{𝑨 : Algebra α 𝑆} where
 πepi : (θ : Con{α}{β} 𝑨) → epi 𝑨 (𝑨 ╱ θ)
 πepi θ = (λ a → ⟪ a ⟫) , (λ _ _ → refl) , cπ-is-epic  where
  cπ-is-epic : IsSurjective (λ a → ⟪ a ⟫)
  cπ-is-epic (C , mkblk a refl ) =  Image_∋_.eq a refl

In may happen that we don’t care about the surjectivity of πepi, in which case would might prefer to work with the homomorphic reduct of πepi. This is obtained by applying epi-to-hom, like so.

 πhom : (θ : Con{α}{β} 𝑨) → hom 𝑨 (𝑨 ╱ θ)
 πhom θ = epi→hom (𝑨 ╱ θ) (πepi θ)

We combine the foregoing to define a function that takes 𝑆-algebras 𝑨 and 𝑩, and a homomorphism h : hom 𝑨 𝑩 and returns the canonical epimorphism from 𝑨 onto 𝑨 [ 𝑩 ]/ker h. (Recall, the latter is the special notation we defined above for the quotient of 𝑨 modulo the kernel of h.)

 πker :  (wd : swelldef 𝓥 β){𝑩 : Algebra β 𝑆}(h : hom 𝑨 𝑩)
  →      epi 𝑨 (ker[ 𝑨 ⇒ 𝑩 ] h ↾ wd)

 πker wd {𝑩} h = πepi (kercon wd {𝑩} h)

The kernel of the canonical projection of 𝑨 onto 𝑨 / θ is equal to θ, but since equality of inhabitants of certain types (like Congruence or Rel) can be a tricky business, we settle for proving the containment 𝑨 / θ ⊆ θ. Of the two containments, this is the easier one to prove; luckily it is also the one we need later.

 open IsCongruence

 ker-in-con :  {wd : swelldef 𝓥 (α ⊔ suc β)}(θ : Con 𝑨)
  →            ∀ {x}{y} → ∣ kercon wd {𝑨 ╱ θ} (πhom θ) ∣ x y →  ∣ θ ∣ x y

 ker-in-con θ hyp = /-≡ θ hyp

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← Base.Homomorphisms.Properties Base.Homomorphisms.Products →