Kernels of homomorphisms of algebras

Content created by Fredrik Bakke, Egbert Rijke, Fernando Chu, Julian KG, fernabnor and louismntnu.

Created on 2023-03-28.
Last modified on 2023-06-25.

module universal-algebra.kernels where
Imports 
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.equivalence-relations
open import foundation.identity-types
open import foundation.universe-levels

open import linear-algebra.functoriality-vectors
open import linear-algebra.vectors

open import universal-algebra.algebraic-theories
open import universal-algebra.algebras-of-theories
open import universal-algebra.congruences
open import universal-algebra.homomorphisms-of-algebras
open import universal-algebra.signatures

Idea

The kernel of a homomorphism f of algebras is the congruence relation given by x ~ y iff f x ~ f y.

Definitions

module _
  { l1 : Level} ( Sg : signature l1)
  { l2 : Level} ( Th : Theory Sg l2)
  { l3 l4 : Level}
  ( Alg1 : Algebra Sg Th l3)
  ( Alg2 : Algebra Sg Th l4)
  ( F : type-hom-Algebra Sg Th Alg1 Alg2)
  where

  rel-prop-kernel-hom-Algebra :
    Relation-Prop l4 (type-Algebra Sg Th Alg1)
  pr1 (rel-prop-kernel-hom-Algebra x y) =
    map-hom-Algebra Sg Th Alg1 Alg2 F x ＝
      map-hom-Algebra Sg Th Alg1 Alg2 F y
  pr2 (rel-prop-kernel-hom-Algebra x y) =
    is-set-Algebra Sg Th Alg2 _ _

  eq-rel-kernel-hom-Algebra :
    Equivalence-Relation l4 (type-Algebra Sg Th Alg1)
  pr1 eq-rel-kernel-hom-Algebra =
    rel-prop-kernel-hom-Algebra
  pr1 (pr2 eq-rel-kernel-hom-Algebra) _ = refl
  pr1 (pr2 (pr2 eq-rel-kernel-hom-Algebra)) _ _ = inv
  pr2 (pr2 (pr2 eq-rel-kernel-hom-Algebra)) _ _ _ f g = g ∙ f

  kernel-hom-Algebra :
    congruence-Algebra Sg Th Alg1 l4
  pr1 kernel-hom-Algebra = eq-rel-kernel-hom-Algebra
  pr2 kernel-hom-Algebra op v v' p =
    equational-reasoning
      f (is-model-set-Algebra Sg Th Alg1 op v)
      ＝ is-model-set-Algebra Sg Th Alg2 op (map-vec f v)
        by preserves-operations-hom-Algebra Sg Th Alg1 Alg2 F op v
      ＝ is-model-set-Algebra Sg Th Alg2 op (map-vec f v')
        by
          ap
            ( is-model-set-Algebra Sg Th Alg2 op)
            ( map-hom-Algebra-lemma (pr2 Sg op) v v' p)
      ＝ f (is-model-set-Algebra Sg Th Alg1 op v')
        by inv (preserves-operations-hom-Algebra Sg Th Alg1 Alg2 F op v')
    where
    f = map-hom-Algebra Sg Th Alg1 Alg2 F
    map-hom-Algebra-lemma :
      ( n : ℕ) →
      ( v v' : vec (type-Algebra Sg Th Alg1) n) →
      ( relation-holds-all-vec Sg Th Alg1 eq-rel-kernel-hom-Algebra v v') →
      (map-vec f v) ＝ (map-vec f v')
    map-hom-Algebra-lemma zero-ℕ empty-vec empty-vec p = refl
    map-hom-Algebra-lemma (succ-ℕ n) (x ∷ v) (x' ∷ v') (p , p') =
      ap-binary (_∷_) p (map-hom-Algebra-lemma n v v' p')

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