Congruence relations on semirings

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Julian KG, fernabnor, louismntnu and maybemabeline.

Created on 2023-03-09.
Last modified on 2023-11-24.

module ring-theory.congruence-relations-semirings where
Imports 
open import foundation.binary-relations
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.equivalence-relations
open import foundation.equivalences
open import foundation.fundamental-theorem-of-identity-types
open import foundation.identity-types
open import foundation.propositions
open import foundation.subtype-identity-principle
open import foundation.torsorial-type-families
open import foundation.universe-levels

open import group-theory.congruence-relations-monoids

open import ring-theory.semirings

Idea

A congruence relation on a semiring R is a congruence relation on the underlying additive monoid of R which is also a congruence relation on the multiplicative monoid of R.

Definition

module _
  {l1 : Level} (R : Semiring l1)
  where

  is-congruence-Semiring :
    {l2 : Level}
    (S : congruence-Monoid l2 (additive-monoid-Semiring R)) → UU (l1 ⊔ l2)
  is-congruence-Semiring S =
    is-congruence-Monoid
      ( multiplicative-monoid-Semiring R)
      ( equivalence-relation-congruence-Monoid (additive-monoid-Semiring R) S)

  is-prop-is-congruence-Semiring :
    {l2 : Level} (S : congruence-Monoid l2 (additive-monoid-Semiring R)) →
    is-prop (is-congruence-Semiring S)
  is-prop-is-congruence-Semiring S =
    is-prop-is-congruence-Monoid
      ( multiplicative-monoid-Semiring R)
      ( equivalence-relation-congruence-Monoid (additive-monoid-Semiring R) S)

  is-congruence-equivalence-relation-Semiring :
    {l2 : Level} (S : equivalence-relation l2 (type-Semiring R)) → UU (l1 ⊔ l2)
  is-congruence-equivalence-relation-Semiring S =
    ( is-congruence-Monoid (additive-monoid-Semiring R) S) ×
    ( is-congruence-Monoid (multiplicative-monoid-Semiring R) S)

congruence-Semiring :
  {l1 : Level} (l2 : Level) (R : Semiring l1) → UU (l1 ⊔ lsuc l2)
congruence-Semiring l2 R =
  Σ ( congruence-Monoid l2 (additive-monoid-Semiring R))
    ( is-congruence-Semiring R)

module _
  {l1 l2 : Level} (R : Semiring l1) (S : congruence-Semiring l2 R)
  where

  congruence-additive-monoid-congruence-Semiring :
    congruence-Monoid l2 (additive-monoid-Semiring R)
  congruence-additive-monoid-congruence-Semiring = pr1 S

  equivalence-relation-congruence-Semiring :
    equivalence-relation l2 (type-Semiring R)
  equivalence-relation-congruence-Semiring =
    equivalence-relation-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  prop-congruence-Semiring : Relation-Prop l2 (type-Semiring R)
  prop-congruence-Semiring =
    prop-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  sim-congruence-Semiring : (x y : type-Semiring R) → UU l2
  sim-congruence-Semiring =
    sim-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  is-prop-sim-congruence-Semiring :
    (x y : type-Semiring R) → is-prop (sim-congruence-Semiring x y)
  is-prop-sim-congruence-Semiring =
    is-prop-sim-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  refl-congruence-Semiring :
    is-reflexive sim-congruence-Semiring
  refl-congruence-Semiring =
    refl-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  symmetric-congruence-Semiring :
    is-symmetric sim-congruence-Semiring
  symmetric-congruence-Semiring =
    symmetric-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  equiv-symmetric-congruence-Semiring :
    (x y : type-Semiring R) →
    sim-congruence-Semiring x y ≃ sim-congruence-Semiring y x
  equiv-symmetric-congruence-Semiring =
    equiv-symmetric-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  transitive-congruence-Semiring :
    is-transitive sim-congruence-Semiring
  transitive-congruence-Semiring =
    transitive-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  add-congruence-Semiring :
    is-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( equivalence-relation-congruence-Semiring)
  add-congruence-Semiring =
    mul-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring)

  mul-congruence-Semiring :
    is-congruence-Monoid
      ( multiplicative-monoid-Semiring R)
      ( equivalence-relation-congruence-Semiring)
  mul-congruence-Semiring = pr2 S

construct-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) →
  (S : equivalence-relation l2 (type-Semiring R)) →
  is-congruence-Monoid (additive-monoid-Semiring R) S →
  is-congruence-Monoid (multiplicative-monoid-Semiring R) S →
  congruence-Semiring l2 R
pr1 (pr1 (construct-congruence-Semiring R S H K)) = S
pr2 (pr1 (construct-congruence-Semiring R S H K)) = H
pr2 (construct-congruence-Semiring R S H K) = K

Properties

Characterizing equality of congruence relations of semirings

relate-same-elements-congruence-Semiring :
  {l1 l2 l3 : Level} (R : Semiring l1) →
  congruence-Semiring l2 R → congruence-Semiring l3 R → UU (l1 ⊔ l2 ⊔ l3)
relate-same-elements-congruence-Semiring R S T =
  relate-same-elements-equivalence-relation
    ( equivalence-relation-congruence-Semiring R S)
    ( equivalence-relation-congruence-Semiring R T)

refl-relate-same-elements-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (S : congruence-Semiring l2 R) →
  relate-same-elements-congruence-Semiring R S S
refl-relate-same-elements-congruence-Semiring R S =
  refl-relate-same-elements-equivalence-relation
    ( equivalence-relation-congruence-Semiring R S)

is-torsorial-relate-same-elements-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (S : congruence-Semiring l2 R) →
  is-torsorial (relate-same-elements-congruence-Semiring R S)
is-torsorial-relate-same-elements-congruence-Semiring R S =
  is-torsorial-Eq-subtype
    ( is-torsorial-relate-same-elements-congruence-Monoid
      ( additive-monoid-Semiring R)
      ( congruence-additive-monoid-congruence-Semiring R S))
    ( is-prop-is-congruence-Semiring R)
    ( congruence-additive-monoid-congruence-Semiring R S)
    ( refl-relate-same-elements-congruence-Semiring R S)
    ( mul-congruence-Semiring R S)

relate-same-elements-eq-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (S T : congruence-Semiring l2 R) →
  S ＝ T → relate-same-elements-congruence-Semiring R S T
relate-same-elements-eq-congruence-Semiring R S .S refl =
  refl-relate-same-elements-congruence-Semiring R S

is-equiv-relate-same-elements-eq-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (S T : congruence-Semiring l2 R) →
  is-equiv (relate-same-elements-eq-congruence-Semiring R S T)
is-equiv-relate-same-elements-eq-congruence-Semiring R S =
    fundamental-theorem-id
      ( is-torsorial-relate-same-elements-congruence-Semiring R S)
      ( relate-same-elements-eq-congruence-Semiring R S)

extensionality-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (S T : congruence-Semiring l2 R) →
  (S ＝ T) ≃ relate-same-elements-congruence-Semiring R S T
pr1 (extensionality-congruence-Semiring R S T) =
  relate-same-elements-eq-congruence-Semiring R S T
pr2 (extensionality-congruence-Semiring R S T) =
  is-equiv-relate-same-elements-eq-congruence-Semiring R S T

eq-relate-same-elements-congruence-Semiring :
  {l1 l2 : Level} (R : Semiring l1) (S T : congruence-Semiring l2 R) →
  relate-same-elements-congruence-Semiring R S T → S ＝ T
eq-relate-same-elements-congruence-Semiring R S T =
  map-inv-equiv (extensionality-congruence-Semiring R S T)

Recent changes