Congruence relations on monoids

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

Created on 2022-08-18.
Last modified on 2023-11-24.

module group-theory.congruence-relations-monoids where
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.equivalence-relations
open import foundation.equivalences
open import foundation.identity-types
open import foundation.propositions
open import foundation.torsorial-type-families
open import foundation.universe-levels

open import group-theory.congruence-relations-semigroups
open import group-theory.monoids


A congruence relation on a monoid M is a congruence relation on the underlying semigroup of M.


is-congruence-prop-Monoid :
  {l1 l2 : Level} (M : Monoid l1) 
  equivalence-relation l2 (type-Monoid M)  Prop (l1  l2)
is-congruence-prop-Monoid M = is-congruence-prop-Semigroup (semigroup-Monoid M)

is-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) 
  equivalence-relation l2 (type-Monoid M)  UU (l1  l2)
is-congruence-Monoid M R =
  is-congruence-Semigroup (semigroup-Monoid M) R

is-prop-is-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1)
  (R : equivalence-relation l2 (type-Monoid M)) 
  is-prop (is-congruence-Monoid M R)
is-prop-is-congruence-Monoid M =
  is-prop-is-congruence-Semigroup (semigroup-Monoid M)

congruence-Monoid : {l : Level} (l2 : Level) (M : Monoid l)  UU (l  lsuc l2)
congruence-Monoid l2 M =
  Σ (equivalence-relation l2 (type-Monoid M)) (is-congruence-Monoid M)

module _
  {l1 l2 : Level} (M : Monoid l1) (R : congruence-Monoid l2 M)

  equivalence-relation-congruence-Monoid :
    equivalence-relation l2 (type-Monoid M)
  equivalence-relation-congruence-Monoid = pr1 R

  prop-congruence-Monoid : Relation-Prop l2 (type-Monoid M)
  prop-congruence-Monoid =
    prop-equivalence-relation equivalence-relation-congruence-Monoid

  sim-congruence-Monoid : (x y : type-Monoid M)  UU l2
  sim-congruence-Monoid =
    sim-equivalence-relation equivalence-relation-congruence-Monoid

  is-prop-sim-congruence-Monoid :
    (x y : type-Monoid M)  is-prop (sim-congruence-Monoid x y)
  is-prop-sim-congruence-Monoid =
    is-prop-sim-equivalence-relation equivalence-relation-congruence-Monoid

  concatenate-sim-eq-congruence-Monoid :
    {x y z : type-Monoid M}  sim-congruence-Monoid x y  y  z 
    sim-congruence-Monoid x z
  concatenate-sim-eq-congruence-Monoid H refl = H

  concatenate-eq-sim-congruence-Monoid :
    {x y z : type-Monoid M}  x  y  sim-congruence-Monoid y z 
    sim-congruence-Monoid x z
  concatenate-eq-sim-congruence-Monoid refl H = H

  concatenate-eq-sim-eq-congruence-Monoid :
    {x y z w : type-Monoid M}  x  y  sim-congruence-Monoid y z 
    z  w  sim-congruence-Monoid x w
  concatenate-eq-sim-eq-congruence-Monoid refl H refl = H

  refl-congruence-Monoid : is-reflexive sim-congruence-Monoid
  refl-congruence-Monoid =
    refl-equivalence-relation equivalence-relation-congruence-Monoid

  symmetric-congruence-Monoid : is-symmetric sim-congruence-Monoid
  symmetric-congruence-Monoid =
    symmetric-equivalence-relation equivalence-relation-congruence-Monoid

  equiv-symmetric-congruence-Monoid :
    (x y : type-Monoid M) 
    sim-congruence-Monoid x y  sim-congruence-Monoid y x
  equiv-symmetric-congruence-Monoid x y =
    equiv-symmetric-equivalence-relation equivalence-relation-congruence-Monoid

  transitive-congruence-Monoid : is-transitive sim-congruence-Monoid
  transitive-congruence-Monoid =
    transitive-equivalence-relation equivalence-relation-congruence-Monoid

  mul-congruence-Monoid :
    is-congruence-Monoid M equivalence-relation-congruence-Monoid
  mul-congruence-Monoid = pr2 R


Extensionality of the type of congruence relations on a monoid

relate-same-elements-congruence-Monoid :
  {l1 l2 l3 : Level} (M : Monoid l1) (R : congruence-Monoid l2 M)
  (S : congruence-Monoid l3 M)  UU (l1  l2  l3)
relate-same-elements-congruence-Monoid M =
    ( semigroup-Monoid M)

refl-relate-same-elements-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) (R : congruence-Monoid l2 M) 
  relate-same-elements-congruence-Monoid M R R
refl-relate-same-elements-congruence-Monoid M =
  refl-relate-same-elements-congruence-Semigroup (semigroup-Monoid M)

is-torsorial-relate-same-elements-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) (R : congruence-Monoid l2 M) 
  is-torsorial (relate-same-elements-congruence-Monoid M R)
is-torsorial-relate-same-elements-congruence-Monoid M =
  is-torsorial-relate-same-elements-congruence-Semigroup (semigroup-Monoid M)

relate-same-elements-eq-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) (R S : congruence-Monoid l2 M) 
  R  S  relate-same-elements-congruence-Monoid M R S
relate-same-elements-eq-congruence-Monoid M =
  relate-same-elements-eq-congruence-Semigroup (semigroup-Monoid M)

is-equiv-relate-same-elements-eq-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) (R S : congruence-Monoid l2 M) 
  is-equiv (relate-same-elements-eq-congruence-Monoid M R S)
is-equiv-relate-same-elements-eq-congruence-Monoid M =
  is-equiv-relate-same-elements-eq-congruence-Semigroup (semigroup-Monoid M)

extensionality-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) (R S : congruence-Monoid l2 M) 
  (R  S)  relate-same-elements-congruence-Monoid M R S
extensionality-congruence-Monoid M =
  extensionality-congruence-Semigroup (semigroup-Monoid M)

eq-relate-same-elements-congruence-Monoid :
  {l1 l2 : Level} (M : Monoid l1) (R S : congruence-Monoid l2 M) 
  relate-same-elements-congruence-Monoid M R S  R  S
eq-relate-same-elements-congruence-Monoid M =
    ( semigroup-Monoid M)

Recent changes