Binary relations with extensions

Content created by Fredrik Bakke.

Created on 2024-04-11.
Last modified on 2024-04-11.

module foundation.binary-relations-with-extensions where
Imports 
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.iterated-dependent-product-types
open import foundation.universe-levels

open import foundation-core.propositions

Idea

We say a relation R has extensions if for every triple x y z : A, there is a binary operation

  R x y → R x z → R y z.

Relations with extensions are closely related to transitive relations. But, instead of giving for every diagram

       y
      ∧ \
     /   \
    /     ∨
  x        z

a horizontal arrow x → z, a binary relation with extensions gives, for every span

       y
      ∧
     /
    /
  x -----> z,

an extension y → z. By symmetry it also gives an extension in the opposite direction z → y.

Dually, a relation R has lifts if for every triple x y z : A, there is a binary operation

  R x z → R y z → R x y.

Definition

The structure on relations of having extensions

module _
  {l1 l2 : Level} {A : UU l1} (R : Relation l2 A)
  where

  has-extensions-Relation : UU (l1 ⊔ l2)
  has-extensions-Relation = {x y z : A} → R x y → R x z → R y z

Properties

If there is an element that relates to y and the relation has extensions, then y relates to y

module _
  {l1 l2 : Level} {A : UU l1} (R : Relation l2 A)
  where

  rel-self-any-rel-has-extensions-Relation :
    has-extensions-Relation R → {x y : A} → R x y → R y y
  rel-self-any-rel-has-extensions-Relation H p = H p p

The reverse of an extension

module _
  {l1 l2 : Level} {A : UU l1} (R : Relation l2 A)
  where

  reverse-has-extensions-Relation :
    has-extensions-Relation R → {x y z : A} → R z x → R z y → R y x
  reverse-has-extensions-Relation H p q = H q p

Reflexive relations with extensions are symmetric

module _
  {l1 l2 : Level} {A : UU l1} (R : Relation l2 A)
  (H : has-extensions-Relation R)
  where

  is-symmetric-is-reflexive-has-extensions-Relation :
    is-reflexive R → is-symmetric R
  is-symmetric-is-reflexive-has-extensions-Relation r x y p = H p (r x)

Reflexive relations with extensions are transitive

module _
  {l1 l2 : Level} {A : UU l1} (R : Relation l2 A)
  (H : has-extensions-Relation R)
  where

  is-transitive-is-symmetric-has-extensions-Relation :
    is-symmetric R → is-transitive R
  is-transitive-is-symmetric-has-extensions-Relation s x y z p q = H (s x y q) p

  is-transitive-is-reflexive-has-extensions-Relation :
    is-reflexive R → is-transitive R
  is-transitive-is-reflexive-has-extensions-Relation r =
    is-transitive-is-symmetric-has-extensions-Relation
      ( is-symmetric-is-reflexive-has-extensions-Relation R H r)

See also

Recent changes