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

Created on 2022-10-18.
Last modified on 2024-01-11.

module order-theory.frames where
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

open import order-theory.greatest-lower-bounds-posets
open import order-theory.least-upper-bounds-posets
open import
open import
open import order-theory.posets
open import order-theory.suplattices


A frame is a meet-suplattice with arbitrary joins in which meets distribute over suprema. The distributive law for meets over suprema states that in any meet-suplattice A, we have

  x ∧ (⋁ᵢ yᵢ) = ⋁ᵢ (x ∧ yᵢ)

for every element x : A and any family y : I → A.


Statement of (instances of) the infinite distributive law

In posets

module _
  {l1 l2 l3 : Level} (P : Poset l1 l2)

  module _
    {I : UU l3} {x : type-Poset P} {y : I  type-Poset P}
    (H : has-least-upper-bound-family-of-elements-Poset P y)
    (K : has-greatest-binary-lower-bound-Poset P x (pr1 H))
    (L : (i : I)  has-greatest-binary-lower-bound-Poset P x (y i))
    (M : has-least-upper-bound-family-of-elements-Poset P  i  (pr1 (L i))))

    instance-distributive-law-meet-sup-Poset-Prop : Prop l1
    instance-distributive-law-meet-sup-Poset-Prop =
      Id-Prop (set-Poset P) (pr1 K) (pr1 M)

    instance-distributive-law-meet-sup-Poset : UU l1
    instance-distributive-law-meet-sup-Poset =
      type-Prop instance-distributive-law-meet-sup-Poset-Prop

    is-prop-instance-distributive-law-meet-sup-Poset :
      is-prop instance-distributive-law-meet-sup-Poset
    is-prop-instance-distributive-law-meet-sup-Poset =
      is-prop-type-Prop instance-distributive-law-meet-sup-Poset-Prop

  module _
    ( H : is-meet-semilattice-Poset P)
    ( K : is-suplattice-Poset l3 P)

    distributive-law-meet-sup-Poset-Prop : Prop (l1  lsuc l3)
    distributive-law-meet-sup-Poset-Prop =
        ( type-Poset P)
        ( λ x 
            ( UU l3)
            ( λ I 
                ( I  type-Poset P)
                ( λ y 
                  instance-distributive-law-meet-sup-Poset-Prop {I} {x} {y}
                    ( K I y)
                    ( H x (pr1 (K I y)))
                    ( λ i  H x (y i))
                    ( K I  i  pr1 (H x (y i)))))))

    distributive-law-meet-sup-Poset : UU (l1  lsuc l3)
    distributive-law-meet-sup-Poset =
      type-Prop distributive-law-meet-sup-Poset-Prop

    is-prop-distributive-law-meet-sup-Poset :
      is-prop distributive-law-meet-sup-Poset
    is-prop-distributive-law-meet-sup-Poset =
      is-prop-type-Prop distributive-law-meet-sup-Poset-Prop

In meet-semilattices

instance-distributive-law-meet-sup-Meet-Semilattice :
  {l1 l2 : Level} (L : Meet-Semilattice l1) {I : UU l2}
  ( x : type-Meet-Semilattice L)
  { y : I  type-Meet-Semilattice L} 
  ( H :
      ( poset-Meet-Semilattice L)
      ( y))
  ( K :
      ( poset-Meet-Semilattice L)
      ( λ i  meet-Meet-Semilattice L x (y i))) 
  UU l1
instance-distributive-law-meet-sup-Meet-Semilattice L x {y} H =
    ( poset-Meet-Semilattice L)
    ( H)
    ( has-greatest-binary-lower-bound-Meet-Semilattice L x (pr1 H))
    ( λ i  has-greatest-binary-lower-bound-Meet-Semilattice L x (y i))

Statement of the distributive law in meet-suplattices

module _
  {l1 l2 : Level} (L : Meet-Suplattice l1 l2)

    _∧_ = meet-Meet-Suplattice L
    ⋁_ = sup-Meet-Suplattice L

  distributive-law-Meet-Suplattice-Prop : Prop (l1  lsuc l2)
  distributive-law-Meet-Suplattice-Prop =
      ( type-Meet-Suplattice L)
      ( λ x 
          ( UU l2)
          ( λ I 
              ( I  type-Meet-Suplattice L)
              ( λ y 
                  ( set-Meet-Suplattice L)
                  ( x  ( y))
                  (   i  (x  (y i)))))))

  distributive-law-Meet-Suplattice : UU (l1  lsuc l2)
  distributive-law-Meet-Suplattice =
    type-Prop distributive-law-Meet-Suplattice-Prop

  is-prop-distributive-law-Meet-Suplattice :
    is-prop distributive-law-Meet-Suplattice
  is-prop-distributive-law-Meet-Suplattice =
    is-prop-type-Prop distributive-law-Meet-Suplattice-Prop

The predicate on meet-suplattices to be a frame

module _
  {l1 l2 : Level} (L : Meet-Suplattice l1 l2)

  is-frame-Meet-Suplattice-Prop : Prop (l1  lsuc l2)
  is-frame-Meet-Suplattice-Prop = distributive-law-Meet-Suplattice-Prop L

  is-frame-Meet-Suplattice : UU (l1  lsuc l2)
  is-frame-Meet-Suplattice = type-Prop is-frame-Meet-Suplattice-Prop

  is-prop-is-frame-Meet-Suplattice : is-prop is-frame-Meet-Suplattice
  is-prop-is-frame-Meet-Suplattice =
    is-prop-type-Prop is-frame-Meet-Suplattice-Prop


Frame : (l1 l2 : Level)  UU (lsuc l1  lsuc l2)
Frame l1 l2 = Σ (Meet-Suplattice l1 l2) is-frame-Meet-Suplattice

module _
  {l1 l2 : Level} (A : Frame l1 l2)

  meet-suplattice-Frame : Meet-Suplattice l1 l2
  meet-suplattice-Frame = pr1 A

  meet-semilattice-Frame : Meet-Semilattice l1
  meet-semilattice-Frame =
    meet-semilattice-Meet-Suplattice meet-suplattice-Frame

  suplattice-Frame : Suplattice l1 l1 l2
  suplattice-Frame = suplattice-Meet-Suplattice meet-suplattice-Frame

  poset-Frame : Poset l1 l1
  poset-Frame = poset-Meet-Suplattice meet-suplattice-Frame

  set-Frame : Set l1
  set-Frame = set-Poset poset-Frame

  type-Frame : UU l1
  type-Frame = type-Poset poset-Frame

  is-set-type-Frame : is-set type-Frame
  is-set-type-Frame = is-set-type-Poset poset-Frame

  leq-Frame-Prop : (x y : type-Frame)  Prop l1
  leq-Frame-Prop = leq-Poset-Prop poset-Frame

  leq-Frame : (x y : type-Frame)  UU l1
  leq-Frame = leq-Poset poset-Frame

  is-prop-leq-Frame : (x y : type-Frame)  is-prop (leq-Frame x y)
  is-prop-leq-Frame = is-prop-leq-Poset poset-Frame

  refl-leq-Frame : is-reflexive leq-Frame
  refl-leq-Frame = refl-leq-Poset poset-Frame

  antisymmetric-leq-Frame : is-antisymmetric leq-Frame
  antisymmetric-leq-Frame = antisymmetric-leq-Poset poset-Frame

  transitive-leq-Frame : is-transitive leq-Frame
  transitive-leq-Frame = transitive-leq-Poset poset-Frame

  meet-Frame : type-Frame  type-Frame  type-Frame
  meet-Frame = meet-Meet-Semilattice meet-semilattice-Frame

  is-greatest-binary-lower-bound-meet-Frame :
    (x y : type-Frame) 
    is-greatest-binary-lower-bound-Poset poset-Frame x y (meet-Frame x y)
  is-greatest-binary-lower-bound-meet-Frame =
    is-greatest-binary-lower-bound-meet-Meet-Semilattice meet-semilattice-Frame

  associative-meet-Frame :
    (x y z : type-Frame) 
    meet-Frame (meet-Frame x y) z  meet-Frame x (meet-Frame y z)
  associative-meet-Frame =
    associative-meet-Meet-Semilattice meet-semilattice-Frame

  commutative-meet-Frame :
    (x y : type-Frame)  meet-Frame x y  meet-Frame y x
  commutative-meet-Frame =
    commutative-meet-Meet-Semilattice meet-semilattice-Frame

  idempotent-meet-Frame :
    (x : type-Frame)  meet-Frame x x  x
  idempotent-meet-Frame =
    idempotent-meet-Meet-Semilattice meet-semilattice-Frame

  is-suplattice-Frame :
    is-suplattice-Poset l2 poset-Frame
  is-suplattice-Frame = is-suplattice-Suplattice suplattice-Frame

  sup-Frame : {I : UU l2}  (I  type-Frame)  type-Frame
  sup-Frame = sup-Suplattice suplattice-Frame

  is-least-upper-bound-sup-Frame :
    {I : UU l2} (x : I  type-Frame) 
    is-least-upper-bound-family-of-elements-Poset poset-Frame x (sup-Frame x)
  is-least-upper-bound-sup-Frame =
    is-least-upper-bound-sup-Suplattice suplattice-Frame

  distributive-meet-sup-Frame :
    distributive-law-Meet-Suplattice meet-suplattice-Frame
  distributive-meet-sup-Frame = pr2 A

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