Greatest lower bounds in large posets

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-05-09.
Last modified on 2023-09-11.

module order-theory.greatest-lower-bounds-large-posets where
open import foundation.logical-equivalences
open import foundation.universe-levels

open import order-theory.dependent-products-large-posets
open import order-theory.large-posets
open import order-theory.lower-bounds-large-posets


A greatest binary lower bound of two elements a and b in a large poset P is an element x such that for every element y in P the logical equivalence

  is-binary-lower-bound-Large-Poset P a b y ↔ y ≤ x



The predicate that an element of a large poset is the greatest lower bound of two elements

module _
  {α : Level  Level} {β : Level  Level  Level} (P : Large-Poset α β)
  {l1 l2 : Level} (a : type-Large-Poset P l1) (b : type-Large-Poset P l2)

  is-greatest-binary-lower-bound-Large-Poset :
    {l3 : Level}  type-Large-Poset P l3  UUω
  is-greatest-binary-lower-bound-Large-Poset x =
    {l4 : Level} (y : type-Large-Poset P l4) 
    is-binary-lower-bound-Large-Poset P a b y  leq-Large-Poset P y x


Any pointwise greatest lower bound of two elements in a dependent product of large posets is a greatest lower bound

module _
  {α : Level  Level} {β : Level  Level  Level}
  {l : Level} {I : UU l} (P : I  Large-Poset α β)
  {l1 l2 l3 : Level}
  (x : type-Π-Large-Poset P l1)
  (y : type-Π-Large-Poset P l2)
  (z : type-Π-Large-Poset P l3)

  is-greatest-binary-lower-bound-Π-Large-Poset :
    ( (i : I) 
      is-greatest-binary-lower-bound-Large-Poset (P i) (x i) (y i) (z i)) 
    is-greatest-binary-lower-bound-Large-Poset (Π-Large-Poset P) x y z
  is-greatest-binary-lower-bound-Π-Large-Poset H u =
      is-binary-lower-bound-Large-Poset (Π-Large-Poset P) x y u
         ((i : I)  is-binary-lower-bound-Large-Poset (P i) (x i) (y i) (u i))
            ( logical-equivalence-is-binary-lower-bound-Π-Large-Poset P x y u)
         leq-Π-Large-Poset P u z
          by iff-Π-iff-family  i  H i (u i))

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