Greatest lower bounds in large posets

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

Idea

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

holds.

Definitions

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)
  where

  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

Properties

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)
  where

  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 =
    logical-equivalence-reasoning
      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))
          by
          inv-iff
            ( logical-equivalence-is-binary-lower-bound-Π-Large-Poset P x y u)
        ↔ leq-Π-Large-Poset P u z
          by iff-Π (λ i → H i (u i))