Lower bounds in large posets

module order-theory.lower-bounds-large-posets where

Imports

open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.logical-equivalences
open import foundation.universe-levels

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

Idea

A binary lower bound of two elements a and b in a large poset P is an element x such that both x ≤ a and x ≤ b hold. Similarly, a lower bound of a family of elements a : I → P in a large poset P is an element x such that x ≤ a i holds for every i : I.

Definitions

The predicate that an element of a large poset is a 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-binary-lower-bound-Large-Poset :
    {l3 : Level} → type-Large-Poset P l3 → UU (β l3 l1 ⊔ β l3 l2)
  is-binary-lower-bound-Large-Poset x =
    leq-Large-Poset P x a × leq-Large-Poset P x b

Properties

An element of a dependent product of large posets is a binary lower bound of two elements if and only if it is a pointwise binary lower bound

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

  is-binary-lower-bound-Π-Large-Poset :
    {l3 : Level} (z : type-Π-Large-Poset P l3) →
    ((i : I) → is-binary-lower-bound-Large-Poset (P i) (x i) (y i) (z i)) →
    is-binary-lower-bound-Large-Poset (Π-Large-Poset P) x y z
  pr1 (is-binary-lower-bound-Π-Large-Poset z H) i = pr1 (H i)
  pr2 (is-binary-lower-bound-Π-Large-Poset z H) i = pr2 (H i)

  is-binary-lower-bound-is-binary-lower-bound-Π-Large-Poset :
    {l3 : Level} (z : type-Π-Large-Poset P l3) →
    is-binary-lower-bound-Large-Poset (Π-Large-Poset P) x y z →
    (i : I) → is-binary-lower-bound-Large-Poset (P i) (x i) (y i) (z i)
  pr1 (is-binary-lower-bound-is-binary-lower-bound-Π-Large-Poset z H i) =
    pr1 H i
  pr2 (is-binary-lower-bound-is-binary-lower-bound-Π-Large-Poset z H i) =
    pr2 H i

  logical-equivalence-is-binary-lower-bound-Π-Large-Poset :
    {l3 : Level} (z : type-Π-Large-Poset P l3) →
    ((i : I) → is-binary-lower-bound-Large-Poset (P i) (x i) (y i) (z i)) ↔
    is-binary-lower-bound-Large-Poset (Π-Large-Poset P) x y z
  pr1 (logical-equivalence-is-binary-lower-bound-Π-Large-Poset z) =
    is-binary-lower-bound-Π-Large-Poset z
  pr2 (logical-equivalence-is-binary-lower-bound-Π-Large-Poset z) =
    is-binary-lower-bound-is-binary-lower-bound-Π-Large-Poset z