distributive-lattices.lagda.md

# Distributive lattices

```agda
module order-theory.distributive-lattices where
```

<details><summary>Imports</summary>

```agda
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.join-semilattices
open import order-theory.lattices
open import order-theory.meet-semilattices
open import order-theory.posets
```

</details>

## Idea

A **distributive lattice** is a lattice in which meets distribute over joins.

## Definition

```agda
module _
  {l1 l2 : Level} (L : Lattice l1 l2)
  where

  is-distributive-lattice-Prop : Prop l1
  is-distributive-lattice-Prop =
    Π-Prop
      ( type-Lattice L)
      ( λ x →
        Π-Prop
          ( type-Lattice L)
          ( λ y →
            Π-Prop
              ( type-Lattice L)
              ( λ z →
                Id-Prop
                  ( set-Lattice L)
                  ( meet-Lattice L x (join-Lattice L y z))
                  ( join-Lattice L (meet-Lattice L x y) (meet-Lattice L x z)))))

  is-distributive-Lattice : UU l1
  is-distributive-Lattice = type-Prop is-distributive-lattice-Prop

  is-prop-is-distributive-Lattice : is-prop is-distributive-Lattice
  is-prop-is-distributive-Lattice =
    is-prop-type-Prop is-distributive-lattice-Prop

Distributive-Lattice : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
Distributive-Lattice l1 l2 = Σ (Lattice l1 l2) is-distributive-Lattice

module _
  {l1 l2 : Level} (L : Distributive-Lattice l1 l2)
  where

  lattice-Distributive-Lattice : Lattice l1 l2
  lattice-Distributive-Lattice = pr1 L

  poset-Distributive-Lattice : Poset l1 l2
  poset-Distributive-Lattice = poset-Lattice lattice-Distributive-Lattice

  type-Distributive-Lattice : UU l1
  type-Distributive-Lattice = type-Lattice lattice-Distributive-Lattice

  leq-Distributive-Lattice-Prop : (x y : type-Distributive-Lattice) → Prop l2
  leq-Distributive-Lattice-Prop = leq-lattice-Prop lattice-Distributive-Lattice

  leq-Distributive-Lattice : (x y : type-Distributive-Lattice) → UU l2
  leq-Distributive-Lattice = leq-Lattice lattice-Distributive-Lattice

  is-prop-leq-Distributive-Lattice :
    (x y : type-Distributive-Lattice) → is-prop (leq-Distributive-Lattice x y)
  is-prop-leq-Distributive-Lattice =
    is-prop-leq-Lattice lattice-Distributive-Lattice

  refl-leq-Distributive-Lattice :
    (x : type-Distributive-Lattice) → leq-Distributive-Lattice x x
  refl-leq-Distributive-Lattice =
    refl-leq-Lattice lattice-Distributive-Lattice

  antisymmetric-leq-Distributive-Lattice :
    (x y : type-Distributive-Lattice) →
    leq-Distributive-Lattice x y → leq-Distributive-Lattice y x → x ＝ y
  antisymmetric-leq-Distributive-Lattice =
    antisymmetric-leq-Lattice lattice-Distributive-Lattice

  transitive-leq-Distributive-Lattice :
    (x y z : type-Distributive-Lattice) →
    leq-Distributive-Lattice y z →
    leq-Distributive-Lattice x y → leq-Distributive-Lattice x z
  transitive-leq-Distributive-Lattice =
    transitive-leq-Lattice lattice-Distributive-Lattice

  is-set-type-Distributive-Lattice : is-set type-Distributive-Lattice
  is-set-type-Distributive-Lattice =
    is-set-type-Lattice lattice-Distributive-Lattice

  set-Distributive-Lattice : Set l1
  set-Distributive-Lattice = set-Lattice lattice-Distributive-Lattice

  is-lattice-Distributive-Lattice :
    is-lattice-Poset poset-Distributive-Lattice
  is-lattice-Distributive-Lattice =
    is-lattice-Lattice lattice-Distributive-Lattice

  is-meet-semilattice-Distributive-Lattice :
    is-meet-semilattice-Poset poset-Distributive-Lattice
  is-meet-semilattice-Distributive-Lattice =
    is-meet-semilattice-Lattice lattice-Distributive-Lattice

  meet-semilattice-Distributive-Lattice : Meet-Semilattice l1
  meet-semilattice-Distributive-Lattice =
    meet-semilattice-Lattice lattice-Distributive-Lattice

  meet-Distributive-Lattice :
    (x y : type-Distributive-Lattice) → type-Distributive-Lattice
  meet-Distributive-Lattice =
    meet-Lattice lattice-Distributive-Lattice

  is-join-semilattice-Distributive-Lattice :
    is-join-semilattice-Poset poset-Distributive-Lattice
  is-join-semilattice-Distributive-Lattice =
    is-join-semilattice-Lattice lattice-Distributive-Lattice

  join-semilattice-Distributive-Lattice : Join-Semilattice l1
  join-semilattice-Distributive-Lattice =
    join-semilattice-Lattice lattice-Distributive-Lattice

  join-Distributive-Lattice :
    (x y : type-Distributive-Lattice) → type-Distributive-Lattice
  join-Distributive-Lattice =
    join-Lattice lattice-Distributive-Lattice

  distributive-meet-join-Distributive-Lattice :
    (x y z : type-Distributive-Lattice) →
    meet-Distributive-Lattice x (join-Distributive-Lattice y z) ＝
    join-Distributive-Lattice
      ( meet-Distributive-Lattice x y)
      ( meet-Distributive-Lattice x z)
  distributive-meet-join-Distributive-Lattice = pr2 L
```