module Order.Lattice where

Lattices🔗

A lattice is a poset which is a semilattice in two compatible, dual ways. We have a “meet” semilattice $(A, \cap, \top)$ , and a “join” semilattice $(A, \cup, \bot)$ — and we take the order on $A$ induced by, for definiteness, the $\cap$ semilattice. The question is then: how do we write, algebraically, a condition on $\cap$ and $\cup$ which guarantees that $\cup$ provides $(A, \le)$ with joins?

The answer is yes! Of course, since if it were not, this page would not be being written. A lattice structure on $(A, \top, \bot, \cap, \cup)$ is a pair of semilattices $(A, \cap, \top)$ and $(A, \cup, \bot)$ , such that the $\cap$ and $\cup$ operations satisfy the two absorption laws with respect to eachother:

  field
    has-meets : is-semilattice ⊤ _∩_
    has-joins : is-semilattice ⊥ _∪_

    ∩-absorbs-∪ : ∀ {x y} → x ∩ (x ∪ y) ≡ x
    ∪-absorbs-∩ : ∀ {x y} → x ∪ (x ∩ y) ≡ x

As usual, a lattice structure on a set is a record packaging together the operations and a proof that they are actually an instance of the algebraic structure we’re talking about.

record Lattice-on {ℓ} (A : Type ℓ) : Type ℓ where
  no-eta-equality
  field
    top : A
    bot : A
    _∩_ : A → A → A
    _∪_ : A → A → A
    has-is-lattice : is-lattice top _∩_ bot _∪_

  open is-lattice has-is-lattice public

  meets : Semilattice-on A
  meets = record { has-is-semilattice = has-meets }

  joins : Semilattice-on A
  joins = record { has-is-semilattice = has-joins }

  open Semilattice-on meets using (⋂) public
  private module x = Semilattice-on joins renaming (⋂ to ⋃)
  open x using (⋃) public

The question then becomes: what is the point of the absorption laws? I’ll tell you! Since we have two semilattice structures on the set, we can imagine defining two orders on it: one is given by $x \le y$ iff $x = x \cap y$ , and the other is given by $y = x \cup y$ . As an exercise, try to work out that these are the same thing in the lattice of subsets of a fixed $A$ .

Really, there are four orderings we could imagine defining on a semilattice: $x \le y$ iff. $x = x \cap y$ , $y = x \cap y$ , $x = x \cup y$ , and $y = x \cup y$ . The absorption laws serve to cut these possibilities in half, because using them we can show that $x = x \cap y$ is the same as $y = x \cup y$ (and, respectively, $y = x \cap y$ is the same as $x = x \cup y$ ).

  ∪-order→∩-order : ∀ {x y} → y ≡ x ∪ y → x ≡ x ∩ y
  ∪-order→∩-order {x} {y} y=x∪y =
    x             ≡˘⟨ ∩-absorbs-∪ ⟩
    x ∩ ⌜ x ∪ y ⌝ ≡˘⟨ ap¡ y=x∪y ⟩
    x ∩ y         ∎

  ∩-order→∪-order : ∀ {x y} → x ≡ x ∩ y → y ≡ x ∪ y
  ∩-order→∪-order {x} {y} p =
    y             ≡⟨ sym ∪-absorbs-∩ ⟩
    y ∪ ⌜ y ∩ x ⌝ ≡⟨ ap! ∩-commutative ⟩
    y ∪ ⌜ x ∩ y ⌝ ≡˘⟨ ap¡ p ⟩
    y ∪ x         ≡⟨ ∪-commutative ⟩
    x ∪ y         ∎

Using the “cup ordering”, we can establish that cups provide a join of two elements. Since the cup ordering and the “cap ordering” agree, we can prove that cups give a join in that case, too!

  ∪-is-join : ∀ {x y} → is-join Lattice→poset x y (x ∪ y)
  ∪-is-join .is-join.l≤join = sym ∩-absorbs-∪
  ∪-is-join {x} {y} .is-join.r≤join =
    y             ≡˘⟨ ∩-absorbs-∪ ⟩
    y ∩ ⌜ y ∪ x ⌝ ≡⟨ ap! ∪-commutative ⟩
    y ∩ (x ∪ y)   ∎
  ∪-is-join {x} {y} .is-join.least ub' x=x∩ub' y=y∩ub' = ∪-order→∩-order $ sym $
    (x ∪ y) ∪ ub'   ≡˘⟨ ∪-associative ⟩
    x ∪ ⌜ y ∪ ub' ⌝ ≡˘⟨ ap¡ (∩-order→∪-order y=y∩ub') ⟩
    x ∪ ub'         ≡˘⟨ ∩-order→∪-order x=x∩ub' ⟩
    ub'             ∎

  ⊥-is-bottom : ∀ {x} → bot P.≤ x
  ⊥-is-bottom = ∪-order→∩-order (sym ∪-idl)

Category of lattices🔗

A lattice homomorphism is a function which is, at the same time, a homomorphism for both of the semilattice monoid structures: A function sending bottom to bottom, top to top, joins to joins, and meets to meets. Put more concisely, a function which preserves finite meets and finite joins.

record
  is-lattice-hom
    {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'}
    (f : A → B)
    (S : Lattice-on A) (T : Lattice-on B)
    : Type (ℓ ⊔ ℓ')
  where

  private
    module S = Lattice-on S
    module T = Lattice-on T

  field
    pres-⊤ : f S.top ≡ T.top
    pres-⊥ : f S.bot ≡ T.bot
    pres-∩ : ∀ {x y} → f (x S.∩ y) ≡ f x T.∩ f y
    pres-∪ : ∀ {x y} → f (x S.∪ y) ≡ f x T.∪ f y

Standard equational nonsense implies that (a) lattices and lattice homomorphisms form a precategory; and (b) this is a univalent category.

Lattice-structure : ∀ ℓ → Thin-structure ℓ Lattice-on

Lattice-structure ℓ .is-hom f S T .∣_∣   = is-lattice-hom f S T
Lattice-structure ℓ .is-hom f S T .is-tr = hlevel!

Lattice-structure ℓ .id-is-hom .pres-⊤ = refl
Lattice-structure ℓ .id-is-hom .pres-⊥ = refl
Lattice-structure ℓ .id-is-hom .pres-∩ = refl
Lattice-structure ℓ .id-is-hom .pres-∪ = refl

Lattice-structure ℓ .∘-is-hom f g α β .pres-⊤ = ap f (β .pres-⊤) ∙ α .pres-⊤
Lattice-structure ℓ .∘-is-hom f g α β .pres-⊥ = ap f (β .pres-⊥) ∙ α .pres-⊥
Lattice-structure ℓ .∘-is-hom f g α β .pres-∩ = ap f (β .pres-∩) ∙ α .pres-∩
Lattice-structure ℓ .∘-is-hom f g α β .pres-∪ = ap f (β .pres-∪) ∙ α .pres-∪

Univalence follows from the special case of considering the identity as a lattice homomorphism $(A, s) \to (A, t)$ — preservation of the operations guarantees that we get $\top_s = \top_t$ (for each operation; $\bot$ , $\cap$ and $\cup$ ), which eventually gives $s = t$ .

Lattice-structure ℓ .id-hom-unique {s = s} {t = t} α _ = p where
  open Lattice-on
  p : s ≡ t
  p i .top = α .pres-⊤ i
  p i .bot = α .pres-⊥ i
  p i ._∩_ x y = α .pres-∩ {x} {y} i
  p i ._∪_ x y = α .pres-∪ {x} {y} i
  p i .has-is-lattice =
    is-prop→pathp
      (λ i → hlevel {T = is-lattice (α .pres-⊤ i) (λ x y → α .pres-∩ i)
                                    (α .pres-⊥ i) (λ x y → α .pres-∪ i)}
        1)
      (s .has-is-lattice) (t .has-is-lattice) i

Lattices : ∀ ℓ → Precategory (lsuc ℓ) ℓ
Lattices ℓ = Structured-objects (Lattice-structure ℓ)