Library UniMath.OrderTheory.Lattice.Complement

Complements


Require Import UniMath.Foundations.Preamble.
Require Import UniMath.Foundations.Sets.

Require Import UniMath.OrderTheory.Lattice.Lattice.
Require Import UniMath.OrderTheory.Lattice.Bounded.

Section Def.
  Context {X : hSet} (L : bounded_lattice X).

The normal "∧", "∨" notation conflicts with that for hProp, whereas "+", "×" conflict with notation for types.
  Local Notation "x ≤ y" := (Lle L x y).
  Local Notation "x ⊗ y" := (Lmin L x y).
  Local Notation "x ⊕ y" := (Lmax L x y).
  Local Notation "⊤" := (Ltop L).
  Local Notation "⊥" := (Lbot L).

  Definition complement (x : X) : UU :=
     y : X, (x y = ) × (x y = ).

  Definition complement_to_element {x : X} (y : complement x) : X := pr1 y.
  Coercion complement_to_element : complement >-> pr1hSet.

  Definition complement_top_axiom (x : X) (y : complement x) : x y = :=
    dirprod_pr1 (pr2 y).

  Definition complement_bottom_axiom (x : X) (y : complement x) : x y = :=
    dirprod_pr2 (pr2 y).

This is not a proposition: complements need not be unique.
  Definition complemented_structure : UU := x : X, complement x.

End Def.