The poset of closed intervals in lattices
Content created by Louis Wasserman.
Created on 2025-10-06.
Last modified on 2025-10-06.
module order-theory.poset-closed-intervals-lattices where
Imports
open import foundation.binary-relations open import foundation.dependent-pair-types open import foundation.logical-equivalences open import foundation.subtypes open import foundation.universe-levels open import order-theory.closed-intervals-lattices open import order-theory.lattices open import order-theory.poset-closed-intervals-posets open import order-theory.posets open import order-theory.preorders
Idea
In a lattice L
, the type of
closed intervals in L
itself
forms a poset, with inequality defined by containment
of the corresponding subtypes. Equivalently, [a,b] ≤ [c,d]
if c ≤ a
and
b ≤ d
.
Definition
module _ {l1 l2 : Level} (L : Lattice l1 l2) where leq-prop-closed-interval-Lattice : Relation-Prop l2 (closed-interval-Lattice L) leq-prop-closed-interval-Lattice = leq-prop-closed-interval-Poset (poset-Lattice L) leq-closed-interval-Lattice : Relation l2 (closed-interval-Lattice L) leq-closed-interval-Lattice = leq-closed-interval-Poset (poset-Lattice L)
Properties
Containment is a preorder
module _ {l1 l2 : Level} (L : Lattice l1 l2) where refl-leq-closed-interval-Lattice : is-reflexive (leq-closed-interval-Lattice L) refl-leq-closed-interval-Lattice = refl-leq-closed-interval-Poset (poset-Lattice L) transitive-leq-closed-interval-Lattice : is-transitive (leq-closed-interval-Lattice L) transitive-leq-closed-interval-Lattice = transitive-leq-closed-interval-Poset (poset-Lattice L) is-preorder-leq-closed-interval-Lattice : is-preorder-Relation-Prop (leq-prop-closed-interval-Lattice L) is-preorder-leq-closed-interval-Lattice = is-preorder-leq-closed-interval-Poset (poset-Lattice L)
Containment is a poset
module _ {l1 l2 : Level} (L : Lattice l1 l2) where antisymmetric-leq-closed-interval-Lattice : is-antisymmetric (leq-closed-interval-Lattice L) antisymmetric-leq-closed-interval-Lattice = antisymmetric-leq-closed-interval-Poset (poset-Lattice L) poset-closed-interval-Lattice : Poset (l1 ⊔ l2) l2 poset-closed-interval-Lattice = poset-closed-interval-Poset (poset-Lattice L)
Containment of closed intervals is equivalent to containment of subtypes
module _ {l1 l2 : Level} (L : Lattice l1 l2) where leq-subtype-leq-closed-interval-Lattice : ([a,b] [c,d] : closed-interval-Lattice L) → leq-closed-interval-Lattice L [a,b] [c,d] → ( subtype-closed-interval-Lattice L [a,b] ⊆ subtype-closed-interval-Lattice L [c,d]) leq-subtype-leq-closed-interval-Lattice = leq-subtype-leq-closed-interval-Poset (poset-Lattice L) leq-closed-interval-leq-subtype-Lattice : ([a,b] [c,d] : closed-interval-Lattice L) → ( subtype-closed-interval-Lattice L [a,b] ⊆ subtype-closed-interval-Lattice L [c,d]) → leq-closed-interval-Lattice L [a,b] [c,d] leq-closed-interval-leq-subtype-Lattice = leq-closed-interval-leq-subtype-Poset (poset-Lattice L) leq-subtype-iff-leq-closed-interval-Lattice : ([a,b] [c,d] : closed-interval-Lattice L) → ( leq-closed-interval-Lattice L [a,b] [c,d] ↔ ( subtype-closed-interval-Lattice L [a,b] ⊆ subtype-closed-interval-Lattice L [c,d])) leq-subtype-iff-leq-closed-interval-Lattice = leq-subtype-iff-leq-closed-interval-Poset (poset-Lattice L)
Recent changes
- 2025-10-06. Louis Wasserman. Poset of closed intervals on lattices (#1565).