Locally finite posets

module order-theory.locally-finite-posets where

open import foundation.propositions
open import foundation.universe-levels

open import order-theory.finite-posets
open import order-theory.interval-subposets
open import order-theory.posets

Idea

A poset X is said to be locally finite if for every x, y ∈ X, the interval subposet [x, y] consisting of z : X such that x ≤ z ≤ y, is finite.

Definition

module _
  {l1 l2 : Level} (X : Poset l1 l2)
  where

  is-locally-finite-Poset-Prop : Prop (l1 ⊔ l2)
  is-locally-finite-Poset-Prop =
    Π-Prop
      ( type-Poset X)
      ( λ x →
        Π-Prop
          ( type-Poset X)
          ( λ y →
            is-finite-Poset-Prop (poset-interval-Subposet X x y)))

  is-locally-finite-Poset : UU (l1 ⊔ l2)
  is-locally-finite-Poset = type-Prop is-locally-finite-Poset-Prop

  is-prop-is-locally-finite-Poset : is-prop is-locally-finite-Poset
  is-prop-is-locally-finite-Poset =
    is-prop-type-Prop is-locally-finite-Poset-Prop