Chains in posets

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2022-03-16.
Last modified on 2024-10-20.

module order-theory.chains-posets where
Imports 
open import foundation.dependent-pair-types
open import foundation.existential-quantification
open import foundation.function-types
open import foundation.propositions
open import foundation.subtypes
open import foundation.universe-levels

open import order-theory.chains-preorders
open import order-theory.posets
open import order-theory.subposets

Idea

A chain in a poset P is a subtype S of P such that the ordering of P restricted to S is linear.

Definitions

The predicate on subsets of posets to be a chain

module _
  {l1 l2 l3 : Level} (X : Poset l1 l2) (S : Subposet l3 X)
  where

  is-chain-prop-Subposet : Prop (l1 ⊔ l2 ⊔ l3)
  is-chain-prop-Subposet = is-chain-prop-Subpreorder (preorder-Poset X) S

  is-chain-Subposet : UU (l1 ⊔ l2 ⊔ l3)
  is-chain-Subposet = is-chain-Subpreorder (preorder-Poset X) S

  is-prop-is-chain-Subposet : is-prop is-chain-Subposet
  is-prop-is-chain-Subposet = is-prop-is-chain-Subpreorder (preorder-Poset X) S

Chains in posets

chain-Poset :
  {l1 l2 : Level} (l : Level) (X : Poset l1 l2) → UU (l1 ⊔ l2 ⊔ lsuc l)
chain-Poset l X = chain-Preorder l (preorder-Poset X)

module _
  {l1 l2 l3 : Level} (X : Poset l1 l2) (C : chain-Poset l3 X)
  where

  subposet-chain-Poset : Subposet l3 X
  subposet-chain-Poset =
    subpreorder-chain-Preorder (preorder-Poset X) C

  is-chain-subposet-chain-Poset :
    is-chain-Subposet X subposet-chain-Poset
  is-chain-subposet-chain-Poset =
    is-chain-subpreorder-chain-Preorder (preorder-Poset X) C

  type-chain-Poset : UU (l1 ⊔ l3)
  type-chain-Poset = type-chain-Preorder (preorder-Poset X) C

  inclusion-type-chain-Poset : type-chain-Poset → type-Poset X
  inclusion-type-chain-Poset =
    inclusion-subpreorder-chain-Preorder (preorder-Poset X) C

module _
  {l1 l2 l3 l4 : Level} (X : Poset l1 l2)
  (C : chain-Poset l3 X) (D : chain-Poset l4 X)
  where

  inclusion-chain-prop-Poset : Prop (l1 ⊔ l3 ⊔ l4)
  inclusion-chain-prop-Poset =
    inclusion-prop-chain-Preorder (preorder-Poset X) C D

  inclusion-chain-Poset : UU (l1 ⊔ l3 ⊔ l4)
  inclusion-chain-Poset = inclusion-chain-Preorder (preorder-Poset X) C D

  is-prop-inclusion-chain-Poset : is-prop inclusion-chain-Poset
  is-prop-inclusion-chain-Poset =
    is-prop-inclusion-chain-Preorder (preorder-Poset X) C D

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