Chains in posets

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

Created on 2022-03-16.
Last modified on 2023-05-05.

module order-theory.chains-posets where
open import foundation.propositions
open import foundation.universe-levels

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


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


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

  is-chain-Subposet-Prop :
    {l3 : Level} (S : type-Poset X  Prop l3)  Prop (l1  l2  l3)
  is-chain-Subposet-Prop = is-chain-Subpreorder-Prop (preorder-Poset X)

  is-chain-Subposet :
    {l3 : Level} (S : type-Poset X  Prop l3)  UU (l1  l2  l3)
  is-chain-Subposet = is-chain-Subpreorder (preorder-Poset X)

  is-prop-is-chain-Subposet :
    {l3 : Level} (S : type-Poset X  Prop l3) 
    is-prop (is-chain-Subposet S)
  is-prop-is-chain-Subposet = is-prop-is-chain-Subpreorder (preorder-Poset X)

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)

  sub-preorder-chain-Poset : type-Poset X  Prop l3
  sub-preorder-chain-Poset =
    sub-preorder-chain-Preorder (preorder-Poset X) C

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

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

  inclusion-chain-Poset-Prop :
    {l3 l4 : Level}  chain-Poset l3 X  chain-Poset l4 X 
    Prop (l1  l3  l4)
  inclusion-chain-Poset-Prop = inclusion-chain-Preorder-Prop (preorder-Poset X)

  inclusion-chain-Poset :
    {l3 l4 : Level}  chain-Poset l3 X  chain-Poset l4 X  UU (l1  l3  l4)
  inclusion-chain-Poset = inclusion-chain-Preorder (preorder-Poset X)

  is-prop-inclusion-chain-Poset :
    {l3 l4 : Level} (C : chain-Poset l3 X) (D : chain-Poset l4 X) 
    is-prop (inclusion-chain-Poset C D)
  is-prop-inclusion-chain-Poset =
    is-prop-inclusion-chain-Preorder (preorder-Poset X)

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