Similarity of elements in strict orders

Content created by Fredrik Bakke.

Created on 2025-05-02.
Last modified on 2025-05-02.

module order-theory.similarity-of-elements-strict-orders where
Imports 
open import foundation.binary-relations
open import foundation.cartesian-product-types
open import foundation.conjunction
open import foundation.dependent-pair-types
open import foundation.equivalence-relations
open import foundation.equivalences
open import foundation.fundamental-theorem-of-identity-types
open import foundation.identity-types
open import foundation.logical-equivalences
open import foundation.propositions
open import foundation.subtypes
open import foundation.torsorial-type-families
open import foundation.universe-levels

open import order-theory.similarity-of-elements-strict-preorders
open import order-theory.strict-orders
open import order-theory.strict-preorders

Idea

Two elements x and y of a strict order A are said to be similar, or indiscernible, if for every z : A we have

We refer to the first condition as similarity from below and the second condition as similarity from above.

In informal writing we will use the notation x ≈ y to assert that x and y are similar elements in a strict order A.

Definitions

Similarity from below of elements in strict orders

module _
  {l1 l2 : Level} (A : Strict-Order l1 l2)
  where

  sim-from-below-Strict-Order :
    (x y : type-Strict-Order A) → UU (l1 ⊔ l2)
  sim-from-below-Strict-Order =
    sim-from-below-Strict-Preorder (strict-preorder-Strict-Order A)

  sim-from-below-prop-Strict-Order :
    (x y : type-Strict-Order A) → Prop (l1 ⊔ l2)
  sim-from-below-prop-Strict-Order =
    sim-from-below-prop-Strict-Preorder (strict-preorder-Strict-Order A)

  is-prop-sim-from-below-Strict-Order :
    (x y : type-Strict-Order A) → is-prop (sim-from-below-Strict-Order x y)
  is-prop-sim-from-below-Strict-Order =
    is-prop-sim-from-below-Strict-Preorder (strict-preorder-Strict-Order A)

Similarity from above of elements in strict orders

module _
  {l1 l2 : Level} (A : Strict-Order l1 l2)
  where

  sim-from-above-Strict-Order :
    (x y : type-Strict-Order A) → UU (l1 ⊔ l2)
  sim-from-above-Strict-Order =
    sim-from-above-Strict-Preorder (strict-preorder-Strict-Order A)

  sim-from-above-prop-Strict-Order :
    (x y : type-Strict-Order A) → Prop (l1 ⊔ l2)
  sim-from-above-prop-Strict-Order =
    sim-from-above-prop-Strict-Preorder (strict-preorder-Strict-Order A)

  is-prop-sim-from-above-Strict-Order :
    (x y : type-Strict-Order A) → is-prop (sim-from-above-Strict-Order x y)
  is-prop-sim-from-above-Strict-Order =
    is-prop-sim-from-above-Strict-Preorder (strict-preorder-Strict-Order A)

Similarity of elements in strict orders

module _
  {l1 l2 : Level} (A : Strict-Order l1 l2)
  where

  sim-Strict-Order :
    (x y : type-Strict-Order A) → UU (l1 ⊔ l2)
  sim-Strict-Order =
    sim-Strict-Preorder (strict-preorder-Strict-Order A)

  sim-prop-Strict-Order :
    (x y : type-Strict-Order A) → Prop (l1 ⊔ l2)
  sim-prop-Strict-Order =
    sim-prop-Strict-Preorder (strict-preorder-Strict-Order A)

  is-prop-sim-Strict-Order :
    (x y : type-Strict-Order A) → is-prop (sim-Strict-Order x y)
  is-prop-sim-Strict-Order =
    is-prop-sim-Strict-Preorder (strict-preorder-Strict-Order A)

Properties

The similarity relation is reflexive

module _
  {l1 l2 : Level} (A : Strict-Order l1 l2)
  where

  refl-sim-from-below-Strict-Order :
    is-reflexive (sim-from-below-Strict-Order A)
  refl-sim-from-below-Strict-Order =
    refl-sim-from-below-Strict-Preorder (strict-preorder-Strict-Order A)

  refl-sim-from-above-Strict-Order :
    is-reflexive (sim-from-above-Strict-Order A)
  refl-sim-from-above-Strict-Order =
    refl-sim-from-above-Strict-Preorder
      ( strict-preorder-Strict-Order A)

  refl-sim-Strict-Order :
    is-reflexive (sim-Strict-Order A)
  refl-sim-Strict-Order =
    refl-sim-Strict-Preorder (strict-preorder-Strict-Order A)

The similarity relation is transitive

module _
  {l1 l2 : Level} (A : Strict-Order l1 l2)
  where

  transitive-sim-from-below-Strict-Order :
    is-transitive (sim-from-below-Strict-Order A)
  transitive-sim-from-below-Strict-Order =
    transitive-sim-from-below-Strict-Preorder (strict-preorder-Strict-Order A)

  transitive-sim-from-above-Strict-Order :
    is-transitive (sim-from-above-Strict-Order A)
  transitive-sim-from-above-Strict-Order =
    transitive-sim-from-above-Strict-Preorder (strict-preorder-Strict-Order A)

  transitive-sim-Strict-Order :
    is-transitive (sim-Strict-Order A)
  transitive-sim-Strict-Order =
    transitive-sim-Strict-Preorder (strict-preorder-Strict-Order A)

The similarity relation is symmetric

module _
  {l1 l2 : Level} (A : Strict-Order l1 l2)
  where

  symmetric-sim-from-below-Strict-Order :
    is-symmetric (sim-from-below-Strict-Order A)
  symmetric-sim-from-below-Strict-Order =
    symmetric-sim-from-below-Strict-Preorder (strict-preorder-Strict-Order A)

  symmetric-sim-from-above-Strict-Order :
    is-symmetric (sim-from-above-Strict-Order A)
  symmetric-sim-from-above-Strict-Order =
    symmetric-sim-from-above-Strict-Preorder (strict-preorder-Strict-Order A)

  symmetric-sim-Strict-Order :
    is-symmetric (sim-Strict-Order A)
  symmetric-sim-Strict-Order =
    symmetric-sim-Strict-Preorder (strict-preorder-Strict-Order A)

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