------------------------------------------------------------------------
-- The Agda standard library
--
-- The identity morphism for algebraic lattice structures
------------------------------------------------------------------------
{-# OPTIONS --safe --cubical-compatible #-}
module Algebra.Lattice.Morphism.Construct.Identity where
open import Algebra.Lattice.Bundles using (RawLattice)
open import Algebra.Lattice.Morphism.Structures
using ( module LatticeMorphisms )
open import Data.Product.Base using (_,_)
open import Function.Base using (id)
import Function.Construct.Identity as Id
open import Level using (Level)
open import Relation.Binary.Morphism.Construct.Identity using (isRelHomomorphism)
open import Relation.Binary.Definitions using (Reflexive)
private
variable
c ℓ : Level
module _ (L : RawLattice c ℓ) (open RawLattice L) (refl : Reflexive _≈_) where
open LatticeMorphisms L L
isLatticeHomomorphism : IsLatticeHomomorphism id
isLatticeHomomorphism = record
{ isRelHomomorphism = isRelHomomorphism _
; ∧-homo = λ _ _ → refl
; ∨-homo = λ _ _ → refl
}
isLatticeMonomorphism : IsLatticeMonomorphism id
isLatticeMonomorphism = record
{ isLatticeHomomorphism = isLatticeHomomorphism
; injective = id
}
isLatticeIsomorphism : IsLatticeIsomorphism id
isLatticeIsomorphism = record
{ isLatticeMonomorphism = isLatticeMonomorphism
; surjective = Id.surjective _
}
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