------------------------------------------------------------------------ -- The Agda standard library -- -- Basic definitions for morphisms between algebraic structures ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} open import Relation.Binary.Core module Algebra.Morphism.Definitions {a} (A : Set a) -- The domain of the morphism {b} (B : Set b) -- The codomain of the morphism {ℓ} (_≈_ : Rel B ℓ) -- The equality relation over the codomain where open import Algebra.Core using (Op₁; Op₂) ------------------------------------------------------------------------ -- Basic definitions Homomorphic₀ : (A → B) → A → B → Set _ Homomorphic₀ ⟦_⟧ ∙ ∘ = ⟦ ∙ ⟧ ≈ ∘ Homomorphic₁ : (A → B) → Op₁ A → Op₁ B → Set _ Homomorphic₁ ⟦_⟧ ∙_ ∘_ = ∀ x → ⟦ ∙ x ⟧ ≈ (∘ ⟦ x ⟧) Homomorphic₂ : (A → B) → Op₂ A → Op₂ B → Set _ Homomorphic₂ ⟦_⟧ _∙_ _∘_ = ∀ x y → ⟦ x ∙ y ⟧ ≈ (⟦ x ⟧ ∘ ⟦ y ⟧) ------------------------------------------------------------------------ -- DEPRECATED NAMES ------------------------------------------------------------------------ -- Please use the new names as continuing support for the old names is -- not guaranteed. -- Version 1.3 Morphism : Set _ Morphism = A → B {-# WARNING_ON_USAGE Morphism "Warning: Morphism was deprecated in v1.3. Please use the standard function notation (e.g. A → B) instead." #-}