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------------------------------------------------------------------------
-- 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
  {} (_≈_ : Rel B )  -- 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."
#-}