{-# OPTIONS --without-K --safe #-}
open import Relation.Binary.Core
open import Relation.Nullary using (¬_)
module Algebra.Definitions
{a ℓ} {A : Set a}
(_≈_ : Rel A ℓ)
where
open import Algebra.Core
open import Data.Product
open import Data.Sum.Base
Congruent₁ : Op₁ A → Set _
Congruent₁ f = f Preserves _≈_ ⟶ _≈_
Congruent₂ : Op₂ A → Set _
Congruent₂ ∙ = ∙ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
LeftCongruent : Op₂ A → Set _
LeftCongruent _∙_ = ∀ {x} → (x ∙_) Preserves _≈_ ⟶ _≈_
RightCongruent : Op₂ A → Set _
RightCongruent _∙_ = ∀ {x} → (_∙ x) Preserves _≈_ ⟶ _≈_
Associative : Op₂ A → Set _
Associative _∙_ = ∀ x y z → ((x ∙ y) ∙ z) ≈ (x ∙ (y ∙ z))
Commutative : Op₂ A → Set _
Commutative _∙_ = ∀ x y → (x ∙ y) ≈ (y ∙ x)
LeftIdentity : A → Op₂ A → Set _
LeftIdentity e _∙_ = ∀ x → (e ∙ x) ≈ x
RightIdentity : A → Op₂ A → Set _
RightIdentity e _∙_ = ∀ x → (x ∙ e) ≈ x
Identity : A → Op₂ A → Set _
Identity e ∙ = (LeftIdentity e ∙) × (RightIdentity e ∙)
LeftZero : A → Op₂ A → Set _
LeftZero z _∙_ = ∀ x → (z ∙ x) ≈ z
RightZero : A → Op₂ A → Set _
RightZero z _∙_ = ∀ x → (x ∙ z) ≈ z
Zero : A → Op₂ A → Set _
Zero z ∙ = (LeftZero z ∙) × (RightZero z ∙)
LeftInverse : A → Op₁ A → Op₂ A → Set _
LeftInverse e _⁻¹ _∙_ = ∀ x → ((x ⁻¹) ∙ x) ≈ e
RightInverse : A → Op₁ A → Op₂ A → Set _
RightInverse e _⁻¹ _∙_ = ∀ x → (x ∙ (x ⁻¹)) ≈ e
Inverse : A → Op₁ A → Op₂ A → Set _
Inverse e ⁻¹ ∙ = (LeftInverse e ⁻¹) ∙ × (RightInverse e ⁻¹ ∙)
LeftInvertible : A → Op₂ A → A → Set _
LeftInvertible e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e
RightInvertible : A → Op₂ A → A → Set _
RightInvertible e _∙_ x = ∃[ x⁻¹ ] (x ∙ x⁻¹) ≈ e
Invertible : A → Op₂ A → A → Set _
Invertible e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e × (x ∙ x⁻¹) ≈ e
LeftConical : A → Op₂ A → Set _
LeftConical e _∙_ = ∀ x y → (x ∙ y) ≈ e → x ≈ e
RightConical : A → Op₂ A → Set _
RightConical e _∙_ = ∀ x y → (x ∙ y) ≈ e → y ≈ e
Conical : A → Op₂ A → Set _
Conical e ∙ = (LeftConical e ∙) × (RightConical e ∙)
_DistributesOverˡ_ : Op₂ A → Op₂ A → Set _
_*_ DistributesOverˡ _+_ =
∀ x y z → (x * (y + z)) ≈ ((x * y) + (x * z))
_DistributesOverʳ_ : Op₂ A → Op₂ A → Set _
_*_ DistributesOverʳ _+_ =
∀ x y z → ((y + z) * x) ≈ ((y * x) + (z * x))
_DistributesOver_ : Op₂ A → Op₂ A → Set _
* DistributesOver + = (* DistributesOverˡ +) × (* DistributesOverʳ +)
_IdempotentOn_ : Op₂ A → A → Set _
_∙_ IdempotentOn x = (x ∙ x) ≈ x
Idempotent : Op₂ A → Set _
Idempotent ∙ = ∀ x → ∙ IdempotentOn x
IdempotentFun : Op₁ A → Set _
IdempotentFun f = ∀ x → f (f x) ≈ f x
Selective : Op₂ A → Set _
Selective _∙_ = ∀ x y → (x ∙ y) ≈ x ⊎ (x ∙ y) ≈ y
_Absorbs_ : Op₂ A → Op₂ A → Set _
_∙_ Absorbs _∘_ = ∀ x y → (x ∙ (x ∘ y)) ≈ x
Absorptive : Op₂ A → Op₂ A → Set _
Absorptive ∙ ∘ = (∙ Absorbs ∘) × (∘ Absorbs ∙)
Involutive : Op₁ A → Set _
Involutive f = ∀ x → f (f x) ≈ x
LeftCancellative : Op₂ A → Set _
LeftCancellative _•_ = ∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z
RightCancellative : Op₂ A → Set _
RightCancellative _•_ = ∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z
Cancellative : Op₂ A → Set _
Cancellative _•_ = (LeftCancellative _•_) × (RightCancellative _•_)
AlmostLeftCancellative : A → Op₂ A → Set _
AlmostLeftCancellative e _•_ = ∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z
AlmostRightCancellative : A → Op₂ A → Set _
AlmostRightCancellative e _•_ = ∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z
AlmostCancellative : A → Op₂ A → Set _
AlmostCancellative e _•_ = AlmostLeftCancellative e _•_ × AlmostRightCancellative e _•_
Interchangable : Op₂ A → Op₂ A → Set _
Interchangable _∘_ _∙_ = ∀ w x y z → ((w ∙ x) ∘ (y ∙ z)) ≈ ((w ∘ y) ∙ (x ∘ z))
LeftDividesˡ : Op₂ A → Op₂ A → Set _
LeftDividesˡ _∙_ _\\_ = ∀ x y → (x ∙ (x \\ y)) ≈ y
LeftDividesʳ : Op₂ A → Op₂ A → Set _
LeftDividesʳ _∙_ _\\_ = ∀ x y → (x \\ (x ∙ y)) ≈ y
RightDividesˡ : Op₂ A → Op₂ A → Set _
RightDividesˡ _∙_ _//_ = ∀ x y → ((y // x) ∙ x) ≈ y
RightDividesʳ : Op₂ A → Op₂ A → Set _
RightDividesʳ _∙_ _//_ = ∀ x y → ((y ∙ x) // x) ≈ y
LeftDivides : Op₂ A → Op₂ A → Set _
LeftDivides ∙ \\ = (LeftDividesˡ ∙ \\) × (LeftDividesʳ ∙ \\)
RightDivides : Op₂ A → Op₂ A → Set _
RightDivides ∙ // = (RightDividesˡ ∙ //) × (RightDividesʳ ∙ //)
StarRightExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _
StarRightExpansive e _+_ _∙_ _* = ∀ x → (e + (x ∙ (x *))) ≈ (x *)
StarLeftExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _
StarLeftExpansive e _+_ _∙_ _* = ∀ x → (e + ((x *) ∙ x)) ≈ (x *)
StarExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _
StarExpansive e _+_ _∙_ _* = (StarLeftExpansive e _+_ _∙_ _*) × (StarRightExpansive e _+_ _∙_ _*)
StarLeftDestructive : Op₂ A → Op₂ A → Op₁ A → Set _
StarLeftDestructive _+_ _∙_ _* = ∀ a b x → (b + (a ∙ x)) ≈ x → ((a *) ∙ b) ≈ x
StarRightDestructive : Op₂ A → Op₂ A → Op₁ A → Set _
StarRightDestructive _+_ _∙_ _* = ∀ a b x → (b + (x ∙ a)) ≈ x → (b ∙ (a *)) ≈ x
StarDestructive : Op₂ A → Op₂ A → Op₁ A → Set _
StarDestructive _+_ _∙_ _* = (StarLeftDestructive _+_ _∙_ _*) × (StarRightDestructive _+_ _∙_ _*)
LeftAlternative : Op₂ A → Set _
LeftAlternative _∙_ = ∀ x y → ((x ∙ x) ∙ y) ≈ (x ∙ (x ∙ y))
RightAlternative : Op₂ A → Set _
RightAlternative _∙_ = ∀ x y → (x ∙ (y ∙ y)) ≈ ((x ∙ y) ∙ y)
Alternative : Op₂ A → Set _
Alternative _∙_ = (LeftAlternative _∙_ ) × (RightAlternative _∙_)
Flexible : Op₂ A → Set _
Flexible _∙_ = ∀ x y → ((x ∙ y) ∙ x) ≈ (x ∙ (y ∙ x))
Medial : Op₂ A → Set _
Medial _∙_ = ∀ x y u z → ((x ∙ y) ∙ (u ∙ z)) ≈ ((x ∙ u) ∙ (y ∙ z))
LeftSemimedial : Op₂ A → Set _
LeftSemimedial _∙_ = ∀ x y z → ((x ∙ x) ∙ (y ∙ z)) ≈ ((x ∙ y) ∙ (x ∙ z))
RightSemimedial : Op₂ A → Set _
RightSemimedial _∙_ = ∀ x y z → ((y ∙ z) ∙ (x ∙ x)) ≈ ((y ∙ x) ∙ (z ∙ x))
Semimedial : Op₂ A → Set _
Semimedial _∙_ = (LeftSemimedial _∙_) × (RightSemimedial _∙_)
LeftBol : Op₂ A → Set _
LeftBol _∙_ = ∀ x y z → (x ∙ (y ∙ (x ∙ z))) ≈ ((x ∙ (y ∙ x)) ∙ z )
RightBol : Op₂ A → Set _
RightBol _∙_ = ∀ x y z → (((z ∙ x) ∙ y) ∙ x) ≈ (z ∙ ((x ∙ y) ∙ x))
Identical : Op₂ A → Set _
Identical _∙_ = ∀ x y z → ((z ∙ x) ∙ (y ∙ z)) ≈ (z ∙ ((x ∙ y) ∙ z))