open import Tools.Relation
module Tools.Algebra {a} (A : Set a) where
open import Tools.Product
open import Tools.PropositionalEquality
open import Algebra.Consequences.Propositional public
using (comm∧idˡ⇒idʳ; comm∧zeˡ⇒zeʳ; comm∧distrˡ⇒distrʳ; comm∧distrʳ⇒distrˡ)
open import Algebra.Core using (Op₁; Op₂) public
open import Algebra.Definitions (_≡_ {A = A})
using (Associative; Commutative; Congruent₂;
_DistributesOver_; _DistributesOverˡ_; _DistributesOverʳ_;
Idempotent; _IdempotentOn_;
Identity; LeftIdentity; RightIdentity;
Zero; LeftZero; RightZero;
Absorptive)
public
open import Algebra.Structures (_≡_ {A = A})
using (IsBand; IsCommutativeMonoid; IsMagma; IsMonoid;
IsSemigroup; IsSemiring;
IsSemiringWithoutAnnihilatingZero; IsCommutativeSemiring)
public
open import Algebra.Lattice.Structures (_≡_ {A = A})
using (IsMeetSemilattice; IsDistributiveLattice)
public
open import Algebra.Bundles using (Semiring) public
open import Algebra.Module.Structures
using (IsLeftSemimodule; IsPreleftSemimodule) public
import Algebra.Lattice.Properties.DistributiveLattice
Op₃ : ∀ {ℓ} → Set ℓ → Set ℓ
Op₃ A = A → A → A → A
_SubDistributesOverˡ_by_ : Op₂ A → Op₂ A → Rel A a → Set a
_*_ SubDistributesOverˡ _+_ by _≤_ =
∀ x y z → (x * (y + z)) ≤ ((x * y) + (x * z))
_SubDistributesOverʳ_by_ : Op₂ A → Op₂ A → Rel A a → Set a
_*_ SubDistributesOverʳ _+_ by _≤_ =
∀ x y z → ((y + z) * x) ≤ ((y * x) + (z * x))
_SubDistributesOver_by_ : Op₂ A → Op₂ A → Rel A a → Set a
* SubDistributesOver + by ≤ =
* SubDistributesOverˡ + by ≤ × * SubDistributesOverʳ + by ≤
_SubInterchangeable_by_ : Op₂ A → Op₂ A → Rel A a → Set _
_∘_ SubInterchangeable _∙_ by _≤_ = ∀ w x y z → ((w ∙ x) ∘ (y ∙ z)) ≤ ((w ∘ y) ∙ (x ∘ z))
module DistributiveLattice
{_∨_ _∧_ : Op₂ A}
(dl : IsDistributiveLattice _∨_ _∧_)
where
open Algebra.Lattice.Properties.DistributiveLattice
(record { isDistributiveLattice = dl })
public