Direct products and minor fixes (#1909)

CHANGELOG.md

@@ -1888,6 +1888,11 @@ Other minor changes
  rightBolLoop : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
  middleBolLoop : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
  moufangLoop : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ rawRingWithoutOne : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ ringWithoutOne : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ nonAssociativeRing : NonAssociativeRing a ℓ₁ → NonAssociativeRing b ℓ₂ → NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ quasiring : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ nearring : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
  ```
 
 * Added new definition to `Algebra.Definitions`:

src/Algebra/Bundles.agda

@@ -892,6 +892,12 @@ record NonAssociativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
  open AbelianGroup +-abelianGroup public
  using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)
 
+ *-unitalMagma : UnitalMagma _ _
+ *-unitalMagma = record { isUnitalMagma = *-isUnitalMagma}
+
+ open UnitalMagma *-unitalMagma public
+ using () renaming (magma to *-magma; identity to *-identity)
+
 record Nearring c ℓ : Set (suc (c ⊔ ℓ)) where
  infixl 7 _*_
  infixl 6 _+_
@@ -1158,3 +1164,4 @@ record MiddleBolLoop c ℓ : Set (suc (c ⊔ ℓ)) where
 
  open Loop loop public
  using (quasigroup)
+

src/Algebra/Construct/DirectProduct.agda

@@ -61,6 +61,16 @@ rawSemiring R S = record
  ; 1# = R.1# , S.1#
  } where module R = RawSemiring R; module S = RawSemiring S
 
+rawRingWithoutOne : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+rawRingWithoutOne R S = record
+ { Carrier = R.Carrier × S.Carrier
+ ; _≈_ = Pointwise R._≈_ S._≈_
+ ; _+_ = zip R._+_ S._+_
+ ; _*_ = zip R._*_ S._*_
+ ; -_ = map R.-_ S.-_
+ ; 0# = R.0# , S.0#
+ } where module R = RawRingWithoutOne R; module S = RawRingWithoutOne S
+
 rawRing : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
 rawRing R S = record
  { Carrier = R.Carrier × S.Carrier
@@ -317,6 +327,59 @@ kleeneAlgebra K L = record
  }
  } where module K = KleeneAlgebra K; module L = KleeneAlgebra L
 
+ringWithoutOne : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ringWithoutOne R S = record
+ { isRingWithoutOne = record
+ { +-isAbelianGroup = AbelianGroup.isAbelianGroup ((abelianGroup R.+-abelianGroup S.+-abelianGroup))
+ ; *-cong = Semigroup.∙-cong (semigroup R.*-semigroup S.*-semigroup)
+ ; *-assoc = Semigroup.assoc (semigroup R.*-semigroup S.*-semigroup)
+ ; distrib = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
+ , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
+ ; zero = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
+ , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
+ }
+
+ } where module R = RingWithoutOne R; module S = RingWithoutOne S
+
+nonAssociativeRing : NonAssociativeRing a ℓ₁ → NonAssociativeRing b ℓ₂ → NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+nonAssociativeRing R S = record
+ { isNonAssociativeRing = record
+ { +-isAbelianGroup = AbelianGroup.isAbelianGroup ((abelianGroup R.+-abelianGroup S.+-abelianGroup))
+ ; *-cong = UnitalMagma.∙-cong (unitalMagma R.*-unitalMagma S.*-unitalMagma)
+ ; *-identity = UnitalMagma.identity (unitalMagma R.*-unitalMagma S.*-unitalMagma)
+ ; distrib = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
+ , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
+ ; zero = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
+ , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
+ }
+
+ } where module R = NonAssociativeRing R; module S = NonAssociativeRing S
+
+quasiring : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+quasiring R S = record
+ { isQuasiring = record
+ { +-isMonoid = Monoid.isMonoid ((monoid R.+-monoid S.+-monoid))
+ ; *-cong = Monoid.∙-cong (monoid R.*-monoid S.*-monoid)
+ ; *-assoc = Monoid.assoc (monoid R.*-monoid S.*-monoid)
+ ; *-identity = Monoid.identity ((monoid R.*-monoid S.*-monoid))
+ ; distrib = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
+ , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
+ ; zero = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
+ , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
+ }
+
+ } where module R = Quasiring R; module S = Quasiring S
+
+nearring : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+nearring R S = record
+ { isNearring = record
+ { isQuasiring = Quasiring.isQuasiring (quasiring R.quasiring S.quasiring)
+ ; +-inverse = (λ x → (R.+-inverseˡ , S.+-inverseˡ) <*> x)
+ , (λ x → (R.+-inverseʳ , S.+-inverseʳ) <*> x)
+ ; ⁻¹-cong = map R.⁻¹-cong S.⁻¹-cong
+ }
+ } where module R = Nearring R; module S = Nearring S
+
 ring : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
 ring R S = record
  { -_ = uncurry (λ x y → R.-_ x , S.-_ y)

src/Algebra/Structures.agda

@@ -594,6 +594,24 @@ record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
  ; isSemigroup to +-isSemigroup
  )
 
+ distribˡ : * DistributesOverˡ +
+ distribˡ = proj₁ distrib
+
+ distribʳ : * DistributesOverʳ +
+ distribʳ = proj₂ distrib
+
+ zeroˡ : LeftZero 0# *
+ zeroˡ = proj₁ zero
+
+ zeroʳ : RightZero 0# *
+ zeroʳ = proj₂ zero
+
+ identityˡ : LeftIdentity 1# *
+ identityˡ = proj₁ *-identity
+
+ identityʳ : RightIdentity 1# *
+ identityʳ = proj₂ *-identity
+
  *-isMagma : IsMagma *
  *-isMagma = record
  { isEquivalence = isEquivalence
@@ -683,11 +701,11 @@ record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ
  ; assoc = *-assoc
  }
 
- open IsMagma *-isMagma public
+ open IsSemigroup *-isSemigroup public
  using ()
  renaming
- ( ∙-congˡ to *-congˡ
- ; ∙-congʳ to *-congʳ
+ ( ∙-congˡ  to *-congˡ
+ ; ∙-congʳ  to *-congʳ
  )
 
 ------------------------------------------------------------------------
@@ -698,7 +716,7 @@ record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a
  field
  +-isAbelianGroup : IsAbelianGroup + 0# -_
  *-cong : Congruent₂ *
- identity  : Identity 1# *
+ *-identity : Identity 1# *
  distrib : * DistributesOver +
  zero : Zero 0# *
 
@@ -728,17 +746,42 @@ record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a
  ; isGroup to +-isGroup
  )
 
+ zeroˡ : LeftZero 0# *
+ zeroˡ = proj₁ zero
+
+ zeroʳ : RightZero 0# *
+ zeroʳ = proj₂ zero
+
+ distribˡ : * DistributesOverˡ +
+ distribˡ = proj₁ distrib
+
+ distribʳ : * DistributesOverʳ +
+ distribʳ = proj₂ distrib
+
  *-isMagma : IsMagma *
  *-isMagma = record
  { isEquivalence = isEquivalence
  ; ∙-cong = *-cong
  }
 
  *-identityˡ : LeftIdentity 1# *
- *-identityˡ = proj₁ identity
+ *-identityˡ = proj₁ *-identity
 
  *-identityʳ : RightIdentity 1# *
- *-identityʳ = proj₂ identity
+ *-identityʳ = proj₂ *-identity
+
+ *-isUnitalMagma : IsUnitalMagma * 1#
+ *-isUnitalMagma = record
+ { isMagma = *-isMagma
+ ; identity = *-identity
+ }
+
+ open IsUnitalMagma *-isUnitalMagma public
+ using ()
+ renaming
+ ( ∙-congˡ to *-congˡ
+ ; ∙-congʳ to *-congʳ
+ )
 
 record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
  field