```------------------------------------------------------------------------
-- The Agda standard library
--
-- Definitions of algebraic structures like monoids and rings
-- (packed in records together with sets, operations, etc.)
------------------------------------------------------------------------

module Algebra where

open import Relation.Binary
open import Algebra.FunctionProperties
open import Algebra.Structures
open import Function
open import Level

------------------------------------------------------------------------
-- Semigroups, (commutative) monoids and (abelian) groups

record Semigroup c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix  4 _≈_
field
Carrier     : Set c
_≈_         : Rel Carrier ℓ
_∙_         : Op₂ Carrier
isSemigroup : IsSemigroup _≈_ _∙_

open IsSemigroup isSemigroup public

setoid : Setoid _ _
setoid = record { isEquivalence = isEquivalence }

-- A raw monoid is a monoid without any laws.

record RawMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix  4 _≈_
field
Carrier : Set c
_≈_     : Rel Carrier ℓ
_∙_     : Op₂ Carrier
ε       : Carrier

record Monoid c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix  4 _≈_
field
Carrier  : Set c
_≈_      : Rel Carrier ℓ
_∙_      : Op₂ Carrier
ε        : Carrier
isMonoid : IsMonoid _≈_ _∙_ ε

open IsMonoid isMonoid public

semigroup : Semigroup _ _
semigroup = record { isSemigroup = isSemigroup }

open Semigroup semigroup public using (setoid)

rawMonoid : RawMonoid _ _
rawMonoid = record
{ _≈_ = _≈_
; _∙_ = _∙_
; ε   = ε
}

record CommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix  4 _≈_
field
Carrier             : Set c
_≈_                 : Rel Carrier ℓ
_∙_                 : Op₂ Carrier
ε                   : Carrier
isCommutativeMonoid : IsCommutativeMonoid _≈_ _∙_ ε

open IsCommutativeMonoid isCommutativeMonoid public

monoid : Monoid _ _
monoid = record { isMonoid = isMonoid }

open Monoid monoid public using (setoid; semigroup; rawMonoid)

record IdempotentCommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix  4 _≈_
field
Carrier                       : Set c
_≈_                           : Rel Carrier ℓ
_∙_                           : Op₂ Carrier
ε                             : Carrier
isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid _≈_ _∙_ ε

open IsIdempotentCommutativeMonoid isIdempotentCommutativeMonoid public

commutativeMonoid : CommutativeMonoid _ _
commutativeMonoid = record { isCommutativeMonoid = isCommutativeMonoid }

open CommutativeMonoid commutativeMonoid public
using (setoid; semigroup; rawMonoid; monoid)

record Group c ℓ : Set (suc (c ⊔ ℓ)) where
infix  8 _⁻¹
infixl 7 _∙_
infix  4 _≈_
field
Carrier : Set c
_≈_     : Rel Carrier ℓ
_∙_     : Op₂ Carrier
ε       : Carrier
_⁻¹     : Op₁ Carrier
isGroup : IsGroup _≈_ _∙_ ε _⁻¹

open IsGroup isGroup public

monoid : Monoid _ _
monoid = record { isMonoid = isMonoid }

open Monoid monoid public using (setoid; semigroup; rawMonoid)

record AbelianGroup c ℓ : Set (suc (c ⊔ ℓ)) where
infix  8 _⁻¹
infixl 7 _∙_
infix  4 _≈_
field
Carrier        : Set c
_≈_            : Rel Carrier ℓ
_∙_            : Op₂ Carrier
ε              : Carrier
_⁻¹            : Op₁ Carrier
isAbelianGroup : IsAbelianGroup _≈_ _∙_ ε _⁻¹

open IsAbelianGroup isAbelianGroup public

group : Group _ _
group = record { isGroup = isGroup }

open Group group public using (setoid; semigroup; monoid; rawMonoid)

commutativeMonoid : CommutativeMonoid _ _
commutativeMonoid =
record { isCommutativeMonoid = isCommutativeMonoid }

------------------------------------------------------------------------
-- Various kinds of semirings

record NearSemiring c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier        : Set c
_≈_            : Rel Carrier ℓ
_+_            : Op₂ Carrier
_*_            : Op₂ Carrier
0#             : Carrier
isNearSemiring : IsNearSemiring _≈_ _+_ _*_ 0#

open IsNearSemiring isNearSemiring public

+-monoid : Monoid _ _
+-monoid = record { isMonoid = +-isMonoid }

open Monoid +-monoid public
using (setoid)
renaming ( semigroup to +-semigroup
; rawMonoid to +-rawMonoid)

*-semigroup : Semigroup _ _
*-semigroup = record { isSemigroup = *-isSemigroup }

record SemiringWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier              : Set c
_≈_                  : Rel Carrier ℓ
_+_                  : Op₂ Carrier
_*_                  : Op₂ Carrier
0#                   : Carrier
isSemiringWithoutOne : IsSemiringWithoutOne _≈_ _+_ _*_ 0#

open IsSemiringWithoutOne isSemiringWithoutOne public

nearSemiring : NearSemiring _ _
nearSemiring = record { isNearSemiring = isNearSemiring }

open NearSemiring nearSemiring public
using ( setoid
; +-semigroup; +-rawMonoid; +-monoid
; *-semigroup
)

+-commutativeMonoid : CommutativeMonoid _ _
+-commutativeMonoid =
record { isCommutativeMonoid = +-isCommutativeMonoid }

record SemiringWithoutAnnihilatingZero c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier                           : Set c
_≈_                               : Rel Carrier ℓ
_+_                               : Op₂ Carrier
_*_                               : Op₂ Carrier
0#                                : Carrier
1#                                : Carrier
isSemiringWithoutAnnihilatingZero :
IsSemiringWithoutAnnihilatingZero _≈_ _+_ _*_ 0# 1#

open IsSemiringWithoutAnnihilatingZero
isSemiringWithoutAnnihilatingZero public

+-commutativeMonoid : CommutativeMonoid _ _
+-commutativeMonoid =
record { isCommutativeMonoid = +-isCommutativeMonoid }

open CommutativeMonoid +-commutativeMonoid public
using (setoid)
renaming ( semigroup to +-semigroup
; rawMonoid to +-rawMonoid
; monoid    to +-monoid
)

*-monoid : Monoid _ _
*-monoid = record { isMonoid = *-isMonoid }

open Monoid *-monoid public
using ()
renaming ( semigroup to *-semigroup
; rawMonoid to *-rawMonoid
)

record Semiring c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier    : Set c
_≈_        : Rel Carrier ℓ
_+_        : Op₂ Carrier
_*_        : Op₂ Carrier
0#         : Carrier
1#         : Carrier
isSemiring : IsSemiring _≈_ _+_ _*_ 0# 1#

open IsSemiring isSemiring public

semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero _ _
semiringWithoutAnnihilatingZero = record
{ isSemiringWithoutAnnihilatingZero =
isSemiringWithoutAnnihilatingZero
}

open SemiringWithoutAnnihilatingZero
semiringWithoutAnnihilatingZero public
using ( setoid
; +-semigroup; +-rawMonoid; +-monoid
; +-commutativeMonoid
; *-semigroup; *-rawMonoid; *-monoid
)

semiringWithoutOne : SemiringWithoutOne _ _
semiringWithoutOne =
record { isSemiringWithoutOne = isSemiringWithoutOne }

open SemiringWithoutOne semiringWithoutOne public
using (nearSemiring)

record CommutativeSemiringWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier                         : Set c
_≈_                             : Rel Carrier ℓ
_+_                             : Op₂ Carrier
_*_                             : Op₂ Carrier
0#                              : Carrier
isCommutativeSemiringWithoutOne :
IsCommutativeSemiringWithoutOne _≈_ _+_ _*_ 0#

open IsCommutativeSemiringWithoutOne
isCommutativeSemiringWithoutOne public

semiringWithoutOne : SemiringWithoutOne _ _
semiringWithoutOne =
record { isSemiringWithoutOne = isSemiringWithoutOne }

open SemiringWithoutOne semiringWithoutOne public
using ( setoid
; +-semigroup; +-rawMonoid; +-monoid
; +-commutativeMonoid
; *-semigroup
; nearSemiring
)

record CommutativeSemiring c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier               : Set c
_≈_                   : Rel Carrier ℓ
_+_                   : Op₂ Carrier
_*_                   : Op₂ Carrier
0#                    : Carrier
1#                    : Carrier
isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1#

open IsCommutativeSemiring isCommutativeSemiring public

semiring : Semiring _ _
semiring = record { isSemiring = isSemiring }

open Semiring semiring public
using ( setoid
; +-semigroup; +-rawMonoid; +-monoid
; +-commutativeMonoid
; *-semigroup; *-rawMonoid; *-monoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
)

*-commutativeMonoid : CommutativeMonoid _ _
*-commutativeMonoid =
record { isCommutativeMonoid = *-isCommutativeMonoid }

commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne _ _
commutativeSemiringWithoutOne = record
{ isCommutativeSemiringWithoutOne = isCommutativeSemiringWithoutOne
}

------------------------------------------------------------------------
-- (Commutative) rings

-- A raw ring is a ring without any laws.

record RawRing c : Set (suc c) where
infix  8 -_
infixl 7 _*_
infixl 6 _+_
field
Carrier : Set c
_+_     : Op₂ Carrier
_*_     : Op₂ Carrier
-_      : Op₁ Carrier
0#      : Carrier
1#      : Carrier

record Ring c ℓ : Set (suc (c ⊔ ℓ)) where
infix  8 -_
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier : Set c
_≈_     : Rel Carrier ℓ
_+_     : Op₂ Carrier
_*_     : Op₂ Carrier
-_      : Op₁ Carrier
0#      : Carrier
1#      : Carrier
isRing  : IsRing _≈_ _+_ _*_ -_ 0# 1#

open IsRing isRing public

+-abelianGroup : AbelianGroup _ _
+-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }

semiring : Semiring _ _
semiring = record { isSemiring = isSemiring }

open Semiring semiring public
using ( setoid
; +-semigroup; +-rawMonoid; +-monoid
; +-commutativeMonoid
; *-semigroup; *-rawMonoid; *-monoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
)

open AbelianGroup +-abelianGroup public
using () renaming (group to +-group)

rawRing : RawRing _
rawRing = record
{ _+_ = _+_
; _*_ = _*_
; -_  = -_
; 0#  = 0#
; 1#  = 1#
}

record CommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
infix  8 -_
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier           : Set c
_≈_               : Rel Carrier ℓ
_+_               : Op₂ Carrier
_*_               : Op₂ Carrier
-_                : Op₁ Carrier
0#                : Carrier
1#                : Carrier
isCommutativeRing : IsCommutativeRing _≈_ _+_ _*_ -_ 0# 1#

open IsCommutativeRing isCommutativeRing public

ring : Ring _ _
ring = record { isRing = isRing }

commutativeSemiring : CommutativeSemiring _ _
commutativeSemiring =
record { isCommutativeSemiring = isCommutativeSemiring }

open Ring ring public using (rawRing; +-group; +-abelianGroup)
open CommutativeSemiring commutativeSemiring public
using ( setoid
; +-semigroup; +-rawMonoid; +-monoid; +-commutativeMonoid
; *-semigroup; *-rawMonoid; *-monoid; *-commutativeMonoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero; semiring
; commutativeSemiringWithoutOne
)

------------------------------------------------------------------------
-- (Distributive) lattices and boolean algebras

record Lattice c ℓ : Set (suc (c ⊔ ℓ)) where
infixr 7 _∧_
infixr 6 _∨_
infix  4 _≈_
field
Carrier   : Set c
_≈_       : Rel Carrier ℓ
_∨_       : Op₂ Carrier
_∧_       : Op₂ Carrier
isLattice : IsLattice _≈_ _∨_ _∧_

open IsLattice isLattice public

setoid : Setoid _ _
setoid = record { isEquivalence = isEquivalence }

record DistributiveLattice c ℓ : Set (suc (c ⊔ ℓ)) where
infixr 7 _∧_
infixr 6 _∨_
infix  4 _≈_
field
Carrier               : Set c
_≈_                   : Rel Carrier ℓ
_∨_                   : Op₂ Carrier
_∧_                   : Op₂ Carrier
isDistributiveLattice : IsDistributiveLattice _≈_ _∨_ _∧_

open IsDistributiveLattice isDistributiveLattice public

lattice : Lattice _ _
lattice = record { isLattice = isLattice }

open Lattice lattice public using (setoid)

record BooleanAlgebra c ℓ : Set (suc (c ⊔ ℓ)) where
infix  8 ¬_
infixr 7 _∧_
infixr 6 _∨_
infix  4 _≈_
field
Carrier          : Set c
_≈_              : Rel Carrier ℓ
_∨_              : Op₂ Carrier
_∧_              : Op₂ Carrier
¬_               : Op₁ Carrier
⊤                : Carrier
⊥                : Carrier
isBooleanAlgebra : IsBooleanAlgebra _≈_ _∨_ _∧_ ¬_ ⊤ ⊥

open IsBooleanAlgebra isBooleanAlgebra public

distributiveLattice : DistributiveLattice _ _
distributiveLattice =
record { isDistributiveLattice = isDistributiveLattice }

open DistributiveLattice distributiveLattice public
using (setoid; lattice)
```