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Bundles.agda
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Definitions of algebraic structures like monoids and rings
-- (packed in records together with sets, operations, etc.)
------------------------------------------------------------------------
-- The contents of this module should be accessed via `Algebra`.
{-# OPTIONS --cubical-compatible --safe #-}
module Algebra.Bundles where
import Algebra.Bundles.Raw as Raw
open import Algebra.Core
open import Algebra.Structures
open import Relation.Binary.Core using (Rel)
open import Level
------------------------------------------------------------------------
-- Re-export definitions of 'raw' bundles
open Raw public
using ( RawSuccessorSet; RawMagma; RawMonoid; RawGroup
; RawNearSemiring; RawSemiring
; RawRingWithoutOne; RawRing
; RawQuasigroup; RawLoop; RawKleeneAlgebra)
------------------------------------------------------------------------
-- Bundles with 1 unary operation & 1 element
------------------------------------------------------------------------
record SuccessorSet c ℓ : Set (suc (c ⊔ ℓ)) where
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
suc# : Op₁ Carrier
zero# : Carrier
isSuccessorSet : IsSuccessorSet _≈_ suc# zero#
open IsSuccessorSet isSuccessorSet public
rawSuccessorSet : RawSuccessorSet _ _
rawSuccessorSet = record { _≈_ = _≈_; suc# = suc#; zero# = zero# }
open RawSuccessorSet rawSuccessorSet public
------------------------------------------------------------------------
-- Bundles with 1 binary operation
------------------------------------------------------------------------
record Magma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isMagma : IsMagma _≈_ _∙_
open IsMagma isMagma public
rawMagma : RawMagma _ _
rawMagma = record { _≈_ = _≈_; _∙_ = _∙_ }
open RawMagma rawMagma public
using (_≉_)
record SelectiveMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isSelectiveMagma : IsSelectiveMagma _≈_ _∙_
open IsSelectiveMagma isSelectiveMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public using (rawMagma)
record CommutativeMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isCommutativeMagma : IsCommutativeMagma _≈_ _∙_
open IsCommutativeMagma isCommutativeMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public using (rawMagma)
record IdempotentMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isIdempotentMagma : IsIdempotentMagma _≈_ _∙_
open IsIdempotentMagma isIdempotentMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (rawMagma)
record AlternativeMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isAlternativeMagma : IsAlternativeMagma _≈_ _∙_
open IsAlternativeMagma isAlternativeMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (rawMagma)
record FlexibleMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isFlexibleMagma : IsFlexibleMagma _≈_ _∙_
open IsFlexibleMagma isFlexibleMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (rawMagma)
record MedialMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isMedialMagma : IsMedialMagma _≈_ _∙_
open IsMedialMagma isMedialMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (rawMagma)
record SemimedialMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isSemimedialMagma : IsSemimedialMagma _≈_ _∙_
open IsSemimedialMagma isSemimedialMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (rawMagma)
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
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (_≉_; rawMagma)
record Band c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isBand : IsBand _≈_ _∙_
open IsBand isBand public
semigroup : Semigroup c ℓ
semigroup = record { isSemigroup = isSemigroup }
open Semigroup semigroup public
using (_≉_; magma; rawMagma)
record CommutativeSemigroup c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isCommutativeSemigroup : IsCommutativeSemigroup _≈_ _∙_
open IsCommutativeSemigroup isCommutativeSemigroup public
semigroup : Semigroup c ℓ
semigroup = record { isSemigroup = isSemigroup }
open Semigroup semigroup public
using (_≉_; magma; rawMagma)
commutativeMagma : CommutativeMagma c ℓ
commutativeMagma = record { isCommutativeMagma = isCommutativeMagma }
------------------------------------------------------------------------
-- Bundles with 1 binary operation & 1 element
------------------------------------------------------------------------
record UnitalMagma c ℓ : Set (suc (c ⊔ ℓ)) where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
ε : Carrier
isUnitalMagma : IsUnitalMagma _≈_ _∙_ ε
open IsUnitalMagma isUnitalMagma public
magma : Magma c ℓ
magma = record { isMagma = isMagma }
open Magma magma public
using (_≉_; rawMagma)
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 (_≉_; rawMagma; magma)
rawMonoid : RawMonoid _ _
rawMonoid = record { _≈_ = _≈_; _∙_ = _∙_; ε = ε}
unitalMagma : UnitalMagma _ _
unitalMagma = record { isUnitalMagma = isUnitalMagma }
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 (_≉_; rawMagma; magma; semigroup; unitalMagma; rawMonoid)
commutativeSemigroup : CommutativeSemigroup _ _
commutativeSemigroup = record { isCommutativeSemigroup = isCommutativeSemigroup }
open CommutativeSemigroup commutativeSemigroup public
using (commutativeMagma)
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
( _≉_; rawMagma; magma; unitalMagma; commutativeMagma
; semigroup; commutativeSemigroup
; rawMonoid; monoid
)
-- Idempotent commutative monoids are also known as bounded lattices.
-- Note that the BoundedLattice necessarily uses the notation inherited
-- from monoids rather than lattices.
BoundedLattice = IdempotentCommutativeMonoid
module BoundedLattice {c ℓ} (idemCommMonoid : IdempotentCommutativeMonoid c ℓ) =
IdempotentCommutativeMonoid idemCommMonoid
------------------------------------------------------------------------
-- Bundles with 1 binary operation, 1 unary operation & 1 element
------------------------------------------------------------------------
record InvertibleMagma 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
isInvertibleMagma : IsInvertibleMagma _≈_ _∙_ ε _⁻¹
open IsInvertibleMagma isInvertibleMagma public
magma : Magma _ _
magma = record { isMagma = isMagma }
open Magma magma public
using (_≉_; rawMagma)
record InvertibleUnitalMagma 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
isInvertibleUnitalMagma : IsInvertibleUnitalMagma _≈_ _∙_ ε _⁻¹
open IsInvertibleUnitalMagma isInvertibleUnitalMagma public
invertibleMagma : InvertibleMagma _ _
invertibleMagma = record { isInvertibleMagma = isInvertibleMagma }
open InvertibleMagma invertibleMagma public
using (_≉_; rawMagma; magma)
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
rawGroup : RawGroup _ _
rawGroup = record { _≈_ = _≈_; _∙_ = _∙_; ε = ε; _⁻¹ = _⁻¹}
monoid : Monoid _ _
monoid = record { isMonoid = isMonoid }
open Monoid monoid public
using (_≉_; rawMagma; magma; semigroup; unitalMagma; rawMonoid)
invertibleMagma : InvertibleMagma c ℓ
invertibleMagma = record
{ isInvertibleMagma = isInvertibleMagma
}
invertibleUnitalMagma : InvertibleUnitalMagma c ℓ
invertibleUnitalMagma = record
{ isInvertibleUnitalMagma = isInvertibleUnitalMagma
}
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
(_≉_; rawMagma; magma; semigroup
; rawMonoid; monoid; rawGroup; invertibleMagma; invertibleUnitalMagma
)
commutativeMonoid : CommutativeMonoid _ _
commutativeMonoid = record { isCommutativeMonoid = isCommutativeMonoid }
open CommutativeMonoid commutativeMonoid public
using (commutativeMagma; commutativeSemigroup)
------------------------------------------------------------------------
-- Bundles with 2 binary operations & 1 element
------------------------------------------------------------------------
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
rawNearSemiring : RawNearSemiring _ _
rawNearSemiring = record
{ _≈_ = _≈_
; _+_ = _+_
; _*_ = _*_
; 0# = 0#
}
+-monoid : Monoid _ _
+-monoid = record { isMonoid = +-isMonoid }
open Monoid +-monoid public
using (_≉_) renaming
( rawMagma to +-rawMagma
; magma to +-magma
; semigroup to +-semigroup
; unitalMagma to +-unitalMagma
; rawMonoid to +-rawMonoid
)
*-semigroup : Semigroup _ _
*-semigroup = record { isSemigroup = *-isSemigroup }
open Semigroup *-semigroup public
using () renaming
( rawMagma to *-rawMagma
; magma to *-magma
)
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
( +-rawMagma; +-magma; +-unitalMagma; +-semigroup
; +-rawMonoid; +-monoid
; *-rawMagma; *-magma; *-semigroup
; rawNearSemiring
)
+-commutativeMonoid : CommutativeMonoid _ _
+-commutativeMonoid = record { isCommutativeMonoid = +-isCommutativeMonoid }
open CommutativeMonoid +-commutativeMonoid public
using () renaming
( commutativeMagma to +-commutativeMagma
; commutativeSemigroup to +-commutativeSemigroup
)
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
( +-rawMagma; +-magma; +-unitalMagma; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-semigroup
; +-rawMonoid; +-monoid; +-commutativeMonoid
; nearSemiring; rawNearSemiring
)
------------------------------------------------------------------------
-- Bundles with 2 binary operations & 2 elements
------------------------------------------------------------------------
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
rawSemiring : RawSemiring c ℓ
rawSemiring = record
{ _≈_ = _≈_
; _+_ = _+_
; _*_ = _*_
; 0# = 0#
; 1# = 1#
}
open RawSemiring rawSemiring public
using (rawNearSemiring)
+-commutativeMonoid : CommutativeMonoid _ _
+-commutativeMonoid =
record { isCommutativeMonoid = +-isCommutativeMonoid }
open CommutativeMonoid +-commutativeMonoid public
using (_≉_) renaming
( rawMagma to +-rawMagma
; magma to +-magma
; unitalMagma to +-unitalMagma
; commutativeMagma to +-commutativeMagma
; semigroup to +-semigroup
; commutativeSemigroup to +-commutativeSemigroup
; rawMonoid to +-rawMonoid
; monoid to +-monoid
)
*-monoid : Monoid _ _
*-monoid = record { isMonoid = *-isMonoid }
open Monoid *-monoid public
using () renaming
( rawMagma to *-rawMagma
; magma to *-magma
; 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
( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma
; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-semigroup
; +-rawMonoid; +-monoid; +-commutativeMonoid
; *-rawMonoid; *-monoid
; rawNearSemiring ; rawSemiring
)
semiringWithoutOne : SemiringWithoutOne _ _
semiringWithoutOne =
record { isSemiringWithoutOne = isSemiringWithoutOne }
open SemiringWithoutOne semiringWithoutOne public
using (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
( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma
; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-semigroup
; +-rawMonoid; +-monoid; +-commutativeMonoid
; *-rawMonoid; *-monoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
; rawSemiring
)
*-commutativeMonoid : CommutativeMonoid _ _
*-commutativeMonoid = record
{ isCommutativeMonoid = *-isCommutativeMonoid
}
open CommutativeMonoid *-commutativeMonoid public
using () renaming
( commutativeMagma to *-commutativeMagma
; commutativeSemigroup to *-commutativeSemigroup
)
commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne _ _
commutativeSemiringWithoutOne = record
{ isCommutativeSemiringWithoutOne = isCommutativeSemiringWithoutOne
}
record CancellativeCommutativeSemiring 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
isCancellativeCommutativeSemiring : IsCancellativeCommutativeSemiring _≈_ _+_ _*_ 0# 1#
open IsCancellativeCommutativeSemiring isCancellativeCommutativeSemiring public
commutativeSemiring : CommutativeSemiring c ℓ
commutativeSemiring = record
{ isCommutativeSemiring = isCommutativeSemiring
}
open CommutativeSemiring commutativeSemiring public
using
( +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma
; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-commutativeMagma; *-semigroup; *-commutativeSemigroup
; +-rawMonoid; +-monoid; +-commutativeMonoid
; *-rawMonoid; *-monoid; *-commutativeMonoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
; rawSemiring
; semiring
; _≉_
)
record IdempotentSemiring 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
isIdempotentSemiring : IsIdempotentSemiring _≈_ _+_ _*_ 0# 1#
open IsIdempotentSemiring isIdempotentSemiring public
semiring : Semiring _ _
semiring = record { isSemiring = isSemiring }
open Semiring semiring public
using
( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma
; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-semigroup
; +-rawMonoid; +-monoid; +-commutativeMonoid
; *-rawMonoid; *-monoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
; rawSemiring
)
record KleeneAlgebra 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
isKleeneAlgebra : IsKleeneAlgebra _≈_ _+_ _*_ _⋆ 0# 1#
open IsKleeneAlgebra isKleeneAlgebra public
idempotentSemiring : IdempotentSemiring _ _
idempotentSemiring = record { isIdempotentSemiring = isIdempotentSemiring }
open IdempotentSemiring idempotentSemiring public
using
( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma
; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-semigroup
; +-rawMonoid; +-monoid; +-commutativeMonoid
; *-rawMonoid; *-monoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
; rawSemiring; semiring
)
record Quasiring 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
isQuasiring : IsQuasiring _≈_ _+_ _*_ 0# 1#
open IsQuasiring isQuasiring public
+-monoid : Monoid _ _
+-monoid = record { isMonoid = +-isMonoid }
open Monoid +-monoid public
using (_≉_) renaming
( rawMagma to +-rawMagma
; magma to +-magma
; semigroup to +-semigroup
; unitalMagma to +-unitalMagma
; rawMonoid to +-rawMonoid
)
*-monoid : Monoid _ _
*-monoid = record { isMonoid = *-isMonoid }
open Monoid *-monoid public
using () renaming
( rawMagma to *-rawMagma
; magma to *-magma
; semigroup to *-semigroup
; rawMonoid to *-rawMonoid
)
------------------------------------------------------------------------
-- Bundles with 2 binary operations, 1 unary operation & 1 element
------------------------------------------------------------------------
record RingWithoutOne 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
isRingWithoutOne : IsRingWithoutOne _≈_ _+_ _*_ -_ 0#
open IsRingWithoutOne isRingWithoutOne public
+-abelianGroup : AbelianGroup _ _
+-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }
*-semigroup : Semigroup _ _
*-semigroup = record { isSemigroup = *-isSemigroup }
open AbelianGroup +-abelianGroup public
using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)
open Semigroup *-semigroup public
using () renaming
( rawMagma to *-rawMagma
; magma to *-magma
)
------------------------------------------------------------------------
-- Bundles with 2 binary operations, 1 unary operation & 2 elements
------------------------------------------------------------------------
record NonAssociativeRing 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
isNonAssociativeRing : IsNonAssociativeRing _≈_ _+_ _*_ -_ 0# 1#
open IsNonAssociativeRing isNonAssociativeRing public
+-abelianGroup : AbelianGroup _ _
+-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }
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 _+_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
1# : Carrier
isNearring : IsNearring _≈_ _+_ _*_ 0# 1# -_
open IsNearring isNearring public
quasiring : Quasiring _ _
quasiring = record { isQuasiring = isQuasiring }
open Quasiring quasiring public
using
(_≉_; +-rawMagma; +-magma; +-unitalMagma; +-semigroup; +-monoid; +-rawMonoid
;*-rawMagma; *-magma; *-semigroup; *-monoid
)
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 }
ringWithoutOne : RingWithoutOne _ _
ringWithoutOne = record { isRingWithoutOne = isRingWithoutOne }
semiring : Semiring _ _
semiring = record { isSemiring = isSemiring }
open Semiring semiring public
using
( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-commutativeMagma
; +-semigroup; +-commutativeSemigroup
; *-rawMagma; *-magma; *-semigroup
; +-rawMonoid; +-monoid ; +-commutativeMonoid
; *-rawMonoid; *-monoid
; nearSemiring; semiringWithoutOne
; semiringWithoutAnnihilatingZero
)
open NearSemiring nearSemiring public
using (rawNearSemiring)
open AbelianGroup +-abelianGroup public
using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)
rawRing : RawRing _ _
rawRing = record
{ _≈_ = _≈_
; _+_ = _+_
; _*_ = _*_
; -_ = -_
; 0# = 0#
; 1# = 1#
}
open RawRing rawRing public
using (rawRingWithoutOne; +-rawGroup)