Numeric.Algebra

Contents

• Additive
  • additive semigroups
  • additive Abelian semigroups
  • additive idempotent semigroups
  • partitionable additive semigroups
  • additive monoids
  • additive groups
• Multiplicative
  • multiplicative semigroups
  • commutative multiplicative semigroups
  • multiplicative monoids
  • idempotent multiplicative semigroups
  • multiplicative groups
  • factorable multiplicative semigroups
  • involutive multiplicative semigroups
• Ring-Structures
  • Semirings
  • Rngs
  • Rigs
• Rings
  • Division Rings
• Modules
• Algebras
  • associative algebras over (non-commutative) semirings
  • unital algebras
  • involutive algebras
  • idempotent algebras
  • commutative algebras
  • division algebras
  • Hopf alegebras
• Ring Properties
  • Characteristic
  • Order
• Natural numbers
• Representable Additive
• Representable Monoidal
• Representable Group
• Representable Multiplicative (via Algebra)
• Representable Unital (via UnitalAlgebra)
• Representable Rig (via Algebra)
• Representable Ring (via Algebra)
• Norm
• Covectors
  • Covectors as linear functionals

Synopsis

• class Additive r where
  • (+) :: r -> r -> r
  • sinnum1p :: Whole n => n -> r -> r
  • sumWith1 :: Foldable1 f => (a -> r) -> f a -> r
• sum1 :: (Foldable1 f, Additive r) => f r -> r
• class Additive r => Abelian r
• class Additive r => Idempotent r
• sinnum1pIdempotent :: Natural -> r -> r
• sinnumIdempotent :: (Integral n, Idempotent r, Monoidal r) => n -> r -> r
• class Additive m => Partitionable m where
  • partitionWith :: (m -> m -> r) -> m -> NonEmpty r
• class (LeftModule Natural m, RightModule Natural m) => Monoidal m where
  • zero :: m
  • sinnum :: Whole n => n -> m -> m
  • sumWith :: Foldable f => (a -> m) -> f a -> m
• sum :: (Foldable f, Monoidal m) => f m -> m
• class (LeftModule Integer r, RightModule Integer r, Monoidal r) => Group r where
  • (-) :: r -> r -> r
  • negate :: r -> r
  • subtract :: r -> r -> r
  • times :: Integral n => n -> r -> r
• class Multiplicative r where
  • (*) :: r -> r -> r
  • pow1p :: Whole n => r -> n -> r
  • productWith1 :: Foldable1 f => (a -> r) -> f a -> r
• product1 :: (Foldable1 f, Multiplicative r) => f r -> r
• class Multiplicative r => Commutative r
• class Multiplicative r => Unital r where
  • one :: r
  • pow :: Whole n => r -> n -> r
  • productWith :: Foldable f => (a -> r) -> f a -> r
• product :: (Foldable f, Unital r) => f r -> r
• class Multiplicative r => Band r
• pow1pBand :: Whole n => r -> n -> r
• powBand :: (Unital r, Whole n) => r -> n -> r
• class Unital r => Division r where
  • recip :: r -> r
  • (/) :: r -> r -> r
  • (\\) :: r -> r -> r
  • (^) :: Integral n => r -> n -> r
• class Multiplicative m => Factorable m where
  • factorWith :: (m -> m -> r) -> m -> NonEmpty r
• class Multiplicative r => InvolutiveMultiplication r where
  • adjoint :: r -> r
• class (Commutative r, InvolutiveMultiplication r) => TriviallyInvolutive r
• class (Additive r, Abelian r, Multiplicative r) => Semiring r
• class (Semiring r, InvolutiveMultiplication r) => InvolutiveSemiring r
• class (Semiring r, Idempotent r) => Dioid r
• class (Group r, Semiring r) => Rng r
• class (Semiring r, Unital r, Monoidal r) => Rig r where
  • fromNatural :: Natural -> r
• class (Rig r, Rng r) => Ring r where
  • fromInteger :: Integer -> r
• class Ring r => LocalRing r
• class (Division r, Ring r) => DivisionRing r
• class (Commutative r, DivisionRing r) => Field r
• class (Semiring r, Additive m) => LeftModule r m where
  • (.*) :: r -> m -> m
• class (Semiring r, Additive m) => RightModule r m where
  • (*.) :: m -> r -> m
• class (LeftModule r m, RightModule r m) => Module r m
• class Semiring r => Algebra r a where
  • mult :: (a -> a -> r) -> a -> r
• class Semiring r => Coalgebra r c where
  • comult :: (c -> r) -> c -> c -> r
• class Algebra r a => UnitalAlgebra r a where
  • unit :: r -> a -> r
• class Coalgebra r c => CounitalCoalgebra r c where
  • counit :: (c -> r) -> r
• class (UnitalAlgebra r a, CounitalCoalgebra r a) => Bialgebra r a
• class (InvolutiveSemiring r, Algebra r a) => InvolutiveAlgebra r a where
  • inv :: (a -> r) -> a -> r
• class (InvolutiveSemiring r, Coalgebra r c) => InvolutiveCoalgebra r c where
  • coinv :: (c -> r) -> c -> r
• class (Bialgebra r h, InvolutiveAlgebra r h, InvolutiveCoalgebra r h) => InvolutiveBialgebra r h
• class (CommutativeAlgebra r a, TriviallyInvolutive r, InvolutiveAlgebra r a) => TriviallyInvolutiveAlgebra r a
• class (CocommutativeCoalgebra r a, TriviallyInvolutive r, InvolutiveCoalgebra r a) => TriviallyInvolutiveCoalgebra r a
• class (InvolutiveBialgebra r h, TriviallyInvolutiveAlgebra r h, TriviallyInvolutiveCoalgebra r h) => TriviallyInvolutiveBialgebra r h
• class Algebra r a => IdempotentAlgebra r a
• class (Bialgebra r h, IdempotentAlgebra r h, IdempotentCoalgebra r h) => IdempotentBialgebra r h
• class Algebra r a => CommutativeAlgebra r a
• class (Bialgebra r h, CommutativeAlgebra r h, CocommutativeCoalgebra r h) => CommutativeBialgebra r h
• class Coalgebra r c => CocommutativeCoalgebra r c
• class UnitalAlgebra r a => DivisionAlgebra r a where
  • recipriocal :: (a -> r) -> a -> r
• class Bialgebra r h => HopfAlgebra r h where
  • antipode :: (h -> r) -> h -> r
• class Rig r => Characteristic r where
  • char :: proxy r -> Natural
• charInt :: (Integral s, Bounded s) => proxy s -> Natural
• charWord :: (Whole s, Bounded s) => proxy s -> Natural
• class Order a where
  • (<~) :: a -> a -> Bool
  • (<) :: a -> a -> Bool
  • (>~) :: a -> a -> Bool
  • (>) :: a -> a -> Bool
  • (~~) :: a -> a -> Bool
  • (/~) :: a -> a -> Bool
  • order :: a -> a -> Maybe Ordering
  • comparable :: a -> a -> Bool
• class (AdditiveOrder r, Rig r) => OrderedRig r
• class (Additive r, Order r) => AdditiveOrder r
• class Order a => LocallyFiniteOrder a
• class Monoidal r => DecidableZero r
• class Unital r => DecidableUnits r
• class Unital r => DecidableAssociates r
• data Natural
• class Integral n => Whole n where
  • toNatural :: n -> Natural
• addRep :: (Applicative m, Additive r) => m r -> m r -> m r
• sinnum1pRep :: (Whole n, Functor m, Additive r) => n -> m r -> m r
• zeroRep :: (Applicative m, Monoidal r) => m r
• sinnumRep :: (Whole n, Functor m, Monoidal r) => n -> m r -> m r
• negateRep :: (Functor m, Group r) => m r -> m r
• minusRep :: (Applicative m, Group r) => m r -> m r -> m r
• subtractRep :: (Applicative m, Group r) => m r -> m r -> m r
• timesRep :: (Integral n, Functor m, Group r) => n -> m r -> m r
• mulRep :: (Representable m, Algebra r (Rep m)) => m r -> m r -> m r
• oneRep :: (Representable m, Unital r, UnitalAlgebra r (Rep m)) => m r
• fromNaturalRep :: (UnitalAlgebra r (Rep m), Representable m, Rig r) => Natural -> m r
• fromIntegerRep :: (UnitalAlgebra r (Rep m), Representable m, Ring r) => Integer -> m r
• class Additive r => Quadrance r m where
  • quadrance :: m -> r
• newtype Covector r a = Covector {
  • ($*) :: (a -> r) -> r
  }
• counitM :: UnitalAlgebra r a => a -> Covector r ()
• unitM :: CounitalCoalgebra r c => Covector r c
• comultM :: Algebra r a => a -> Covector r (a, a)
• multM :: Coalgebra r c => c -> c -> Covector r c
• invM :: InvolutiveAlgebra r h => h -> Covector r h
• coinvM :: InvolutiveCoalgebra r h => h -> Covector r h
• antipodeM :: HopfAlgebra r h => h -> Covector r h
• convolveM :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a

Additive

additive semigroups

 (a + b) + c = a + (b + c)
 sinnum 1 a = a
 sinnum (2 * n) a = sinnum n a + sinnum n a
 sinnum (2 * n + 1) a = sinnum n a + sinnum n a + a

additive Abelian semigroups

additive idempotent semigroups

 a + a = a

partitionable additive semigroups

additive monoids

 zero + a = a = a + zero

additive groups

Multiplicative

multiplicative semigroups

commutative multiplicative semigroups

multiplicative monoids

idempotent multiplicative semigroups

 a * a = a

multiplicative groups

factorable multiplicative semigroups

involutive multiplicative semigroups

 adjoint a * adjoint b = adjoint (b * a)

 adjoint = id

Ring-Structures

Semirings

 a(b + c) = ab + ac -- left distribution (we are a LeftNearSemiring)
 (a + b)c = ac + bc -- right distribution (we are a [Right]NearSemiring)

 class (LeftModule r r, RightModule r r, Additive r, Abelian r, Multiplicative r) => Semiring r

Rngs

Rigs

Rings

Division Rings

Modules

Algebras

associative algebras over (non-commutative) semirings

unital algebras

involutive algebras

idempotent algebras

commutative algebras

division algebras

Hopf alegebras

Ring Properties

Characteristic

Order

Natural numbers