Numeric.Algebra.Unital

Unital Multiplication (Multiplicative monoid)

class Multiplicative r => Unital r where

Minimal complete definition

one

Methods

one :: r

pow :: r -> Natural -> r infixr 8

productWith :: Foldable f => (a -> r) -> f a -> r

Instances

Unital Bool

Methods

one :: Bool

pow :: Bool -> Natural -> Bool

productWith :: Foldable f => (a -> Bool) -> f a -> Bool

Unital Int

Methods

one :: Int

pow :: Int -> Natural -> Int

productWith :: Foldable f => (a -> Int) -> f a -> Int

Unital Int8

Methods

one :: Int8

pow :: Int8 -> Natural -> Int8

productWith :: Foldable f => (a -> Int8) -> f a -> Int8

Unital Int16

Methods

one :: Int16

pow :: Int16 -> Natural -> Int16

productWith :: Foldable f => (a -> Int16) -> f a -> Int16

Unital Int32

Methods

one :: Int32

pow :: Int32 -> Natural -> Int32

productWith :: Foldable f => (a -> Int32) -> f a -> Int32

Unital Int64

Methods

one :: Int64

pow :: Int64 -> Natural -> Int64

productWith :: Foldable f => (a -> Int64) -> f a -> Int64

Unital Integer

Methods

one :: Integer

pow :: Integer -> Natural -> Integer

productWith :: Foldable f => (a -> Integer) -> f a -> Integer

Unital Natural

Methods

one :: Natural

pow :: Natural -> Natural -> Natural

productWith :: Foldable f => (a -> Natural) -> f a -> Natural

Unital Word

Methods

one :: Word

pow :: Word -> Natural -> Word

productWith :: Foldable f => (a -> Word) -> f a -> Word

Unital Word8

Methods

one :: Word8

pow :: Word8 -> Natural -> Word8

productWith :: Foldable f => (a -> Word8) -> f a -> Word8

Unital Word16

Methods

one :: Word16

pow :: Word16 -> Natural -> Word16

productWith :: Foldable f => (a -> Word16) -> f a -> Word16

Unital Word32

Methods

one :: Word32

pow :: Word32 -> Natural -> Word32

productWith :: Foldable f => (a -> Word32) -> f a -> Word32

Unital Word64

Methods

one :: Word64

pow :: Word64 -> Natural -> Word64

productWith :: Foldable f => (a -> Word64) -> f a -> Word64

Unital ()

Methods

one :: ()

pow :: () -> Natural -> ()

productWith :: Foldable f => (a -> ()) -> f a -> ()

Unital Euclidean

Methods

one :: Euclidean

pow :: Euclidean -> Natural -> Euclidean

productWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean

Rng r => Unital (RngRing r)

Methods

one :: RngRing r

pow :: RngRing r -> Natural -> RngRing r

productWith :: Foldable f => (a -> RngRing r) -> f a -> RngRing r

Unital r => Unital (Opposite r)

Methods

one :: Opposite r

pow :: Opposite r -> Natural -> Opposite r

productWith :: Foldable f => (a -> Opposite r) -> f a -> Opposite r

Unital (End r)

Methods

one :: End r

pow :: End r -> Natural -> End r

productWith :: Foldable f => (a -> End r) -> f a -> End r

Monoidal r => Unital (Exp r)

Methods

one :: Exp r

pow :: Exp r -> Natural -> Exp r

productWith :: Foldable f => (a -> Exp r) -> f a -> Exp r

(Commutative k, Ring k) => Unital (Trig k)

Methods

one :: Trig k

pow :: Trig k -> Natural -> Trig k

productWith :: Foldable f => (a -> Trig k) -> f a -> Trig k

(TriviallyInvolutive r, Ring r) => Unital (Quaternion' r)

Methods

one :: Quaternion' r

pow :: Quaternion' r -> Natural -> Quaternion' r

productWith :: Foldable f => (a -> Quaternion' r) -> f a -> Quaternion' r

(Commutative k, Rig k) => Unital (Hyper k)

Methods

one :: Hyper k

pow :: Hyper k -> Natural -> Hyper k

productWith :: Foldable f => (a -> Hyper k) -> f a -> Hyper k

Unital (BasisCoblade m)

Methods

one :: BasisCoblade m

pow :: BasisCoblade m -> Natural -> BasisCoblade m

productWith :: Foldable f => (a -> BasisCoblade m) -> f a -> BasisCoblade m

(Commutative r, Ring r) => Unital (Dual' r)

Methods

one :: Dual' r

pow :: Dual' r -> Natural -> Dual' r

productWith :: Foldable f => (a -> Dual' r) -> f a -> Dual' r

(TriviallyInvolutive r, Ring r) => Unital (Quaternion r)

Methods

one :: Quaternion r

pow :: Quaternion r -> Natural -> Quaternion r

productWith :: Foldable f => (a -> Quaternion r) -> f a -> Quaternion r

(Commutative k, Rig k) => Unital (Hyper' k)

Methods

one :: Hyper' k

pow :: Hyper' k -> Natural -> Hyper' k

productWith :: Foldable f => (a -> Hyper' k) -> f a -> Hyper' k

(Commutative r, Ring r) => Unital (Dual r)

Methods

one :: Dual r

pow :: Dual r -> Natural -> Dual r

productWith :: Foldable f => (a -> Dual r) -> f a -> Dual r

(Commutative r, Ring r) => Unital (Complex r)

Methods

one :: Complex r

pow :: Complex r -> Natural -> Complex r

productWith :: Foldable f => (a -> Complex r) -> f a -> Complex r

GCDDomain d => Unital (Fraction d)

Methods

one :: Fraction d

pow :: Fraction d -> Natural -> Fraction d

productWith :: Foldable f => (a -> Fraction d) -> f a -> Fraction d

(Unital r, UnitalAlgebra r a) => Unital (a -> r)

Methods

one :: a -> r

pow :: (a -> r) -> Natural -> a -> r

productWith :: Foldable f => (a -> a -> r) -> f a -> a -> r

(Unital a, Unital b) => Unital (a, b)

Methods

one :: (a, b)

pow :: (a, b) -> Natural -> (a, b)

productWith :: Foldable f => (a -> (a, b)) -> f a -> (a, b)

CounitalCoalgebra r m => Unital (Covector r m)

Methods

one :: Covector r m

pow :: Covector r m -> Natural -> Covector r m

productWith :: Foldable f => (a -> Covector r m) -> f a -> Covector r m

(Unital a, Unital b, Unital c) => Unital (a, b, c)

Methods

one :: (a, b, c)

pow :: (a, b, c) -> Natural -> (a, b, c)

productWith :: Foldable f => (a -> (a, b, c)) -> f a -> (a, b, c)

CounitalCoalgebra r m => Unital (Map r b m)

Methods

one :: Map r b m

pow :: Map r b m -> Natural -> Map r b m

productWith :: Foldable f => (a -> Map r b m) -> f a -> Map r b m

(Unital a, Unital b, Unital c, Unital d) => Unital (a, b, c, d)

Methods

one :: (a, b, c, d)

pow :: (a, b, c, d) -> Natural -> (a, b, c, d)

productWith :: Foldable f => (a -> (a, b, c, d)) -> f a -> (a, b, c, d)

(Unital a, Unital b, Unital c, Unital d, Unital e) => Unital (a, b, c, d, e)

Methods

one :: (a, b, c, d, e)

pow :: (a, b, c, d, e) -> Natural -> (a, b, c, d, e)

productWith :: Foldable f => (a -> (a, b, c, d, e)) -> f a -> (a, b, c, d, e)

product :: (Foldable f, Unital r) => f r -> r

Unital Associative Algebra

class Algebra r a => UnitalAlgebra r a where

An associative unital algebra over a semiring, built using a free module

Minimal complete definition

unit

Methods

unit :: r -> a -> r

Instances

Semiring r => UnitalAlgebra r ()

Methods

unit :: r -> () -> r

(Commutative k, Rng k) => UnitalAlgebra k TrigBasis

Methods

unit :: k -> TrigBasis -> k

(TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis'

Methods

unit :: r -> QuaternionBasis' -> r

Semiring k => UnitalAlgebra k HyperBasis

Methods

unit :: k -> HyperBasis -> k

Semiring k => UnitalAlgebra k DualBasis'

Methods

unit :: k -> DualBasis' -> k

(TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis

Methods

unit :: r -> QuaternionBasis -> r

(Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis'

Methods

unit :: k -> HyperBasis' -> k

Rng k => UnitalAlgebra k DualBasis

Methods

unit :: k -> DualBasis -> k

Rng k => UnitalAlgebra k ComplexBasis

Methods

unit :: k -> ComplexBasis -> k

(Monoidal r, Semiring r) => UnitalAlgebra r (Seq a)

Methods

unit :: r -> Seq a -> r

(Monoidal r, Semiring r) => UnitalAlgebra r [a]

Methods

unit :: r -> [a] -> r

(Commutative r, Monoidal r, Semiring r, LocallyFiniteOrder a) => UnitalAlgebra r (Interval a)

Methods

unit :: r -> Interval a -> r

(UnitalAlgebra r a, UnitalAlgebra r b) => UnitalAlgebra r (a, b)

Methods

unit :: r -> (a, b) -> r

(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c) => UnitalAlgebra r (a, b, c)

Methods

unit :: r -> (a, b, c) -> r

(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c, UnitalAlgebra r d) => UnitalAlgebra r (a, b, c, d)

Methods

unit :: r -> (a, b, c, d) -> r

(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c, UnitalAlgebra r d, UnitalAlgebra r e) => UnitalAlgebra r (a, b, c, d, e)

Methods

unit :: r -> (a, b, c, d, e) -> r

Unital Coassociative Coalgebra

class Coalgebra r c => CounitalCoalgebra r c where

Minimal complete definition

counit

Methods

counit :: (c -> r) -> r

Instances

Semiring r => CounitalCoalgebra r ()

Methods

counit :: (() -> r) -> r

(Commutative k, Rng k) => CounitalCoalgebra k TrigBasis

Methods

counit :: (TrigBasis -> k) -> k

(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis'

Methods

counit :: (QuaternionBasis' -> r) -> r

(Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis

Methods

counit :: (HyperBasis -> k) -> k

Rng k => CounitalCoalgebra k DualBasis'

Methods

counit :: (DualBasis' -> k) -> k

(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis

Methods

counit :: (QuaternionBasis -> r) -> r

(Commutative k, Monoidal k, Semiring k) => CounitalCoalgebra k HyperBasis'

Methods

counit :: (HyperBasis' -> k) -> k

Rng k => CounitalCoalgebra k DualBasis

Methods

counit :: (DualBasis -> k) -> k

Rng k => CounitalCoalgebra k ComplexBasis

Methods

counit :: (ComplexBasis -> k) -> k

Semiring r => CounitalCoalgebra r (Seq a)

Methods

counit :: (Seq a -> r) -> r

Semiring r => CounitalCoalgebra r [a]

Methods

counit :: ([a] -> r) -> r

(Commutative r, Monoidal r, Semiring r, PartialMonoid a) => CounitalCoalgebra r (Morphism a)

Methods

counit :: (Morphism a -> r) -> r

(Eq a, Bounded a, Commutative r, Monoidal r, Semiring r) => CounitalCoalgebra r (Interval' a)

Methods

counit :: (Interval' a -> r) -> r

Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m)

Methods

counit :: (BasisCoblade m -> r) -> r

(CounitalCoalgebra r a, CounitalCoalgebra r b) => CounitalCoalgebra r (a, b)

Methods

counit :: ((a, b) -> r) -> r

(Unital r, UnitalAlgebra r m) => CounitalCoalgebra r (m -> r)

Methods

counit :: ((m -> r) -> r) -> r

(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c) => CounitalCoalgebra r (a, b, c)

Methods

counit :: ((a, b, c) -> r) -> r

(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c, CounitalCoalgebra r d) => CounitalCoalgebra r (a, b, c, d)

Methods

counit :: ((a, b, c, d) -> r) -> r

(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c, CounitalCoalgebra r d, CounitalCoalgebra r e) => CounitalCoalgebra r (a, b, c, d, e)

Methods

counit :: ((a, b, c, d, e) -> r) -> r

Bialgebra

class (UnitalAlgebra r a, CounitalCoalgebra r a) => Bialgebra r a

A bialgebra is both a unital algebra and counital coalgebra where the mult and unit are compatible in some sense with the comult and counit. That is to say that mult and unit are a coalgebra homomorphisms or (equivalently) that comult and counit are an algebra homomorphisms.

Instances

Semiring r => Bialgebra r ()

(Commutative k, Rng k) => Bialgebra k TrigBasis

(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis'

(Commutative k, Semiring k) => Bialgebra k HyperBasis

Rng k => Bialgebra k DualBasis'

(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis

(Commutative k, Monoidal k, Semiring k) => Bialgebra k HyperBasis'

Rng k => Bialgebra k DualBasis

Rng k => Bialgebra k ComplexBasis

(Monoidal r, Semiring r) => Bialgebra r (Seq a)

(Monoidal r, Semiring r) => Bialgebra r [a]

(Bialgebra r a, Bialgebra r b) => Bialgebra r (a, b)

(Bialgebra r a, Bialgebra r b, Bialgebra r c) => Bialgebra r (a, b, c)

(Bialgebra r a, Bialgebra r b, Bialgebra r c, Bialgebra r d) => Bialgebra r (a, b, c, d)

(Bialgebra r a, Bialgebra r b, Bialgebra r c, Bialgebra r d, Bialgebra r e) => Bialgebra r (a, b, c, d, e)