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Numeric.Algebra.Unital

Contents

Unital Multiplication (Multiplicative monoid)
Unital Associative Algebra
Unital Coassociative Coalgebra
Bialgebra

Synopsis

class Multiplicative r => Unital r where
product :: (Foldable f, Unital r) => f r -> r
class Algebra r a => UnitalAlgebra r a where
class Coalgebra r c => CounitalCoalgebra r c where
class (UnitalAlgebra r a, CounitalCoalgebra r a) => Bialgebra r a

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 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Bool # pow :: Bool -> Natural -> Bool # productWith :: Foldable f => (a -> Bool) -> f a -> Bool #
Unital Int #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int # pow :: Int -> Natural -> Int # productWith :: Foldable f => (a -> Int) -> f a -> Int #
Unital Int8 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int8 # pow :: Int8 -> Natural -> Int8 # productWith :: Foldable f => (a -> Int8) -> f a -> Int8 #
Unital Int16 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int16 # pow :: Int16 -> Natural -> Int16 # productWith :: Foldable f => (a -> Int16) -> f a -> Int16 #
Unital Int32 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int32 # pow :: Int32 -> Natural -> Int32 # productWith :: Foldable f => (a -> Int32) -> f a -> Int32 #
Unital Int64 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int64 # pow :: Int64 -> Natural -> Int64 # productWith :: Foldable f => (a -> Int64) -> f a -> Int64 #
Unital Integer #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Integer # pow :: Integer -> Natural -> Integer # productWith :: Foldable f => (a -> Integer) -> f a -> Integer #
Unital Natural #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Natural # pow :: Natural -> Natural -> Natural # productWith :: Foldable f => (a -> Natural) -> f a -> Natural #
Unital Word #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word # pow :: Word -> Natural -> Word # productWith :: Foldable f => (a -> Word) -> f a -> Word #
Unital Word8 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word8 # pow :: Word8 -> Natural -> Word8 # productWith :: Foldable f => (a -> Word8) -> f a -> Word8 #
Unital Word16 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word16 # pow :: Word16 -> Natural -> Word16 # productWith :: Foldable f => (a -> Word16) -> f a -> Word16 #
Unital Word32 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word32 # pow :: Word32 -> Natural -> Word32 # productWith :: Foldable f => (a -> Word32) -> f a -> Word32 #
Unital Word64 #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word64 # pow :: Word64 -> Natural -> Word64 # productWith :: Foldable f => (a -> Word64) -> f a -> Word64 #
Unital () #
Instance details Defined in Numeric.Algebra.Unital Methods one :: () # pow :: () -> Natural -> () # productWith :: Foldable f => (a -> ()) -> f a -> () #
Unital Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods one :: Euclidean # pow :: Euclidean -> Natural -> Euclidean # productWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean #
Rng r => Unital (RngRing r) #
Instance details Defined in Numeric.Ring.Rng 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) #
Instance details Defined in Numeric.Ring.Opposite Methods one :: Opposite r # pow :: Opposite r -> Natural -> Opposite r # productWith :: Foldable f => (a -> Opposite r) -> f a -> Opposite r #
Unital (End r) #
Instance details Defined in Numeric.Ring.Endomorphism 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) #
Instance details Defined in Numeric.Exp 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) #
Instance details Defined in Numeric.Coalgebra.Trigonometric 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) #
Instance details Defined in Numeric.Coalgebra.Quaternion 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) #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods one :: Hyper k # pow :: Hyper k -> Natural -> Hyper k # productWith :: Foldable f => (a -> Hyper k) -> f a -> Hyper k #
Unital (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Dual 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) #
Instance details Defined in Numeric.Algebra.Quaternion 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) #
Instance details Defined in Numeric.Algebra.Hyperbolic 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) #
Instance details Defined in Numeric.Algebra.Dual 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) #
Instance details Defined in Numeric.Algebra.Complex 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) #
Instance details Defined in Numeric.Field.Fraction 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) #
Instance details Defined in Numeric.Algebra.Unital Methods one :: a -> r # pow :: (a -> r) -> Natural -> a -> r # productWith :: Foldable f => (a0 -> a -> r) -> f a0 -> a -> r #
(Unital a, Unital b) => Unital (a, b) #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b) # pow :: (a, b) -> Natural -> (a, b) # productWith :: Foldable f => (a0 -> (a, b)) -> f a0 -> (a, b) #
CounitalCoalgebra r m => Unital (Covector r m) #
Instance details Defined in Numeric.Covector 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) #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b, c) # pow :: (a, b, c) -> Natural -> (a, b, c) # productWith :: Foldable f => (a0 -> (a, b, c)) -> f a0 -> (a, b, c) #
CounitalCoalgebra r m => Unital (Map r b m) #
Instance details Defined in Numeric.Map 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) #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b, c, d) # pow :: (a, b, c, d) -> Natural -> (a, b, c, d) # productWith :: Foldable f => (a0 -> (a, b, c, d)) -> f a0 -> (a, b, c, d) #
(Unital a, Unital b, Unital c, Unital d, Unital e) => Unital (a, b, c, d, e) #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b, c, d, e) # pow :: (a, b, c, d, e) -> Natural -> (a, b, c, d, e) # productWith :: Foldable f => (a0 -> (a, b, c, d, e)) -> f a0 -> (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 () #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> () -> r #
(Commutative k, Rng k) => UnitalAlgebra k TrigBasis #
Instance details Defined in Numeric.Coalgebra.Trigonometric Methods unit :: k -> TrigBasis -> k #
(TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis' #
Instance details Defined in Numeric.Coalgebra.Quaternion Methods unit :: r -> QuaternionBasis' -> r #
Semiring k => UnitalAlgebra k HyperBasis #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods unit :: k -> HyperBasis -> k #
Semiring k => UnitalAlgebra k DualBasis' #
Instance details Defined in Numeric.Coalgebra.Dual Methods unit :: k -> DualBasis' -> k #
(TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis #
Instance details Defined in Numeric.Algebra.Quaternion Methods unit :: r -> QuaternionBasis -> r #
(Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis' #
Instance details Defined in Numeric.Algebra.Hyperbolic Methods unit :: k -> HyperBasis' -> k #
Rng k => UnitalAlgebra k DualBasis #
Instance details Defined in Numeric.Algebra.Dual Methods unit :: k -> DualBasis -> k #
Rng k => UnitalAlgebra k ComplexBasis #
Instance details Defined in Numeric.Algebra.Complex Methods unit :: k -> ComplexBasis -> k #
(Monoidal r, Semiring r) => UnitalAlgebra r (Seq a) #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> Seq a -> r #
(Monoidal r, Semiring r) => UnitalAlgebra r [a] #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> [a] -> r #
(Commutative r, Monoidal r, Semiring r, LocallyFiniteOrder a) => UnitalAlgebra r (Interval a) #
Instance details Defined in Numeric.Algebra.Incidence Methods unit :: r -> Interval a -> r #
(UnitalAlgebra r a, UnitalAlgebra r b) => UnitalAlgebra r (a, b) #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> (a, b) -> r #
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c) => UnitalAlgebra r (a, b, c) #
Instance details Defined in Numeric.Algebra.Unital 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) #
Instance details Defined in Numeric.Algebra.Unital 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) #
Instance details Defined in Numeric.Algebra.Unital 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 () #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: (() -> r) -> r #
(Commutative k, Rng k) => CounitalCoalgebra k TrigBasis #
Instance details Defined in Numeric.Coalgebra.Trigonometric Methods counit :: (TrigBasis -> k) -> k #
(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis' #
Instance details Defined in Numeric.Coalgebra.Quaternion Methods counit :: (QuaternionBasis' -> r) -> r #
(Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods counit :: (HyperBasis -> k) -> k #
Rng k => CounitalCoalgebra k DualBasis' #
Instance details Defined in Numeric.Coalgebra.Dual Methods counit :: (DualBasis' -> k) -> k #
(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis #
Instance details Defined in Numeric.Algebra.Quaternion Methods counit :: (QuaternionBasis -> r) -> r #
(Commutative k, Monoidal k, Semiring k) => CounitalCoalgebra k HyperBasis' #
Instance details Defined in Numeric.Algebra.Hyperbolic Methods counit :: (HyperBasis' -> k) -> k #
Rng k => CounitalCoalgebra k DualBasis #
Instance details Defined in Numeric.Algebra.Dual Methods counit :: (DualBasis -> k) -> k #
Rng k => CounitalCoalgebra k ComplexBasis #
Instance details Defined in Numeric.Algebra.Complex Methods counit :: (ComplexBasis -> k) -> k #
Semiring r => CounitalCoalgebra r (Seq a) #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: (Seq a -> r) -> r #
Semiring r => CounitalCoalgebra r [a] #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ([a] -> r) -> r #
(Commutative r, Monoidal r, Semiring r, PartialMonoid a) => CounitalCoalgebra r (Morphism a) #
Instance details Defined in Numeric.Coalgebra.Categorical Methods counit :: (Morphism a -> r) -> r #
(Eq a, Bounded a, Commutative r, Monoidal r, Semiring r) => CounitalCoalgebra r (Interval' a) #
Instance details Defined in Numeric.Coalgebra.Incidence Methods counit :: (Interval' a -> r) -> r #
Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods counit :: (BasisCoblade m -> r) -> r #
(CounitalCoalgebra r a, CounitalCoalgebra r b) => CounitalCoalgebra r (a, b) #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((a, b) -> r) -> r #
(Unital r, UnitalAlgebra r m) => CounitalCoalgebra r (m -> r) #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((m -> r) -> r) -> r #
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c) => CounitalCoalgebra r (a, b, c) #
Instance details Defined in Numeric.Algebra.Unital 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) #
Instance details Defined in Numeric.Algebra.Unital 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) #
Instance details Defined in Numeric.Algebra.Unital 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 () #
Instance details Defined in Numeric.Algebra.Unital
(Commutative k, Rng k) => Bialgebra k TrigBasis #
Instance details Defined in Numeric.Coalgebra.Trigonometric
(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis' #
Instance details Defined in Numeric.Coalgebra.Quaternion
(Commutative k, Semiring k) => Bialgebra k HyperBasis #
Instance details Defined in Numeric.Coalgebra.Hyperbolic
Rng k => Bialgebra k DualBasis' #
Instance details Defined in Numeric.Coalgebra.Dual
(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis #
Instance details Defined in Numeric.Algebra.Quaternion
(Commutative k, Monoidal k, Semiring k) => Bialgebra k HyperBasis' #
Instance details Defined in Numeric.Algebra.Hyperbolic
Rng k => Bialgebra k DualBasis #
Instance details Defined in Numeric.Algebra.Dual
Rng k => Bialgebra k ComplexBasis #
Instance details Defined in Numeric.Algebra.Complex
(Monoidal r, Semiring r) => Bialgebra r (Seq a) #
Instance details Defined in Numeric.Algebra.Unital
(Monoidal r, Semiring r) => Bialgebra r [a] #
Instance details Defined in Numeric.Algebra.Unital
(Bialgebra r a, Bialgebra r b) => Bialgebra r (a, b) #
Instance details Defined in Numeric.Algebra.Unital
(Bialgebra r a, Bialgebra r b, Bialgebra r c) => Bialgebra r (a, b, c) #
Instance details Defined in Numeric.Algebra.Unital
(Bialgebra r a, Bialgebra r b, Bialgebra r c, Bialgebra r d) => Bialgebra r (a, b, c, d) #
Instance details Defined in Numeric.Algebra.Unital
(Bialgebra r a, Bialgebra r b, Bialgebra r c, Bialgebra r d, Bialgebra r e) => Bialgebra r (a, b, c, d, e) #
Instance details Defined in Numeric.Algebra.Unital