Safe Haskell	Safe-Inferred
Language	Haskell98

Numeric.Algebra.Unital

Contents

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

Synopsis

class Multiplicative r => Unital r where
- one :: r
- pow :: r -> Natural -> r
- productWith :: Foldable f => (a -> r) -> f a -> r
product :: (Foldable f, Unital r) => f r -> 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

Unital Multiplication (Multiplicative monoid)

class Multiplicative r => Unital r where Source #

Minimal complete definition

one

Methods

one :: r Source #

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

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

Instances

Instances details

Unital Euclidean Source #
Instance details Defined in Numeric.Coalgebra.Geometric Methods one :: Euclidean Source # pow :: Euclidean -> Natural -> Euclidean Source # productWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean Source #
Unital Int16 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int16 Source # pow :: Int16 -> Natural -> Int16 Source # productWith :: Foldable f => (a -> Int16) -> f a -> Int16 Source #
Unital Int32 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int32 Source # pow :: Int32 -> Natural -> Int32 Source # productWith :: Foldable f => (a -> Int32) -> f a -> Int32 Source #
Unital Int64 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int64 Source # pow :: Int64 -> Natural -> Int64 Source # productWith :: Foldable f => (a -> Int64) -> f a -> Int64 Source #
Unital Int8 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int8 Source # pow :: Int8 -> Natural -> Int8 Source # productWith :: Foldable f => (a -> Int8) -> f a -> Int8 Source #
Unital Word16 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word16 Source # pow :: Word16 -> Natural -> Word16 Source # productWith :: Foldable f => (a -> Word16) -> f a -> Word16 Source #
Unital Word32 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word32 Source # pow :: Word32 -> Natural -> Word32 Source # productWith :: Foldable f => (a -> Word32) -> f a -> Word32 Source #
Unital Word64 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word64 Source # pow :: Word64 -> Natural -> Word64 Source # productWith :: Foldable f => (a -> Word64) -> f a -> Word64 Source #
Unital Word8 Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word8 Source # pow :: Word8 -> Natural -> Word8 Source # productWith :: Foldable f => (a -> Word8) -> f a -> Word8 Source #
Unital Integer Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Integer Source # pow :: Integer -> Natural -> Integer Source # productWith :: Foldable f => (a -> Integer) -> f a -> Integer Source #
Unital Natural Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Natural Source # pow :: Natural -> Natural -> Natural Source # productWith :: Foldable f => (a -> Natural) -> f a -> Natural Source #
Unital () Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: () Source # pow :: () -> Natural -> () Source # productWith :: Foldable f => (a -> ()) -> f a -> () Source #
Unital Bool Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Bool Source # pow :: Bool -> Natural -> Bool Source # productWith :: Foldable f => (a -> Bool) -> f a -> Bool Source #
Unital Int Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Int Source # pow :: Int -> Natural -> Int Source # productWith :: Foldable f => (a -> Int) -> f a -> Int Source #
Unital Word Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: Word Source # pow :: Word -> Natural -> Word Source # productWith :: Foldable f => (a -> Word) -> f a -> Word Source #
(Commutative r, Ring r) => Unital (Complex r) Source #
Instance details Defined in Numeric.Algebra.Complex Methods one :: Complex r Source # pow :: Complex r -> Natural -> Complex r Source # productWith :: Foldable f => (a -> Complex r) -> f a -> Complex r Source #
(Commutative r, Ring r) => Unital (Dual r) Source #
Instance details Defined in Numeric.Algebra.Dual Methods one :: Dual r Source # pow :: Dual r -> Natural -> Dual r Source # productWith :: Foldable f => (a -> Dual r) -> f a -> Dual r Source #
(Commutative k, Rig k) => Unital (Hyper' k) Source #
Instance details Defined in Numeric.Algebra.Hyperbolic Methods one :: Hyper' k Source # pow :: Hyper' k -> Natural -> Hyper' k Source # productWith :: Foldable f => (a -> Hyper' k) -> f a -> Hyper' k Source #
(TriviallyInvolutive r, Ring r) => Unital (Quaternion r) Source #
Instance details Defined in Numeric.Algebra.Quaternion Methods one :: Quaternion r Source # pow :: Quaternion r -> Natural -> Quaternion r Source # productWith :: Foldable f => (a -> Quaternion r) -> f a -> Quaternion r Source #
(Commutative r, Ring r) => Unital (Dual' r) Source #
Instance details Defined in Numeric.Coalgebra.Dual Methods one :: Dual' r Source # pow :: Dual' r -> Natural -> Dual' r Source # productWith :: Foldable f => (a -> Dual' r) -> f a -> Dual' r Source #
Unital (BasisCoblade m) Source #
Instance details Defined in Numeric.Coalgebra.Geometric Methods one :: BasisCoblade m Source # pow :: BasisCoblade m -> Natural -> BasisCoblade m Source # productWith :: Foldable f => (a -> BasisCoblade m) -> f a -> BasisCoblade m Source #
(Commutative k, Rig k) => Unital (Hyper k) Source #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods one :: Hyper k Source # pow :: Hyper k -> Natural -> Hyper k Source # productWith :: Foldable f => (a -> Hyper k) -> f a -> Hyper k Source #
(TriviallyInvolutive r, Ring r) => Unital (Quaternion' r) Source #
Instance details Defined in Numeric.Coalgebra.Quaternion Methods one :: Quaternion' r Source # pow :: Quaternion' r -> Natural -> Quaternion' r Source # productWith :: Foldable f => (a -> Quaternion' r) -> f a -> Quaternion' r Source #
(Commutative k, Ring k) => Unital (Trig k) Source #
Instance details Defined in Numeric.Coalgebra.Trigonometric Methods one :: Trig k Source # pow :: Trig k -> Natural -> Trig k Source # productWith :: Foldable f => (a -> Trig k) -> f a -> Trig k Source #
Monoidal r => Unital (Exp r) Source #
Instance details Defined in Numeric.Exp Methods one :: Exp r Source # pow :: Exp r -> Natural -> Exp r Source # productWith :: Foldable f => (a -> Exp r) -> f a -> Exp r Source #
GCDDomain d => Unital (Fraction d) Source #
Instance details Defined in Numeric.Field.Fraction Methods one :: Fraction d Source # pow :: Fraction d -> Natural -> Fraction d Source # productWith :: Foldable f => (a -> Fraction d) -> f a -> Fraction d Source #
Unital (End r) Source #
Instance details Defined in Numeric.Ring.Endomorphism Methods one :: End r Source # pow :: End r -> Natural -> End r Source # productWith :: Foldable f => (a -> End r) -> f a -> End r Source #
Unital r => Unital (Opposite r) Source #
Instance details Defined in Numeric.Ring.Opposite Methods one :: Opposite r Source # pow :: Opposite r -> Natural -> Opposite r Source # productWith :: Foldable f => (a -> Opposite r) -> f a -> Opposite r Source #
Rng r => Unital (RngRing r) Source #
Instance details Defined in Numeric.Ring.Rng Methods one :: RngRing r Source # pow :: RngRing r -> Natural -> RngRing r Source # productWith :: Foldable f => (a -> RngRing r) -> f a -> RngRing r Source #
CounitalCoalgebra r m => Unital (Covector r m) Source #
Instance details Defined in Numeric.Covector Methods one :: Covector r m Source # pow :: Covector r m -> Natural -> Covector r m Source # productWith :: Foldable f => (a -> Covector r m) -> f a -> Covector r m Source #
(Unital a, Unital b) => Unital (a, b) Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b) Source # pow :: (a, b) -> Natural -> (a, b) Source # productWith :: Foldable f => (a0 -> (a, b)) -> f a0 -> (a, b) Source #
(Unital r, UnitalAlgebra r a) => Unital (a -> r) Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: a -> r Source # pow :: (a -> r) -> Natural -> a -> r Source # productWith :: Foldable f => (a0 -> a -> r) -> f a0 -> a -> r Source #
CounitalCoalgebra r m => Unital (Map r b m) Source #
Instance details Defined in Numeric.Map Methods one :: Map r b m Source # pow :: Map r b m -> Natural -> Map r b m Source # productWith :: Foldable f => (a -> Map r b m) -> f a -> Map r b m Source #
(Unital a, Unital b, Unital c) => Unital (a, b, c) Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b, c) Source # pow :: (a, b, c) -> Natural -> (a, b, c) Source # productWith :: Foldable f => (a0 -> (a, b, c)) -> f a0 -> (a, b, c) Source #
(Unital a, Unital b, Unital c, Unital d) => Unital (a, b, c, d) Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b, c, d) Source # pow :: (a, b, c, d) -> Natural -> (a, b, c, d) Source # productWith :: Foldable f => (a0 -> (a, b, c, d)) -> f a0 -> (a, b, c, d) Source #
(Unital a, Unital b, Unital c, Unital d, Unital e) => Unital (a, b, c, d, e) Source #
Instance details Defined in Numeric.Algebra.Unital Methods one :: (a, b, c, d, e) Source # pow :: (a, b, c, d, e) -> Natural -> (a, b, c, d, e) Source # productWith :: Foldable f => (a0 -> (a, b, c, d, e)) -> f a0 -> (a, b, c, d, e) Source #

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

Unital Associative Algebra

class Algebra r a => UnitalAlgebra r a where Source #

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

Methods

unit :: r -> a -> r Source #

Instances

Instances details

Rng k => UnitalAlgebra k ComplexBasis Source #
Instance details Defined in Numeric.Algebra.Complex Methods unit :: k -> ComplexBasis -> k Source #
Rng k => UnitalAlgebra k DualBasis Source #
Instance details Defined in Numeric.Algebra.Dual Methods unit :: k -> DualBasis -> k Source #
(Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis' Source #
Instance details Defined in Numeric.Algebra.Hyperbolic Methods unit :: k -> HyperBasis' -> k Source #
Semiring k => UnitalAlgebra k DualBasis' Source #
Instance details Defined in Numeric.Coalgebra.Dual Methods unit :: k -> DualBasis' -> k Source #
Semiring k => UnitalAlgebra k HyperBasis Source #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods unit :: k -> HyperBasis -> k Source #
(Commutative k, Rng k) => UnitalAlgebra k TrigBasis Source #
Instance details Defined in Numeric.Coalgebra.Trigonometric Methods unit :: k -> TrigBasis -> k Source #
(TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis Source #
Instance details Defined in Numeric.Algebra.Quaternion Methods unit :: r -> QuaternionBasis -> r Source #
(TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis' Source #
Instance details Defined in Numeric.Coalgebra.Quaternion Methods unit :: r -> QuaternionBasis' -> r Source #
Semiring r => UnitalAlgebra r () Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> () -> r Source #
(Commutative r, Monoidal r, Semiring r, LocallyFiniteOrder a) => UnitalAlgebra r (Interval a) Source #
Instance details Defined in Numeric.Algebra.Incidence Methods unit :: r -> Interval a -> r Source #
(Monoidal r, Semiring r) => UnitalAlgebra r (Seq a) Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> Seq a -> r Source #
(Monoidal r, Semiring r) => UnitalAlgebra r [a] Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> [a] -> r Source #
(UnitalAlgebra r a, UnitalAlgebra r b) => UnitalAlgebra r (a, b) Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> (a, b) -> r Source #
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c) => UnitalAlgebra r (a, b, c) Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> (a, b, c) -> r Source #
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c, UnitalAlgebra r d) => UnitalAlgebra r (a, b, c, d) Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> (a, b, c, d) -> r Source #
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c, UnitalAlgebra r d, UnitalAlgebra r e) => UnitalAlgebra r (a, b, c, d, e) Source #
Instance details Defined in Numeric.Algebra.Unital Methods unit :: r -> (a, b, c, d, e) -> r Source #

Unital Coassociative Coalgebra

class Coalgebra r c => CounitalCoalgebra r c where Source #

Methods

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

Instances

Instances details

Rng k => CounitalCoalgebra k ComplexBasis Source #
Instance details Defined in Numeric.Algebra.Complex Methods counit :: (ComplexBasis -> k) -> k Source #
Rng k => CounitalCoalgebra k DualBasis Source #
Instance details Defined in Numeric.Algebra.Dual Methods counit :: (DualBasis -> k) -> k Source #
(Commutative k, Monoidal k, Semiring k) => CounitalCoalgebra k HyperBasis' Source #
Instance details Defined in Numeric.Algebra.Hyperbolic Methods counit :: (HyperBasis' -> k) -> k Source #
Rng k => CounitalCoalgebra k DualBasis' Source #
Instance details Defined in Numeric.Coalgebra.Dual Methods counit :: (DualBasis' -> k) -> k Source #
(Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis Source #
Instance details Defined in Numeric.Coalgebra.Hyperbolic Methods counit :: (HyperBasis -> k) -> k Source #
(Commutative k, Rng k) => CounitalCoalgebra k TrigBasis Source #
Instance details Defined in Numeric.Coalgebra.Trigonometric Methods counit :: (TrigBasis -> k) -> k Source #
(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis Source #
Instance details Defined in Numeric.Algebra.Quaternion Methods counit :: (QuaternionBasis -> r) -> r Source #
(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis' Source #
Instance details Defined in Numeric.Coalgebra.Quaternion Methods counit :: (QuaternionBasis' -> r) -> r Source #
Semiring r => CounitalCoalgebra r () Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: (() -> r) -> r Source #
(Commutative r, Monoidal r, Semiring r, PartialMonoid a) => CounitalCoalgebra r (Morphism a) Source #
Instance details Defined in Numeric.Coalgebra.Categorical Methods counit :: (Morphism a -> r) -> r Source #
Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m) Source #
Instance details Defined in Numeric.Coalgebra.Geometric Methods counit :: (BasisCoblade m -> r) -> r Source #
(Eq a, Bounded a, Commutative r, Monoidal r, Semiring r) => CounitalCoalgebra r (Interval' a) Source #
Instance details Defined in Numeric.Coalgebra.Incidence Methods counit :: (Interval' a -> r) -> r Source #
Semiring r => CounitalCoalgebra r (Seq a) Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: (Seq a -> r) -> r Source #
Semiring r => CounitalCoalgebra r [a] Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ([a] -> r) -> r Source #
(CounitalCoalgebra r a, CounitalCoalgebra r b) => CounitalCoalgebra r (a, b) Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((a, b) -> r) -> r Source #
(Unital r, UnitalAlgebra r m) => CounitalCoalgebra r (m -> r) Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((m -> r) -> r) -> r Source #
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c) => CounitalCoalgebra r (a, b, c) Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((a, b, c) -> r) -> r Source #
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c, CounitalCoalgebra r d) => CounitalCoalgebra r (a, b, c, d) Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((a, b, c, d) -> r) -> r Source #
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c, CounitalCoalgebra r d, CounitalCoalgebra r e) => CounitalCoalgebra r (a, b, c, d, e) Source #
Instance details Defined in Numeric.Algebra.Unital Methods counit :: ((a, b, c, d, e) -> r) -> r Source #

Bialgebra

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

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

Instances details

Rng k => Bialgebra k ComplexBasis Source #
Instance details Defined in Numeric.Algebra.Complex
Rng k => Bialgebra k DualBasis Source #
Instance details Defined in Numeric.Algebra.Dual
(Commutative k, Monoidal k, Semiring k) => Bialgebra k HyperBasis' Source #
Instance details Defined in Numeric.Algebra.Hyperbolic
Rng k => Bialgebra k DualBasis' Source #
Instance details Defined in Numeric.Coalgebra.Dual
(Commutative k, Semiring k) => Bialgebra k HyperBasis Source #
Instance details Defined in Numeric.Coalgebra.Hyperbolic
(Commutative k, Rng k) => Bialgebra k TrigBasis Source #
Instance details Defined in Numeric.Coalgebra.Trigonometric
(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis Source #
Instance details Defined in Numeric.Algebra.Quaternion
(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis' Source #
Instance details Defined in Numeric.Coalgebra.Quaternion
Semiring r => Bialgebra r () Source #
Instance details Defined in Numeric.Algebra.Unital
(Monoidal r, Semiring r) => Bialgebra r (Seq a) Source #
Instance details Defined in Numeric.Algebra.Unital
(Monoidal r, Semiring r) => Bialgebra r [a] Source #
Instance details Defined in Numeric.Algebra.Unital
(Bialgebra r a, Bialgebra r b) => Bialgebra r (a, b) Source #
Instance details Defined in Numeric.Algebra.Unital
(Bialgebra r a, Bialgebra r b, Bialgebra r c) => Bialgebra r (a, b, c) Source #
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) Source #
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) Source #
Instance details Defined in Numeric.Algebra.Unital