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Numeric.Algebra.Unital

Contents

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

Synopsis

Unital Multiplication (Multiplicative monoid)

class Multiplicative r => Unital r whereSource

Methods

one :: rSource

pow :: Whole n => r -> n -> rSource

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

Instances

Unital Bool
Unital Int
Unital Int8
Unital Int16
Unital Int32
Unital Int64
Unital Integer
Unital Word
Unital Word8
Unital Word16
Unital Word32
Unital Word64
Unital ()
Unital Natural
Unital Euclidean
(Commutative r, Ring r) => Unital (Complex r)
(TriviallyInvolutive r, Ring r) => Unital (Quaternion r)
(Commutative r, Ring r) => Unital (Dual r)
(Commutative k, Rig k) => Unital (Hyper' k)
(Commutative k, Rig k) => Unital (Hyper k)
(Commutative r, Ring r) => Unital (Dual' r)
Unital (BasisCoblade m)
(TriviallyInvolutive r, Ring r) => Unital (Quaternion' r)
(Commutative k, Ring k) => Unital (Trig k)
Monoidal r => Unital (Exp r)
Unital (End r)
Unital r => Unital (Opposite r)
Rng r => Unital (RngRing r)
Euclidean d => Unital (Fraction d)
(Unital r, UnitalAlgebra r a) => Unital (a -> r)
(Unital a, Unital b) => Unital (a, b)
CounitalCoalgebra r m => Unital (Covector r m)
(Unital a, Unital b, Unital c) => Unital (a, b, c)
CounitalCoalgebra r m => Unital (Map r b m)
(Unital a, Unital b, Unital c, Unital d) => Unital (a, b, c, d)
(Unital a, Unital b, Unital c, Unital d, Unital e) => Unital (a, b, c, d, e)

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

Unital Associative Algebra

class Algebra r a => UnitalAlgebra r a whereSource

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

Methods

unit :: r -> a -> rSource

Instances

Semiring r => UnitalAlgebra r ()
Rng k => UnitalAlgebra k ComplexBasis
(TriviallyInvolutive r, Rng r) => UnitalAlgebra r QuaternionBasis
Rng k => UnitalAlgebra k DualBasis
(Commutative k, Monoidal k, Semiring k) => UnitalAlgebra k HyperBasis'
Semiring k => UnitalAlgebra k HyperBasis
Semiring k => UnitalAlgebra k DualBasis'
(TriviallyInvolutive r, Semiring r) => UnitalAlgebra r QuaternionBasis'
(Commutative k, Rng k) => UnitalAlgebra k TrigBasis
(Monoidal r, Semiring r) => UnitalAlgebra r (Seq a)
(Monoidal r, Semiring r) => UnitalAlgebra r [a]
(Commutative r, Monoidal r, Semiring r, LocallyFiniteOrder a) => UnitalAlgebra r (Interval a)
(UnitalAlgebra r a, UnitalAlgebra r b) => UnitalAlgebra r (a, b)
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c) => UnitalAlgebra r (a, b, c)
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c, UnitalAlgebra r d) => UnitalAlgebra r (a, b, c, d)
(UnitalAlgebra r a, UnitalAlgebra r b, UnitalAlgebra r c, UnitalAlgebra r d, UnitalAlgebra r e) => UnitalAlgebra r (a, b, c, d, e)

Unital Coassociative Coalgebra

class Coalgebra r c => CounitalCoalgebra r c whereSource

Methods

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

Instances

Semiring r => CounitalCoalgebra r ()
Rng k => CounitalCoalgebra k ComplexBasis
(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis
Rng k => CounitalCoalgebra k DualBasis
(Commutative k, Monoidal k, Semiring k) => CounitalCoalgebra k HyperBasis'
(Commutative k, Semiring k) => CounitalCoalgebra k HyperBasis
Rng k => CounitalCoalgebra k DualBasis'
(TriviallyInvolutive r, Rng r) => CounitalCoalgebra r QuaternionBasis'
(Commutative k, Rng k) => CounitalCoalgebra k TrigBasis
Semiring r => CounitalCoalgebra r (Seq a)
Semiring r => CounitalCoalgebra r [a]
(Commutative r, Monoidal r, Semiring r, PartialMonoid a) => CounitalCoalgebra r (Morphism a)
(Eq a, Bounded a, Commutative r, Monoidal r, Semiring r) => CounitalCoalgebra r (Interval' a)
Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m)
(CounitalCoalgebra r a, CounitalCoalgebra r b) => CounitalCoalgebra r (a, b)
(Unital r, UnitalAlgebra r m) => CounitalCoalgebra r (m -> r)
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c) => CounitalCoalgebra r (a, b, c)
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c, CounitalCoalgebra r d) => CounitalCoalgebra r (a, b, c, d)
(CounitalCoalgebra r a, CounitalCoalgebra r b, CounitalCoalgebra r c, CounitalCoalgebra r d, CounitalCoalgebra r e) => CounitalCoalgebra r (a, b, c, d, e)

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

Semiring r => Bialgebra r ()
Rng k => Bialgebra k ComplexBasis
(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis
Rng k => Bialgebra k DualBasis
(Commutative k, Monoidal k, Semiring k) => Bialgebra k HyperBasis'
(Commutative k, Semiring k) => Bialgebra k HyperBasis
Rng k => Bialgebra k DualBasis'
(TriviallyInvolutive r, Rng r) => Bialgebra r QuaternionBasis'
(Commutative k, Rng k) => Bialgebra k TrigBasis
(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)