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

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 -> r

Unital Associative Algebra

class Algebra r a => UnitalAlgebra r a where

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

Methods

unit :: r -> a -> r

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 where

Methods

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

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

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)