Numeric.Algebra.Unital

Unital Multiplication (Multiplicative monoid)

class Multiplicative r => Unital r where

Methods

one :: r

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

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

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

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

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

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

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

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

Multiplicative (BasisCoblade m) => Unital (BasisCoblade m)

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

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

(Multiplicative (Exp r), Monoidal r) => Unital (Exp r)

Multiplicative (End r) => Unital (End r)

(Multiplicative (Opposite r), Unital r) => Unital (Opposite r)

(Multiplicative (RngRing r), Rng r) => Unital (RngRing r)

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

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

(Multiplicative (Covector r m), CounitalCoalgebra r m) => Unital (Covector r m)

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

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

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

(Multiplicative (a, b, c, d, e), 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

(Algebra r (), Semiring r) => UnitalAlgebra r ()

(Algebra k ComplexBasis, Rng k) => UnitalAlgebra k ComplexBasis

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

(Algebra k DualBasis, Rng k) => UnitalAlgebra k DualBasis

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

(Algebra k HyperBasis, Semiring k) => UnitalAlgebra k HyperBasis

(Algebra k DualBasis', Semiring k) => UnitalAlgebra k DualBasis'

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

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

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

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

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

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

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

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

(Algebra r (a, b, c, d, e), 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

(Coalgebra r (), Semiring r) => CounitalCoalgebra r ()

(Coalgebra k ComplexBasis, Rng k) => CounitalCoalgebra k ComplexBasis

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

(Coalgebra k DualBasis, Rng k) => CounitalCoalgebra k DualBasis

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

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

(Coalgebra k DualBasis', Rng k) => CounitalCoalgebra k DualBasis'

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

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

(Coalgebra r (Seq a), Semiring r) => CounitalCoalgebra r (Seq a)

(Coalgebra r [a], Semiring r) => CounitalCoalgebra r [a]

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

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

(Coalgebra r (BasisCoblade m), Eigenmetric r m) => CounitalCoalgebra r (BasisCoblade m)

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

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

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

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

(Coalgebra r (a, b, c, d, e), 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

(UnitalAlgebra r (), CounitalCoalgebra r (), Semiring r) => Bialgebra r ()

(UnitalAlgebra k ComplexBasis, CounitalCoalgebra k ComplexBasis, Rng k) => Bialgebra k ComplexBasis

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

(UnitalAlgebra k DualBasis, CounitalCoalgebra k DualBasis, Rng k) => Bialgebra k DualBasis

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

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

(UnitalAlgebra k DualBasis', CounitalCoalgebra k DualBasis', Rng k) => Bialgebra k DualBasis'

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

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

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

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

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

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

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

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