Numeric.Algebra.Free

Documentation

class Semiring r => FreeAlgebra r a where

An associative algebra built with a free module over a semiring

Methods

join :: (a -> a -> r) -> a -> r

Instances

FreeAlgebra () a

(FreeAlgebra r a, FreeAlgebra r b) => FreeAlgebra r (a, b)

(FreeAlgebra r a, FreeAlgebra r b, FreeAlgebra r c) => FreeAlgebra r (a, b, c)

(FreeAlgebra r a, FreeAlgebra r b, FreeAlgebra r c, FreeAlgebra r d) => FreeAlgebra r (a, b, c, d)

(FreeAlgebra r a, FreeAlgebra r b, FreeAlgebra r c, FreeAlgebra r d, FreeAlgebra r e) => FreeAlgebra r (a, b, c, d, e)

(FreeAlgebra r b, FreeAlgebra r a) => FreeAlgebra (b -> r) a

class (Unital r, FreeAlgebra r a) => FreeUnitalAlgebra r a where

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

Methods

unit :: r -> a -> r

Instances

FreeUnitalAlgebra () a

(FreeUnitalAlgebra r a, FreeUnitalAlgebra r b) => FreeUnitalAlgebra r (a, b)

(FreeUnitalAlgebra r a, FreeUnitalAlgebra r b, FreeUnitalAlgebra r c) => FreeUnitalAlgebra r (a, b, c)

(FreeUnitalAlgebra r a, FreeUnitalAlgebra r b, FreeUnitalAlgebra r c, FreeUnitalAlgebra r d) => FreeUnitalAlgebra r (a, b, c, d)

(FreeUnitalAlgebra r a, FreeUnitalAlgebra r b, FreeUnitalAlgebra r c, FreeUnitalAlgebra r d, FreeUnitalAlgebra r e) => FreeUnitalAlgebra r (a, b, c, d, e)

(FreeUnitalAlgebra r a, FreeUnitalAlgebra r b) => FreeUnitalAlgebra (a -> r) b

class Semiring r => FreeCoalgebra r c where

Methods

cojoin :: (c -> r) -> c -> c -> r

Instances

FreeCoalgebra () c

(FreeCoalgebra r a, FreeCoalgebra r b) => FreeCoalgebra r (a, b)

FreeAlgebra r m => FreeCoalgebra r (m -> r)

Every coalgebra gives rise to an algebra by vector space duality classically. Sadly, it requires vector space duality, which we cannot use constructively. This is the dual, which relies in the fact that any constructive coalgebra can only inspect a finite number of coefficients.

(FreeCoalgebra r a, FreeCoalgebra r b, FreeCoalgebra r c) => FreeCoalgebra r (a, b, c)

(FreeCoalgebra r a, FreeCoalgebra r b, FreeCoalgebra r c, FreeCoalgebra r d) => FreeCoalgebra r (a, b, c, d)

(FreeCoalgebra r a, FreeCoalgebra r b, FreeCoalgebra r c, FreeCoalgebra r d, FreeCoalgebra r e) => FreeCoalgebra r (a, b, c, d, e)

(FreeAlgebra r b, FreeCoalgebra r c) => FreeCoalgebra (b -> r) c

class FreeCoalgebra r c => FreeCounitalCoalgebra r c where

Methods

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

Instances

FreeCounitalCoalgebra () a

(FreeCounitalCoalgebra r a, FreeCounitalCoalgebra r b) => FreeCounitalCoalgebra r (a, b)

FreeUnitalAlgebra r m => FreeCounitalCoalgebra r (m -> r)

(FreeCounitalCoalgebra r a, FreeCounitalCoalgebra r b, FreeCounitalCoalgebra r c) => FreeCounitalCoalgebra r (a, b, c)

(FreeCounitalCoalgebra r a, FreeCounitalCoalgebra r b, FreeCounitalCoalgebra r c, FreeCounitalCoalgebra r d) => FreeCounitalCoalgebra r (a, b, c, d)

(FreeCounitalCoalgebra r a, FreeCounitalCoalgebra r b, FreeCounitalCoalgebra r c, FreeCounitalCoalgebra r d, FreeCounitalCoalgebra r e) => FreeCounitalCoalgebra r (a, b, c, d, e)

(FreeUnitalAlgebra r a, FreeCounitalCoalgebra r c) => FreeCounitalCoalgebra (a -> r) c

class (FreeUnitalAlgebra r h, FreeCounitalCoalgebra r h) => Hopf r h where

a Hopf algebra on a semiring, where the module is a free.

If antipode . antipode = id then we are Involutive

Methods

antipode :: (h -> r) -> h -> r

Instances

Hopf () h

(Hopf r a, Hopf r b) => Hopf r (a, b)

(Hopf r a, Hopf r b, Hopf r c) => Hopf r (a, b, c)

(Hopf r a, Hopf r b, Hopf r c, Hopf r d) => Hopf r (a, b, c, d)

(Hopf r a, Hopf r b, Hopf r c, Hopf r d, Hopf r e) => Hopf r (a, b, c, d, e)

(FreeUnitalAlgebra r a, Hopf r h) => Hopf (a -> r) h