| Safe Haskell | None |
|---|
Numeric.Algebra.Class
Contents
- class Multiplicative r where
- (*) :: r -> r -> r
- pow1p :: Whole n => r -> n -> r
- productWith1 :: Foldable1 f => (a -> r) -> f a -> r
- pow1pIntegral :: (Integral r, Integral n) => r -> n -> r
- product1 :: (Foldable1 f, Multiplicative r) => f r -> r
- class (Additive r, Abelian r, Multiplicative r) => Semiring r
- class (Semiring r, Additive m) => LeftModule r m where
- (.*) :: r -> m -> m
- class (Semiring r, Additive m) => RightModule r m where
- (*.) :: m -> r -> m
- class (LeftModule r m, RightModule r m) => Module r m
- class (LeftModule Natural m, RightModule Natural m) => Monoidal m where
- sum :: (Foldable f, Monoidal m) => f m -> m
- sinnumIdempotent :: (Integral n, Idempotent r, Monoidal r) => n -> r -> r
- class Semiring r => Algebra r a where
- mult :: (a -> a -> r) -> a -> r
- class Semiring r => Coalgebra r c where
- comult :: (c -> r) -> c -> c -> r
Multiplicative Semigroups
class Multiplicative r whereSource
A multiplicative semigroup
Methods
pow1p :: Whole n => r -> n -> rSource
productWith1 :: Foldable1 f => (a -> r) -> f a -> rSource
Instances
pow1pIntegral :: (Integral r, Integral n) => r -> n -> rSource
product1 :: (Foldable1 f, Multiplicative r) => f r -> rSource
Semirings
class (Additive r, Abelian r, Multiplicative r) => Semiring r Source
A pair of an additive abelian semigroup, and a multiplicative semigroup, with the distributive laws:
a(b + c) = ab + ac -- left distribution (we are a LeftNearSemiring) (a + b)c = ac + bc -- right distribution (we are a [Right]NearSemiring)
Common notation includes the laws for additive and multiplicative identity in semiring.
If you want that, look at Rig instead.
Ideally we'd use the cyclic definition:
class (LeftModule r r, RightModule r r, Additive r, Abelian r, Multiplicative r) => Semiring r
to enforce that every semiring r is an r-module over itself, but Haskell doesn't like that.
Instances
Left and Right Modules
class (Semiring r, Additive m) => LeftModule r m whereSource
Instances
class (Semiring r, Additive m) => RightModule r m whereSource
Instances
class (LeftModule r m, RightModule r m) => Module r m Source
Instances
| (LeftModule r m, RightModule r m) => Module r m |
Additive Monoids
class (LeftModule Natural m, RightModule Natural m) => Monoidal m whereSource
An additive monoid
zero + a = a = a + zero
Instances
sinnumIdempotent :: (Integral n, Idempotent r, Monoidal r) => n -> r -> rSource
Associative algebras
class Semiring r => Algebra r a whereSource
An associative algebra built with a free module over a semiring
Instances
Coassociative coalgebras
class Semiring r => Coalgebra r c whereSource
Instances
| Semiring r => Coalgebra r IntSet | the free commutative band coalgebra over Int |
| Semiring r => Coalgebra r () | |
| (Semiring k, Rng k) => Coalgebra k ComplexBasis | |
| (Semiring r, TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis | the trivial diagonal coalgebra |
| (Semiring k, Rng k) => Coalgebra k DualBasis | |
| (Commutative k, Monoidal k, Semiring k) => Coalgebra k HyperBasis' | |
| (Commutative k, Semiring k) => Coalgebra k HyperBasis | the hyperbolic trigonometric coalgebra |
| (Semiring k, Rng k) => Coalgebra k DualBasis' | |
| (Semiring r, TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis' | dual quaternion comultiplication |
| (Semiring k, Commutative k, Rng k) => Coalgebra k TrigBasis | |
| (Semiring r, Additive b) => Coalgebra r (IntMap b) | the free commutative coalgebra over a set and Int |
| (Semiring r, Ord a) => Coalgebra r (Set a) | the free commutative band coalgebra |
| Semiring r => Coalgebra r (Seq a) | The tensor Hopf algebra |
| Semiring r => Coalgebra r [a] | The tensor Hopf algebra |
| (Commutative r, Monoidal r, Semiring r, PartialSemigroup a) => Coalgebra r (Morphism a) | |
| (Eq a, Commutative r, Monoidal r, Semiring r) => Coalgebra r (Interval' a) | |
| (Semiring r, Eigenmetric r m) => Coalgebra r (BasisCoblade m) | |
| (Semiring r, Ord a, Additive b) => Coalgebra r (Map a b) | the free commutative coalgebra over a set and a given semigroup |
| (Semiring r, Coalgebra r a, Coalgebra r b) => Coalgebra r (a, b) | |
| (Semiring r, HasTrie m, Algebra r m) => Coalgebra r (:->: m r) | |
| (Semiring r, Algebra r m) => Coalgebra 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. The dual argument only relies in the fact that any constructive coalgebra can only inspect a finite number of coefficients, which we CAN exploit. |
| (Semiring r, Coalgebra r a, Coalgebra r b, Coalgebra r c) => Coalgebra r (a, b, c) | |
| (Semiring r, Coalgebra r a, Coalgebra r b, Coalgebra r c, Coalgebra r d) => Coalgebra r (a, b, c, d) | |
| (Semiring r, Coalgebra r a, Coalgebra r b, Coalgebra r c, Coalgebra r d, Coalgebra r e) => Coalgebra r (a, b, c, d, e) |