Safe Haskell	Safe-Inferred
Language	Haskell98

Numeric.Algebra.Class

Contents

Multiplicative Semigroups
Semirings
Left and Right Modules
Additive Monoids
Associative algebras
Coassociative coalgebras

Synopsis

Multiplicative Semigroups

class Multiplicative r where Source

A multiplicative semigroup

Minimal complete definition

(*)

Methods

(*) :: r -> r -> r infixl 7 Source

pow1p :: r -> Natural -> r infixr 8 Source

productWith1 :: Foldable1 f => (a -> r) -> f a -> r Source

Instances

Multiplicative Bool
Multiplicative Int
Multiplicative Int8
Multiplicative Int16
Multiplicative Int32
Multiplicative Int64
Multiplicative Integer
Multiplicative Word
Multiplicative Word8
Multiplicative Word16
Multiplicative Word32
Multiplicative Word64
Multiplicative ()
Multiplicative Natural
Multiplicative Euclidean
(Commutative r, Rng r) => Multiplicative (Complex r)
(TriviallyInvolutive r, Rng r) => Multiplicative (Quaternion r)
(Commutative r, Rng r) => Multiplicative (Dual r)
(Commutative k, Semiring k) => Multiplicative (Hyper' k)
(Commutative k, Semiring k) => Multiplicative (Hyper k)
(Commutative r, Rng r) => Multiplicative (Dual' r)
Multiplicative (BasisCoblade m)
(TriviallyInvolutive r, Semiring r) => Multiplicative (Quaternion' r)
(Commutative k, Rng k) => Multiplicative (Trig k)
Additive r => Multiplicative (Exp r)
Multiplicative (End r)
Multiplicative r => Multiplicative (Opposite r)
Rng r => Multiplicative (RngRing r)
Monoidal r => Multiplicative (ZeroRng r)
Euclidean d => Multiplicative (Fraction d)
Algebra r a => Multiplicative (a -> r)
(Multiplicative a, Multiplicative b) => Multiplicative (a, b)
Multiplicative (Rect i j)
Coalgebra r m => Multiplicative (Covector r m)
(Multiplicative a, Multiplicative b, Multiplicative c) => Multiplicative (a, b, c)
Coalgebra r m => Multiplicative (Map r b m)
(Multiplicative a, Multiplicative b, Multiplicative c, Multiplicative d) => Multiplicative (a, b, c, d)
(Multiplicative a, Multiplicative b, Multiplicative c, Multiplicative d, Multiplicative e) => Multiplicative (a, b, c, d, e)

pow1pIntegral :: (Integral r, Integral n) => r -> n -> r Source

product1 :: (Foldable1 f, Multiplicative r) => f r -> r Source

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

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

Left and Right Modules

class (Semiring r, Additive m) => LeftModule r m where Source

Methods

(.*) :: r -> m -> m infixl 7 Source

Instances

LeftModule Integer Int
LeftModule Integer Int8
LeftModule Integer Int16
LeftModule Integer Int32
LeftModule Integer Int64
LeftModule Integer Integer
LeftModule Integer Word
LeftModule Integer Word8
LeftModule Integer Word16
LeftModule Integer Word32
LeftModule Integer Word64
LeftModule Integer Euclidean
Additive m => LeftModule () m
Semiring r => LeftModule r ()
LeftModule Natural Bool
LeftModule Natural Int
LeftModule Natural Int8
LeftModule Natural Int16
LeftModule Natural Int32
LeftModule Natural Int64
LeftModule Natural Integer
LeftModule Natural Word
LeftModule Natural Word8
LeftModule Natural Word16
LeftModule Natural Word32
LeftModule Natural Word64
LeftModule Natural Natural
LeftModule Natural Euclidean
Division r => LeftModule Integer (Log r)
(Abelian r, Group r) => LeftModule Integer (RngRing r)
Group r => LeftModule Integer (ZeroRng r)
Euclidean d => LeftModule Integer (Fraction d)
LeftModule r s => LeftModule r (Complex s)
LeftModule r s => LeftModule r (Quaternion s)
LeftModule r s => LeftModule r (Dual s)
LeftModule r s => LeftModule r (Hyper' s)
LeftModule r s => LeftModule r (Hyper s)
LeftModule r s => LeftModule r (Dual' s)
LeftModule r s => LeftModule r (Quaternion' s)
LeftModule r s => LeftModule r (Trig s)
LeftModule r m => LeftModule r (End m)
RightModule r s => LeftModule r (Opposite s)
LeftModule Natural (BasisCoblade m)
Unital r => LeftModule Natural (Log r)
(Abelian r, Monoidal r) => LeftModule Natural (RngRing r)
Monoidal r => LeftModule Natural (ZeroRng r)
Euclidean d => LeftModule Natural (Fraction d)
(LeftModule r a, LeftModule r b) => LeftModule r (a, b)
LeftModule r m => LeftModule r (e -> m)
LeftModule r s => LeftModule r (Covector s m)
(LeftModule r a, LeftModule r b, LeftModule r c) => LeftModule r (a, b, c)
LeftModule r s => LeftModule r (Map s b m)
(LeftModule r a, LeftModule r b, LeftModule r c, LeftModule r d) => LeftModule r (a, b, c, d)
(LeftModule r a, LeftModule r b, LeftModule r c, LeftModule r d, LeftModule r e) => LeftModule r (a, b, c, d, e)
(Commutative r, Rng r) => LeftModule (Complex r) (Complex r)
(TriviallyInvolutive r, Rng r) => LeftModule (Quaternion r) (Quaternion r)
(Commutative r, Rng r) => LeftModule (Dual r) (Dual r)
(Commutative r, Semiring r) => LeftModule (Hyper' r) (Hyper' r)
(Commutative r, Semiring r) => LeftModule (Hyper r) (Hyper r)
(Commutative r, Rng r) => LeftModule (Dual' r) (Dual' r)
(TriviallyInvolutive r, Rng r) => LeftModule (Quaternion' r) (Quaternion' r)
(Commutative r, Rng r) => LeftModule (Trig r) (Trig r)
(Monoidal m, Abelian m) => LeftModule (End m) (End m)
Semiring r => LeftModule (Opposite r) (Opposite r)
Rng s => LeftModule (RngRing s) (RngRing s)
Coalgebra r m => LeftModule (Covector r m) (Covector r m)
Coalgebra r m => LeftModule (Map r b m) (Map r b m)

class (Semiring r, Additive m) => RightModule r m where Source

Methods

(*.) :: m -> r -> m infixl 7 Source

Instances

RightModule Integer Int
RightModule Integer Int8
RightModule Integer Int16
RightModule Integer Int32
RightModule Integer Int64
RightModule Integer Integer
RightModule Integer Word
RightModule Integer Word8
RightModule Integer Word16
RightModule Integer Word32
RightModule Integer Word64
RightModule Integer Euclidean
Additive m => RightModule () m
Semiring r => RightModule r ()
RightModule Natural Bool
RightModule Natural Int
RightModule Natural Int8
RightModule Natural Int16
RightModule Natural Int32
RightModule Natural Int64
RightModule Natural Integer
RightModule Natural Word
RightModule Natural Word8
RightModule Natural Word16
RightModule Natural Word32
RightModule Natural Word64
RightModule Natural Natural
RightModule Natural Euclidean
Division r => RightModule Integer (Log r)
(Abelian r, Group r) => RightModule Integer (RngRing r)
Group r => RightModule Integer (ZeroRng r)
Euclidean d => RightModule Integer (Fraction d)
RightModule r s => RightModule r (Complex s)
RightModule r s => RightModule r (Quaternion s)
RightModule r s => RightModule r (Dual s)
RightModule r s => RightModule r (Hyper' s)
RightModule r s => RightModule r (Hyper s)
RightModule r s => RightModule r (Dual' s)
RightModule r s => RightModule r (Quaternion' s)
RightModule r s => RightModule r (Trig s)
RightModule r m => RightModule r (End m)
LeftModule r s => RightModule r (Opposite s)
RightModule Natural (BasisCoblade m)
Unital r => RightModule Natural (Log r)
(Abelian r, Monoidal r) => RightModule Natural (RngRing r)
Monoidal r => RightModule Natural (ZeroRng r)
Euclidean d => RightModule Natural (Fraction d)
(RightModule r a, RightModule r b) => RightModule r (a, b)
RightModule r m => RightModule r (e -> m)
RightModule r s => RightModule r (Covector s m)
(RightModule r a, RightModule r b, RightModule r c) => RightModule r (a, b, c)
RightModule r s => RightModule r (Map s b m)
(RightModule r a, RightModule r b, RightModule r c, RightModule r d) => RightModule r (a, b, c, d)
(RightModule r a, RightModule r b, RightModule r c, RightModule r d, RightModule r e) => RightModule r (a, b, c, d, e)
(Commutative r, Rng r) => RightModule (Complex r) (Complex r)
(TriviallyInvolutive r, Rng r) => RightModule (Quaternion r) (Quaternion r)
(Commutative r, Rng r) => RightModule (Dual r) (Dual r)
(Commutative r, Semiring r) => RightModule (Hyper' r) (Hyper' r)
(Commutative r, Semiring r) => RightModule (Hyper r) (Hyper r)
(Commutative r, Rng r) => RightModule (Dual' r) (Dual' r)
(TriviallyInvolutive r, Rng r) => RightModule (Quaternion' r) (Quaternion' r)
(Commutative r, Rng r) => RightModule (Trig r) (Trig r)
(Monoidal m, Abelian m) => RightModule (End m) (End m)
Semiring r => RightModule (Opposite r) (Opposite r)
Rng s => RightModule (RngRing s) (RngRing s)
Coalgebra r m => RightModule (Covector r m) (Covector r m)
Coalgebra r m => RightModule (Map r b m) (Map r b m)

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 where Source

An additive monoid

zero + a = a = a + zero

Minimal complete definition

zero

Methods

zero :: m Source

sinnum :: Natural -> m -> m Source

sumWith :: Foldable f => (a -> m) -> f a -> m Source

Instances

Monoidal Bool
Monoidal Int
Monoidal Int8
Monoidal Int16
Monoidal Int32
Monoidal Int64
Monoidal Integer
Monoidal Word
Monoidal Word8
Monoidal Word16
Monoidal Word32
Monoidal Word64
Monoidal ()
Monoidal Natural
Monoidal Euclidean
Monoidal r => Monoidal (Complex r)
Monoidal r => Monoidal (Quaternion r)
Monoidal r => Monoidal (Dual r)
Monoidal r => Monoidal (Hyper' r)
Monoidal r => Monoidal (Hyper r)
Monoidal r => Monoidal (Dual' r)
Monoidal (BasisCoblade m)
Monoidal r => Monoidal (Quaternion' r)
Monoidal r => Monoidal (Trig r)
Unital r => Monoidal (Log r)
Monoidal r => Monoidal (End r)
Monoidal r => Monoidal (Opposite r)
(Abelian r, Monoidal r) => Monoidal (RngRing r)
Monoidal r => Monoidal (ZeroRng r)
Euclidean d => Monoidal (Fraction d)
Monoidal r => Monoidal (e -> r)
(Monoidal a, Monoidal b) => Monoidal (a, b)
Monoidal s => Monoidal (Covector s a)
(Monoidal a, Monoidal b, Monoidal c) => Monoidal (a, b, c)
Monoidal s => Monoidal (Map s b a)
(Monoidal a, Monoidal b, Monoidal c, Monoidal d) => Monoidal (a, b, c, d)
(Monoidal a, Monoidal b, Monoidal c, Monoidal d, Monoidal e) => Monoidal (a, b, c, d, e)

sum :: (Foldable f, Monoidal m) => f m -> m Source

sinnumIdempotent :: (Integral n, Idempotent r, Monoidal r) => n -> r -> r Source

Associative algebras

class Semiring r => Algebra r a where Source

An associative algebra built with a free module over a semiring

Methods

mult :: (a -> a -> r) -> a -> r Source

Instances

Algebra () a
Semiring r => Algebra r IntSet
Semiring r => Algebra r ()
Rng k => Algebra k ComplexBasis
(TriviallyInvolutive r, Rng r) => Algebra r QuaternionBasis	the quaternion algebra
Rng k => Algebra k DualBasis
(Commutative k, Semiring k) => Algebra k HyperBasis'
Semiring k => Algebra k HyperBasis	the trivial diagonal algebra
Semiring k => Algebra k DualBasis'
(TriviallyInvolutive r, Semiring r) => Algebra r QuaternionBasis'	the trivial diagonal algebra
(Commutative k, Rng k) => Algebra k TrigBasis
(Semiring r, Monoidal r, Partitionable a) => Algebra r (IntMap a)
(Semiring r, Ord a) => Algebra r (Set a)
Semiring r => Algebra r (Seq a)	The tensor algebra
Semiring r => Algebra r [a]	The tensor algebra
(Commutative r, Monoidal r, Semiring r, LocallyFiniteOrder a) => Algebra r (Interval a)
(Algebra r a, Algebra r b) => Algebra r (a, b)
(Semiring r, Monoidal r, Ord a, Partitionable b) => Algebra r (Map a b)
(Algebra r a, Algebra r b, Algebra r c) => Algebra r (a, b, c)
(Algebra r a, Algebra r b, Algebra r c, Algebra r d) => Algebra r (a, b, c, d)
(Algebra r a, Algebra r b, Algebra r c, Algebra r d, Algebra r e) => Algebra r (a, b, c, d, e)

Coassociative coalgebras

class Semiring r => Coalgebra r c where Source

Methods

comult :: (c -> r) -> c -> c -> r Source

Instances

Semiring r => Coalgebra r IntSet	the free commutative band coalgebra over Int
Semiring r => Coalgebra r ()
Rng k => Coalgebra k ComplexBasis
(TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis	the trivial diagonal coalgebra
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
Rng k => Coalgebra k DualBasis'
(TriviallyInvolutive r, Rng r) => Coalgebra r QuaternionBasis'	dual quaternion comultiplication
(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)
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
(Coalgebra r a, Coalgebra r b) => Coalgebra r (a, b)
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.
(Coalgebra r a, Coalgebra r b, Coalgebra r c) => Coalgebra r (a, b, c)
(Coalgebra r a, Coalgebra r b, Coalgebra r c, Coalgebra r d) => Coalgebra r (a, b, c, d)
(Coalgebra r a, Coalgebra r b, Coalgebra r c, Coalgebra r d, Coalgebra r e) => Coalgebra r (a, b, c, d, e)