Numeric.Algebra

Additive

additive semigroups

class Additive r where

 (a + b) + c = a + (b + c)
 sinnum 1 a = a
 sinnum (2 * n) a = sinnum n a + sinnum n a
 sinnum (2 * n + 1) a = sinnum n a + sinnum n a + a

Methods

(+) :: r -> r -> r

sinnum1p :: Whole n => n -> r -> r

sinnum1p n r = sinnum (1 + n) r

sumWith1 :: Foldable1 f => (a -> r) -> f a -> r

Instances

Additive Bool

Additive Int

Additive Int8

Additive Int16

Additive Int32

Additive Int64

Additive Integer

Additive Word

Additive Word8

Additive Word16

Additive Word32

Additive Word64

Additive ()

Additive Natural

Additive Euclidean

Additive r => Additive (Complex r)

Additive r => Additive (Quaternion r)

Additive r => Additive (Dual r)

Additive r => Additive (Hyper' r)

Additive r => Additive (Hyper r)

Additive r => Additive (Dual' r)

Additive (BasisCoblade m)

Additive r => Additive (Quaternion' r)

Additive r => Additive (Trig r)

Multiplicative r => Additive (Log r)

Additive r => Additive (End r)

Additive r => Additive (Opposite r)

Abelian r => Additive (RngRing r)

Additive r => Additive (ZeroRng r)

Additive r => Additive (b -> r)

(Additive a, Additive b) => Additive (a, b)

(HasTrie b, Additive r) => Additive (:->: b r)

Additive r => Additive (Covector r a)

(Additive a, Additive b, Additive c) => Additive (a, b, c)

Additive r => Additive (Map r b a)

(Additive a, Additive b, Additive c, Additive d) => Additive (a, b, c, d)

(Additive a, Additive b, Additive c, Additive d, Additive e) => Additive (a, b, c, d, e)

sum1 :: (Foldable1 f, Additive r) => f r -> r

additive Abelian semigroups

class Additive r => Abelian r

an additive abelian semigroup

a + b = b + a

Instances

Abelian Bool

Abelian Int

Abelian Int8

Abelian Int16

Abelian Int32

Abelian Int64

Abelian Integer

Abelian Word

Abelian Word8

Abelian Word16

Abelian Word32

Abelian Word64

Abelian ()

Abelian Natural

Abelian Euclidean

Abelian r => Abelian (Complex r)

Abelian r => Abelian (Quaternion r)

Abelian r => Abelian (Dual r)

Abelian r => Abelian (Hyper' r)

Abelian r => Abelian (Hyper r)

Abelian r => Abelian (Dual' r)

Abelian (BasisCoblade m)

Abelian r => Abelian (Quaternion' r)

Abelian r => Abelian (Trig r)

Commutative r => Abelian (Log r)

Abelian r => Abelian (End r)

Abelian r => Abelian (Opposite r)

Abelian r => Abelian (RngRing r)

Abelian r => Abelian (ZeroRng r)

Abelian r => Abelian (e -> r)

(Abelian a, Abelian b) => Abelian (a, b)

(HasTrie e, Abelian r) => Abelian (:->: e r)

Abelian s => Abelian (Covector s a)

(Abelian a, Abelian b, Abelian c) => Abelian (a, b, c)

Abelian s => Abelian (Map s b a)

(Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d)

(Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e)

additive idempotent semigroups

class Additive r => Idempotent r

An additive semigroup with idempotent addition.

 a + a = a

Instances

Idempotent Bool

Idempotent ()

Idempotent r => Idempotent (Complex r)

Idempotent r => Idempotent (Quaternion r)

Idempotent r => Idempotent (Dual r)

Idempotent r => Idempotent (Hyper' r)

Idempotent r => Idempotent (Hyper r)

Idempotent r => Idempotent (Dual' r)

Idempotent r => Idempotent (Quaternion' r)

Idempotent r => Idempotent (Trig r)

Band r => Idempotent (Log r)

Idempotent r => Idempotent (Opposite r)

Idempotent r => Idempotent (ZeroRng r)

Idempotent r => Idempotent (e -> r)

(Idempotent a, Idempotent b) => Idempotent (a, b)

(HasTrie e, Idempotent r) => Idempotent (:->: e r)

Idempotent r => Idempotent (Covector r a)

(Idempotent a, Idempotent b, Idempotent c) => Idempotent (a, b, c)

(Idempotent a, Idempotent b, Idempotent c, Idempotent d) => Idempotent (a, b, c, d)

(Idempotent a, Idempotent b, Idempotent c, Idempotent d, Idempotent e) => Idempotent (a, b, c, d, e)

sinnum1pIdempotent :: Natural -> r -> r

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

partitionable additive semigroups

class Additive m => Partitionable m where

Methods

partitionWith :: (m -> m -> r) -> m -> NonEmpty r

partitionWith f c returns a list containing f a b for each a b such that a + b = c,

Instances

Partitionable Bool

Partitionable ()

Partitionable Natural

Partitionable r => Partitionable (Complex r)

Partitionable r => Partitionable (Quaternion r)

Partitionable r => Partitionable (Dual r)

Partitionable r => Partitionable (Hyper' r)

Partitionable r => Partitionable (Hyper r)

Partitionable r => Partitionable (Dual' r)

Partitionable r => Partitionable (Quaternion' r)

Partitionable r => Partitionable (Trig r)

Factorable r => Partitionable (Log r)

(Partitionable a, Partitionable b) => Partitionable (a, b)

(Partitionable a, Partitionable b, Partitionable c) => Partitionable (a, b, c)

(Partitionable a, Partitionable b, Partitionable c, Partitionable d) => Partitionable (a, b, c, d)

(Partitionable a, Partitionable b, Partitionable c, Partitionable d, Partitionable e) => Partitionable (a, b, c, d, e)

additive monoids

class (LeftModule Natural m, RightModule Natural m) => Monoidal m where

An additive monoid

 zero + a = a = a + zero

Methods

zero :: m

sinnum :: Whole n => n -> m -> m

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

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)

Monoidal r => Monoidal (e -> r)

(Monoidal a, Monoidal b) => Monoidal (a, b)

(HasTrie e, Monoidal r) => Monoidal (:->: e r)

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

additive groups

class (LeftModule Integer r, RightModule Integer r, Monoidal r) => Group r where

Methods

(-) :: r -> r -> r

negate :: r -> r

subtract :: r -> r -> r

times :: Integral n => n -> r -> r

Instances

Group Int

Group Int8

Group Int16

Group Int32

Group Int64

Group Integer

Group Word

Group Word8

Group Word16

Group Word32

Group Word64

Group ()

Group Euclidean

Group r => Group (Complex r)

Group r => Group (Quaternion r)

Group r => Group (Dual r)

Group r => Group (Hyper' r)

Group r => Group (Hyper r)

Group r => Group (Dual' r)

Group r => Group (Quaternion' r)

Group r => Group (Trig r)

Division r => Group (Log r)

Group r => Group (End r)

Group r => Group (Opposite r)

(Abelian r, Group r) => Group (RngRing r)

Group r => Group (ZeroRng r)

Group r => Group (e -> r)

(Group a, Group b) => Group (a, b)

(HasTrie e, Group r) => Group (:->: e r)

Group s => Group (Covector s a)

(Group a, Group b, Group c) => Group (a, b, c)

Group s => Group (Map s b a)

(Group a, Group b, Group c, Group d) => Group (a, b, c, d)

(Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e)

Multiplicative

multiplicative semigroups

class Multiplicative r where

A multiplicative semigroup

Methods

(*) :: r -> r -> r

pow1p :: Whole n => r -> n -> r

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

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)

Algebra r a => Multiplicative (a -> r)

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

(HasTrie a, Algebra r a) => Multiplicative (:->: a r)

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)

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

commutative multiplicative semigroups

class Multiplicative r => Commutative r

A commutative multiplicative semigroup

Instances

Commutative Bool

Commutative Int

Commutative Int8

Commutative Int16

Commutative Int32

Commutative Int64

Commutative Integer

Commutative Word

Commutative Word8

Commutative Word16

Commutative Word32

Commutative Word64

Commutative ()

Commutative Natural

Commutative Euclidean

(TriviallyInvolutive r, Rng r) => Commutative (Complex r)

(TriviallyInvolutive r, Rng r) => Commutative (Dual r)

(Commutative k, Semiring k) => Commutative (Hyper' k)

(Commutative k, Semiring k) => Commutative (Hyper k)

(TriviallyInvolutive r, Rng r) => Commutative (Dual' r)

Commutative (BasisCoblade m)

(Commutative k, Rng k) => Commutative (Trig k)

Abelian r => Commutative (Exp r)

(Abelian r, Commutative r) => Commutative (End r)

Commutative r => Commutative (Opposite r)

(Commutative r, Rng r) => Commutative (RngRing r)

Monoidal r => Commutative (ZeroRng r)

CommutativeAlgebra r a => Commutative (a -> r)

(Commutative a, Commutative b) => Commutative (a, b)

(HasTrie a, CommutativeAlgebra r a) => Commutative (:->: a r)

(Commutative m, Coalgebra r m) => Commutative (Covector r m)

(Commutative a, Commutative b, Commutative c) => Commutative (a, b, c)

(Commutative m, Coalgebra r m) => Commutative (Map r b m)

(Commutative a, Commutative b, Commutative c, Commutative d) => Commutative (a, b, c, d)

(Commutative a, Commutative b, Commutative c, Commutative d, Commutative e) => Commutative (a, b, c, d, e)

multiplicative monoids

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

(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)

(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

idempotent multiplicative semigroups

class Multiplicative r => Band r

An multiplicative semigroup with idempotent multiplication.

 a * a = a

Instances

Band Bool

Band ()

Idempotent r => Band (Exp r)

Band r => Band (Opposite r)

(Band a, Band b) => Band (a, b)

Band (Rect i j)

(Idempotent r, IdempotentCoalgebra r a) => Band (Covector r a)

(Band a, Band b, Band c) => Band (a, b, c)

(Band a, Band b, Band c, Band d) => Band (a, b, c, d)

(Band a, Band b, Band c, Band d, Band e) => Band (a, b, c, d, e)

pow1pBand :: Whole n => r -> n -> r

powBand :: (Unital r, Whole n) => r -> n -> r

multiplicative groups

class Unital r => Division r where

Methods

recip :: r -> r

(/) :: r -> r -> r

(\\) :: r -> r -> r

(^) :: Integral n => r -> n -> r

Instances

Division ()

(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Complex r)

(TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion r)

(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Dual r)

(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Hyper' r)

(Commutative r, InvolutiveSemiring r, DivisionRing r) => Division (Dual' r)

(TriviallyInvolutive r, Ring r, Division r) => Division (Quaternion' r)

Group r => Division (Exp r)

Division r => Division (Opposite r)

(Rng r, Division r) => Division (RngRing r)

(Unital r, DivisionAlgebra r a) => Division (a -> r)

(Division a, Division b) => Division (a, b)

(Division a, Division b, Division c) => Division (a, b, c)

(Division a, Division b, Division c, Division d) => Division (a, b, c, d)

(Division a, Division b, Division c, Division d, Division e) => Division (a, b, c, d, e)

factorable multiplicative semigroups

class Multiplicative m => Factorable m where

`factorWith f c` returns a non-empty list containing `f a b` for all `a, b` such that `a * b = c`.

Results of factorWith f 0 are undefined and may result in either an error or an infinite list.

Methods

factorWith :: (m -> m -> r) -> m -> NonEmpty r

Instances

Factorable Bool

Factorable ()

Partitionable r => Factorable (Exp r)

(Factorable a, Factorable b) => Factorable (a, b)

(Factorable a, Factorable b, Factorable c) => Factorable (a, b, c)

(Factorable a, Factorable b, Factorable c, Factorable d) => Factorable (a, b, c, d)

(Factorable a, Factorable b, Factorable c, Factorable d, Factorable e) => Factorable (a, b, c, d, e)

involutive multiplicative semigroups

class Multiplicative r => InvolutiveMultiplication r where

An semigroup with involution

 adjoint a * adjoint b = adjoint (b * a)

Methods

adjoint :: r -> r

Instances

InvolutiveMultiplication Bool

InvolutiveMultiplication Int

InvolutiveMultiplication Int8

InvolutiveMultiplication Int16

InvolutiveMultiplication Int32

InvolutiveMultiplication Int64

InvolutiveMultiplication Integer

InvolutiveMultiplication Word

InvolutiveMultiplication Word8

InvolutiveMultiplication Word16

InvolutiveMultiplication Word32

InvolutiveMultiplication Word64

InvolutiveMultiplication ()

InvolutiveMultiplication Natural

InvolutiveMultiplication Euclidean

(Commutative r, Rng r, InvolutiveMultiplication r) => InvolutiveMultiplication (Complex r)

(TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion r)

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveMultiplication (Dual r)

(Commutative r, InvolutiveSemiring r, Rng r) => InvolutiveMultiplication (Hyper' r)

(Commutative r, Group r, InvolutiveSemiring r) => InvolutiveMultiplication (Hyper r)

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveMultiplication (Dual' r)

(TriviallyInvolutive r, Rng r) => InvolutiveMultiplication (Quaternion' r)

(Commutative r, Rng r, InvolutiveMultiplication r) => InvolutiveMultiplication (Trig r)

InvolutiveAlgebra r h => InvolutiveMultiplication (h -> r)

(InvolutiveMultiplication a, InvolutiveMultiplication b) => InvolutiveMultiplication (a, b)

(HasTrie h, InvolutiveAlgebra r h) => InvolutiveMultiplication (:->: h r)

(InvolutiveMultiplication a, InvolutiveMultiplication b, InvolutiveMultiplication c) => InvolutiveMultiplication (a, b, c)

(InvolutiveMultiplication a, InvolutiveMultiplication b, InvolutiveMultiplication c, InvolutiveMultiplication d) => InvolutiveMultiplication (a, b, c, d)

(InvolutiveMultiplication a, InvolutiveMultiplication b, InvolutiveMultiplication c, InvolutiveMultiplication d, InvolutiveMultiplication e) => InvolutiveMultiplication (a, b, c, d, e)

class (Commutative r, InvolutiveMultiplication r) => TriviallyInvolutive r

 adjoint = id

Instances

TriviallyInvolutive Bool

TriviallyInvolutive Int

TriviallyInvolutive Int8

TriviallyInvolutive Int16

TriviallyInvolutive Int32

TriviallyInvolutive Int64

TriviallyInvolutive Integer

TriviallyInvolutive Word

TriviallyInvolutive Word8

TriviallyInvolutive Word16

TriviallyInvolutive Word32

TriviallyInvolutive Word64

TriviallyInvolutive ()

TriviallyInvolutive Natural

TriviallyInvolutive Euclidean

(TriviallyInvolutive r, TriviallyInvolutiveAlgebra r a) => TriviallyInvolutive (a -> r)

(TriviallyInvolutive a, TriviallyInvolutive b) => TriviallyInvolutive (a, b)

(HasTrie a, TriviallyInvolutive r, TriviallyInvolutiveAlgebra r a) => TriviallyInvolutive (:->: a r)

(TriviallyInvolutive a, TriviallyInvolutive b, TriviallyInvolutive c) => TriviallyInvolutive (a, b, c)

(TriviallyInvolutive a, TriviallyInvolutive b, TriviallyInvolutive c, TriviallyInvolutive d) => TriviallyInvolutive (a, b, c, d)

(TriviallyInvolutive a, TriviallyInvolutive b, TriviallyInvolutive c, TriviallyInvolutive d, TriviallyInvolutive e) => TriviallyInvolutive (a, b, c, d, e)

Ring-Structures

Semirings

class (Additive r, Abelian r, Multiplicative r) => Semiring r

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)

Algebra r a => Semiring (a -> r)

(Semiring a, Semiring b) => Semiring (a, b)

(HasTrie a, Algebra r a) => Semiring (:->: a r)

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)

class (Semiring r, InvolutiveMultiplication r) => InvolutiveSemiring r

adjoint (x + y) = adjoint x + adjoint y

Instances

InvolutiveSemiring Bool

InvolutiveSemiring Int

InvolutiveSemiring Int8

InvolutiveSemiring Int16

InvolutiveSemiring Int32

InvolutiveSemiring Int64

InvolutiveSemiring Integer

InvolutiveSemiring Word

InvolutiveSemiring Word8

InvolutiveSemiring Word16

InvolutiveSemiring Word32

InvolutiveSemiring Word64

InvolutiveSemiring ()

InvolutiveSemiring Natural

InvolutiveSemiring Euclidean

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Complex r)

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Dual r)

(Commutative r, InvolutiveSemiring r, Rng r) => InvolutiveSemiring (Hyper' r)

(Commutative r, Group r, InvolutiveSemiring r) => InvolutiveSemiring (Hyper r)

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Dual' r)

(Commutative r, Rng r, InvolutiveSemiring r) => InvolutiveSemiring (Trig r)

(InvolutiveSemiring a, InvolutiveSemiring b) => InvolutiveSemiring (a, b)

(InvolutiveSemiring a, InvolutiveSemiring b, InvolutiveSemiring c) => InvolutiveSemiring (a, b, c)

(InvolutiveSemiring a, InvolutiveSemiring b, InvolutiveSemiring c, InvolutiveSemiring d) => InvolutiveSemiring (a, b, c, d)

(InvolutiveSemiring a, InvolutiveSemiring b, InvolutiveSemiring c, InvolutiveSemiring d, InvolutiveSemiring e) => InvolutiveSemiring (a, b, c, d, e)

class (Semiring r, Idempotent r) => Dioid r

Instances

(Semiring r, Idempotent r) => Dioid r

Rngs

class (Group r, Semiring r) => Rng r

A Ring without an identity.

Instances

(Group r, Semiring r) => Rng r

(Group r, Abelian r) => Rng (ZeroRng r)

Rigs

class (Semiring r, Unital r, Monoidal r) => Rig r where

A Ring without (n)egation

Methods

fromNatural :: Natural -> r

Instances

Rig Bool

Rig Int

Rig Int8

Rig Int16

Rig Int32

Rig Int64

Rig Integer

Rig Word

Rig Word8

Rig Word16

Rig Word32

Rig Word64

Rig ()

Rig Natural

Rig Euclidean

(Commutative r, Ring r) => Rig (Complex r)

(TriviallyInvolutive r, Ring r) => Rig (Quaternion r)

(Commutative r, Ring r) => Rig (Dual r)

(Commutative r, Rig r) => Rig (Hyper' r)

(Commutative r, Rig r) => Rig (Hyper r)

(Commutative r, Ring r) => Rig (Dual' r)

Rig (BasisCoblade m)

(TriviallyInvolutive r, Ring r) => Rig (Quaternion' r)

(Commutative r, Ring r) => Rig (Trig r)

(Abelian r, Monoidal r) => Rig (End r)

Rig r => Rig (Opposite r)

Rng r => Rig (RngRing r)

(Rig a, Rig b) => Rig (a, b)

(Rig r, CounitalCoalgebra r m) => Rig (Covector r m)

(Rig a, Rig b, Rig c) => Rig (a, b, c)

(Rig r, CounitalCoalgebra r m) => Rig (Map r b m)

(Rig a, Rig b, Rig c, Rig d) => Rig (a, b, c, d)

(Rig a, Rig b, Rig c, Rig d, Rig e) => Rig (a, b, c, d, e)

Rings

class (Rig r, Rng r) => Ring r where

Methods

fromInteger :: Integer -> r

Instances

Ring Int

Ring Int8

Ring Int16

Ring Int32

Ring Int64

Ring Integer

Ring Word

Ring Word8

Ring Word16

Ring Word32

Ring Word64

Ring ()

Ring Euclidean

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

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

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

(Commutative r, Ring r) => Ring (Hyper' r)

(Commutative r, Ring r) => Ring (Hyper r)

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

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

(Commutative r, Ring r) => Ring (Trig r)

(Abelian r, Group r) => Ring (End r)

Ring r => Ring (Opposite r)

Rng r => Ring (RngRing r)

(Ring a, Ring b) => Ring (a, b)

(Ring r, CounitalCoalgebra r m) => Ring (Covector r m)

(Ring a, Ring b, Ring c) => Ring (a, b, c)

(Ring r, CounitalCoalgebra r m) => Ring (Map r a m)

(Ring a, Ring b, Ring c, Ring d) => Ring (a, b, c, d)

(Ring a, Ring b, Ring c, Ring d, Ring e) => Ring (a, b, c, d, e)

Division Rings

class Ring r => LocalRing r

class (Division r, Ring r) => DivisionRing r

Instances

(Division r, Ring r) => DivisionRing r

class (Commutative r, DivisionRing r) => Field r

Instances

(Commutative r, DivisionRing r) => Field r

Modules

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

Methods

(.*) :: r -> m -> m

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)

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)

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

(HasTrie e, LeftModule r m) => LeftModule r (:->: e m)

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

Methods

(*.) :: m -> r -> m

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)

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)

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

(HasTrie e, RightModule r m) => RightModule r (:->: e m)

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

Instances

(LeftModule r m, RightModule r m) => Module r m

Algebras

associative algebras over (non-commutative) semirings

class Semiring r => Algebra r a where

An associative algebra built with a free module over a semiring

Methods

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

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)

class Semiring r => Coalgebra r c where

Methods

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

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)

(HasTrie m, Algebra r m) => Coalgebra r (:->: m 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.

(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)

unital algebras

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)

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)

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)

involutive algebras

class (InvolutiveSemiring r, Algebra r a) => InvolutiveAlgebra r a where

Methods

inv :: (a -> r) -> a -> r

Instances

InvolutiveSemiring r => InvolutiveAlgebra r ()

(InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k ComplexBasis

(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveAlgebra r QuaternionBasis

(InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k DualBasis

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k HyperBasis'

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k HyperBasis

(InvolutiveSemiring k, Rng k) => InvolutiveAlgebra k DualBasis'

(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveAlgebra r QuaternionBasis'

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveAlgebra k TrigBasis

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

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

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

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

class (InvolutiveSemiring r, Coalgebra r c) => InvolutiveCoalgebra r c where

Methods

coinv :: (c -> r) -> c -> r

Instances

InvolutiveSemiring r => InvolutiveCoalgebra r ()

(InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k ComplexBasis

(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis

(InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k DualBasis

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k HyperBasis'

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k HyperBasis

(InvolutiveSemiring k, Rng k) => InvolutiveCoalgebra k DualBasis'

(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => InvolutiveCoalgebra r QuaternionBasis'

(Commutative k, Group k, InvolutiveSemiring k) => InvolutiveCoalgebra k TrigBasis

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

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

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

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

class (Bialgebra r h, InvolutiveAlgebra r h, InvolutiveCoalgebra r h) => InvolutiveBialgebra r h

Instances

(Bialgebra r h, InvolutiveAlgebra r h, InvolutiveCoalgebra r h) => InvolutiveBialgebra r h

class (CommutativeAlgebra r a, TriviallyInvolutive r, InvolutiveAlgebra r a) => TriviallyInvolutiveAlgebra r a

Instances

(TriviallyInvolutive r, InvolutiveSemiring r) => TriviallyInvolutiveAlgebra r ()

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

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

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

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

class (CocommutativeCoalgebra r a, TriviallyInvolutive r, InvolutiveCoalgebra r a) => TriviallyInvolutiveCoalgebra r a

Instances

(TriviallyInvolutive r, InvolutiveSemiring r) => TriviallyInvolutiveCoalgebra r ()

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

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

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

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

class (InvolutiveBialgebra r h, TriviallyInvolutiveAlgebra r h, TriviallyInvolutiveCoalgebra r h) => TriviallyInvolutiveBialgebra r h

Instances

(InvolutiveBialgebra r h, TriviallyInvolutiveAlgebra r h, TriviallyInvolutiveCoalgebra r h) => TriviallyInvolutiveBialgebra r h

idempotent algebras

class Algebra r a => IdempotentAlgebra r a

Instances

(Semiring r, Band r) => IdempotentAlgebra r ()

(Semiring r, Band r) => IdempotentAlgebra r IntSet

(Semiring r, Band r, Ord a) => IdempotentAlgebra r (Set a)

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

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

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

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

class (Bialgebra r h, IdempotentAlgebra r h, IdempotentCoalgebra r h) => IdempotentBialgebra r h

Instances

(Bialgebra r h, IdempotentAlgebra r h, IdempotentCoalgebra r h) => IdempotentBialgebra r h

commutative algebras

class Algebra r a => CommutativeAlgebra r a

Instances

(Commutative r, Semiring r) => CommutativeAlgebra r IntSet

(Commutative r, Semiring r) => CommutativeAlgebra r ()

(Commutative r, Monoidal r, Semiring r, Abelian b, Partitionable b) => CommutativeAlgebra r (IntMap b)

(Commutative r, Semiring r, Ord a) => CommutativeAlgebra r (Set a)

(Commutative r, Monoidal r, Semiring r, Ord a, Abelian b, Partitionable b) => CommutativeAlgebra r (Map a b)

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

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

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

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

class (Bialgebra r h, CommutativeAlgebra r h, CocommutativeCoalgebra r h) => CommutativeBialgebra r h

Instances

(Bialgebra r h, CommutativeAlgebra r h, CocommutativeCoalgebra r h) => CommutativeBialgebra r h

class Coalgebra r c => CocommutativeCoalgebra r c

Instances

(Commutative r, Semiring r) => CocommutativeCoalgebra r IntSet

(Commutative r, Semiring r) => CocommutativeCoalgebra r ()

(Commutative r, Semiring r, Abelian b) => CocommutativeCoalgebra r (IntMap b)

(Commutative r, Semiring r, Ord a) => CocommutativeCoalgebra r (Set a)

(Commutative r, Semiring r, Ord a, Abelian b) => CocommutativeCoalgebra r (Map a b)

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

(HasTrie m, CommutativeAlgebra r m) => CocommutativeCoalgebra r (:->: m r)

CommutativeAlgebra r m => CocommutativeCoalgebra r (m -> r)

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

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

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

division algebras

class UnitalAlgebra r a => DivisionAlgebra r a where

Methods

recipriocal :: (a -> r) -> a -> r

Hopf alegebras

class Bialgebra r h => HopfAlgebra r h where

A HopfAlgebra algebra on a semiring, where the module is free.

When antipode . antipode = id and antipode is an antihomomorphism then we are an InvolutiveBialgebra with inv = antipode as well

Methods

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

Instances

(InvolutiveSemiring k, Rng k) => HopfAlgebra k ComplexBasis

(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis

(InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis

(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis'

(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k HyperBasis

(InvolutiveSemiring k, Rng k) => HopfAlgebra k DualBasis'

(TriviallyInvolutive r, InvolutiveSemiring r, Rng r) => HopfAlgebra r QuaternionBasis'

(Commutative k, Group k, InvolutiveSemiring k) => HopfAlgebra k TrigBasis

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

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

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

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

Ring Properties

Characteristic

class Rig r => Characteristic r where

Methods

char :: proxy r -> Natural

Instances

Characteristic Bool

NB: we're using the boolean semiring, not the boolean ring

Characteristic Int

Characteristic Int8

Characteristic Int16

Characteristic Int32

Characteristic Int64

Characteristic Integer

Characteristic Word

Characteristic Word8

Characteristic Word16

Characteristic Word32

Characteristic Word64

Characteristic ()

Characteristic Natural

(Characteristic a, Characteristic b) => Characteristic (a, b)

(Characteristic a, Characteristic b, Characteristic c) => Characteristic (a, b, c)

(Characteristic a, Characteristic b, Characteristic c, Characteristic d) => Characteristic (a, b, c, d)

(Characteristic a, Characteristic b, Characteristic c, Characteristic d, Characteristic e) => Characteristic (a, b, c, d, e)

charInt :: (Integral s, Bounded s) => proxy s -> Natural

charWord :: (Whole s, Bounded s) => proxy s -> Natural

Order

class Order a where

Methods

(<~) :: a -> a -> Bool

(<) :: a -> a -> Bool

(>~) :: a -> a -> Bool

(>) :: a -> a -> Bool

(~~) :: a -> a -> Bool

(/~) :: a -> a -> Bool

order :: a -> a -> Maybe Ordering

comparable :: a -> a -> Bool

Instances

Order Bool

Order Int

Order Int8

Order Int16

Order Int32

Order Int64

Order Integer

Order Word

Order Word8

Order Word16

Order Word32

Order Word64

Order ()

Order Natural

Ord a => Order (Set a)

(Order a, Order b) => Order (a, b)

(Order a, Order b, Order c) => Order (a, b, c)

(Order a, Order b, Order c, Order d) => Order (a, b, c, d)

(Order a, Order b, Order c, Order d, Order e) => Order (a, b, c, d, e)

class (AdditiveOrder r, Rig r) => OrderedRig r

Instances

OrderedRig Bool

OrderedRig Integer

OrderedRig ()

OrderedRig Natural

(OrderedRig a, OrderedRig b) => OrderedRig (a, b)

(OrderedRig a, OrderedRig b, OrderedRig c) => OrderedRig (a, b, c)

(OrderedRig a, OrderedRig b, OrderedRig c, OrderedRig d) => OrderedRig (a, b, c, d)

(OrderedRig a, OrderedRig b, OrderedRig c, OrderedRig d, OrderedRig e) => OrderedRig (a, b, c, d, e)

class (Additive r, Order r) => AdditiveOrder r

z + x <= z + y = x <= y = x + z <= y + z

Instances

AdditiveOrder Bool

AdditiveOrder Integer

AdditiveOrder ()

AdditiveOrder Natural

(AdditiveOrder a, AdditiveOrder b) => AdditiveOrder (a, b)

(AdditiveOrder a, AdditiveOrder b, AdditiveOrder c) => AdditiveOrder (a, b, c)

(AdditiveOrder a, AdditiveOrder b, AdditiveOrder c, AdditiveOrder d) => AdditiveOrder (a, b, c, d)

(AdditiveOrder a, AdditiveOrder b, AdditiveOrder c, AdditiveOrder d, AdditiveOrder e) => AdditiveOrder (a, b, c, d, e)

class Order a => LocallyFiniteOrder a

Instances

LocallyFiniteOrder Bool

LocallyFiniteOrder Int

LocallyFiniteOrder Int8

LocallyFiniteOrder Int16

LocallyFiniteOrder Int32

LocallyFiniteOrder Int64

LocallyFiniteOrder Integer

LocallyFiniteOrder Word

LocallyFiniteOrder Word8

LocallyFiniteOrder Word16

LocallyFiniteOrder Word32

LocallyFiniteOrder Word64

LocallyFiniteOrder ()

LocallyFiniteOrder Natural

Ord a => LocallyFiniteOrder (Set a)

(LocallyFiniteOrder a, LocallyFiniteOrder b) => LocallyFiniteOrder (a, b)

(LocallyFiniteOrder a, LocallyFiniteOrder b, LocallyFiniteOrder c) => LocallyFiniteOrder (a, b, c)

(LocallyFiniteOrder a, LocallyFiniteOrder b, LocallyFiniteOrder c, LocallyFiniteOrder d) => LocallyFiniteOrder (a, b, c, d)

(LocallyFiniteOrder a, LocallyFiniteOrder b, LocallyFiniteOrder c, LocallyFiniteOrder d, LocallyFiniteOrder e) => LocallyFiniteOrder (a, b, c, d, e)

class Monoidal r => DecidableZero r

Instances

DecidableZero Bool

DecidableZero Int

DecidableZero Int8

DecidableZero Int16

DecidableZero Int32

DecidableZero Int64

DecidableZero Integer

DecidableZero Word

DecidableZero Word8

DecidableZero Word16

DecidableZero Word32

DecidableZero Word64

DecidableZero ()

DecidableZero Natural

DecidableZero (BasisCoblade m)

DecidableZero r => DecidableZero (Opposite r)

(DecidableZero a, DecidableZero b) => DecidableZero (a, b)

(DecidableZero a, DecidableZero b, DecidableZero c) => DecidableZero (a, b, c)

(DecidableZero a, DecidableZero b, DecidableZero c, DecidableZero d) => DecidableZero (a, b, c, d)

(DecidableZero a, DecidableZero b, DecidableZero c, DecidableZero d, DecidableZero e) => DecidableZero (a, b, c, d, e)

class Unital r => DecidableUnits r

Instances

DecidableUnits Bool

DecidableUnits Int

DecidableUnits Int8

DecidableUnits Int16

DecidableUnits Int32

DecidableUnits Int64

DecidableUnits Integer

DecidableUnits Word

DecidableUnits Word8

DecidableUnits Word16

DecidableUnits Word32

DecidableUnits Word64

DecidableUnits ()

DecidableUnits Natural

DecidableUnits (BasisCoblade m)

DecidableUnits r => DecidableUnits (Opposite r)

(DecidableUnits a, DecidableUnits b) => DecidableUnits (a, b)

(DecidableUnits a, DecidableUnits b, DecidableUnits c) => DecidableUnits (a, b, c)

(DecidableUnits a, DecidableUnits b, DecidableUnits c, DecidableUnits d) => DecidableUnits (a, b, c, d)

(DecidableUnits a, DecidableUnits b, DecidableUnits c, DecidableUnits d, DecidableUnits e) => DecidableUnits (a, b, c, d, e)

class Unital r => DecidableAssociates r

Instances

DecidableAssociates Bool

DecidableAssociates Int

DecidableAssociates Int8

DecidableAssociates Int16

DecidableAssociates Int32

DecidableAssociates Int64

DecidableAssociates Integer

DecidableAssociates Word

DecidableAssociates Word8

DecidableAssociates Word16

DecidableAssociates Word32

DecidableAssociates Word64

DecidableAssociates ()

DecidableAssociates Natural

DecidableAssociates (BasisCoblade m)

DecidableAssociates r => DecidableAssociates (Opposite r)

(DecidableAssociates a, DecidableAssociates b) => DecidableAssociates (a, b)

(DecidableAssociates a, DecidableAssociates b, DecidableAssociates c) => DecidableAssociates (a, b, c)

(DecidableAssociates a, DecidableAssociates b, DecidableAssociates c, DecidableAssociates d) => DecidableAssociates (a, b, c, d)

(DecidableAssociates a, DecidableAssociates b, DecidableAssociates c, DecidableAssociates d, DecidableAssociates e) => DecidableAssociates (a, b, c, d, e)

Natural numbers

data Natural

Instances

Enum Natural

Eq Natural

Integral Natural

Num Natural

Ord Natural

Read Natural

Real Natural

Show Natural

Ix Natural

Bits Natural

Whole Natural

PartialSemigroup Natural

PartialMonoid Natural

PartialGroup Natural

Order Natural

Abelian Natural

Partitionable Natural

Additive Natural

Monoidal Natural

Semiring Natural

Multiplicative Natural

Unital Natural

DecidableAssociates Natural

DecidableUnits Natural

DecidableZero Natural

Rig Natural

Characteristic Natural

IntegralSemiring Natural

Commutative Natural

TriviallyInvolutive Natural

InvolutiveSemiring Natural

InvolutiveMultiplication Natural

AdditiveOrder Natural

OrderedRig Natural

LocallyFiniteOrder Natural

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

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

Rig r => Quadrance r Natural

RightModule Natural (BasisCoblade m)

Unital r => RightModule Natural (Log r)

(Abelian r, Monoidal r) => RightModule Natural (RngRing r)

Monoidal r => RightModule Natural (ZeroRng r)

LeftModule Natural (BasisCoblade m)

Unital r => LeftModule Natural (Log r)

(Abelian r, Monoidal r) => LeftModule Natural (RngRing r)

Monoidal r => LeftModule Natural (ZeroRng r)

class Integral n => Whole n where

Methods

toNatural :: n -> Natural

Instances

Whole Word

Whole Word8

Whole Word16

Whole Word32

Whole Word64

Whole Natural

Representable Additive

addRep :: (Zip m, Additive r) => m r -> m r -> m r

`Additive.(+)` default definition

sinnum1pRep :: (Whole n, Functor m, Additive r) => n -> m r -> m r

Additive.sinnum1p default definition

Representable Monoidal

zeroRep :: (Applicative m, Monoidal r) => m r

Monoidal.zero default definition

sinnumRep :: (Whole n, Functor m, Monoidal r) => n -> m r -> m r

Monoidal.sinnum default definition

Representable Group

negateRep :: (Functor m, Group r) => m r -> m r

Group.negate default definition

minusRep :: (Zip m, Group r) => m r -> m r -> m r

`Group.(-)` default definition

subtractRep :: (Zip m, Group r) => m r -> m r -> m r

Group.subtract default definition

timesRep :: (Integral n, Functor m, Group r) => n -> m r -> m r

Group.times default definition

Representable Multiplicative (via Algebra)

mulRep :: (Representable m, Algebra r (Key m)) => m r -> m r -> m r

`Multiplicative.(*)` default definition

Representable Unital (via UnitalAlgebra)

oneRep :: (Representable m, Unital r, UnitalAlgebra r (Key m)) => m r

Unital.one default definition

Representable Rig (via Algebra)

fromNaturalRep :: (UnitalAlgebra r (Key m), Representable m, Rig r) => Natural -> m r

Rig.fromNatural default definition

Representable Ring (via Algebra)

fromIntegerRep :: (UnitalAlgebra r (Key m), Representable m, Ring r) => Integer -> m r

Ring.fromInteger default definition

Norm

class Additive r => Quadrance r m where

Methods

quadrance :: m -> r

Instances

Quadrance () a

Rig r => Quadrance r Word64

Rig r => Quadrance r Word32

Rig r => Quadrance r Word16

Rig r => Quadrance r Word8

Rig r => Quadrance r Int64

Rig r => Quadrance r Int32

Rig r => Quadrance r Int16

Rig r => Quadrance r Int8

Rig r => Quadrance r Integer

Rig r => Quadrance r Natural

Rig r => Quadrance r Word

Rig r => Quadrance r Int

Rig r => Quadrance r Bool

Monoidal r => Quadrance r ()

(Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Complex r)

(TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion r)

(Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Dual r)

(Commutative r, InvolutiveSemiring r, Rng r) => Quadrance r (Hyper' r)

(Commutative r, Rng r, InvolutiveSemiring r) => Quadrance r (Dual' r)

(TriviallyInvolutive r, Rng r) => Quadrance r (Quaternion' r)

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

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

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

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

Covectors

newtype Covector r a

Linear functionals from elements of an (infinite) free module to a scalar

Constructors

Covector ((a -> r) -> r)

Instances

RightModule r s => RightModule r (Covector s m)

LeftModule r s => LeftModule r (Covector s m)

Monad (Covector r)

Functor (Covector r)

Monoidal r => MonadPlus (Covector r)

Applicative (Covector r)

Monoidal r => Alternative (Covector r)

Monoidal r => Plus (Covector r)

Additive r => Alt (Covector r)

Apply (Covector r)

Bind (Covector r)

Idempotent r => Idempotent (Covector r a)

Abelian s => Abelian (Covector s a)

Additive r => Additive (Covector r a)

Monoidal s => Monoidal (Covector s a)

Coalgebra r m => Semiring (Covector r m)

Coalgebra r m => Multiplicative (Covector r m)

Group s => Group (Covector s a)

CounitalCoalgebra r m => Unital (Covector r m)

(Idempotent r, IdempotentCoalgebra r a) => Band (Covector r a)

(Rig r, CounitalCoalgebra r m) => Rig (Covector r m)

(Ring r, CounitalCoalgebra r m) => Ring (Covector r m)

(Commutative m, Coalgebra r m) => Commutative (Covector r m)

Distinguished a => Distinguished (Covector r a)

Complicated a => Complicated (Covector r a)

Hamiltonian a => Hamiltonian (Covector r a)

Infinitesimal a => Infinitesimal (Covector r a)

Hyperbolic a => Hyperbolic (Covector r a)

Trigonometric a => Trigonometric (Covector r a)

Coalgebra r m => RightModule (Covector r m) (Covector r m)

Coalgebra r m => LeftModule (Covector r m) (Covector r m)

Covectors as linear functionals

counitM :: UnitalAlgebra r a => a -> Covector r ()

unitM :: CounitalCoalgebra r c => Covector r c

comultM :: Algebra r a => a -> Covector r (a, a)

multM :: Coalgebra r c => c -> c -> Covector r c

invM :: InvolutiveAlgebra r h => h -> Covector r h

coinvM :: InvolutiveCoalgebra r h => h -> Covector r h

antipodeM :: HopfAlgebra r h => h -> Covector r h

convolveM antipodeM return = convolveM return antipodeM = comultM >=> uncurry joinM

convolveM :: (Algebra r c, Coalgebra r a) => (c -> Covector r a) -> (c -> Covector r a) -> c -> Covector r a

memoM :: HasTrie a => a -> Covector s a