Numeric.Algebra.Involutive

Involution

class Multiplicative r => InvolutiveMultiplication r where

An semigroup with involution

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

Minimal complete definition

adjoint

Methods

adjoint :: r -> r

Instances

InvolutiveMultiplication Bool

Methods

adjoint :: Bool -> Bool

InvolutiveMultiplication Int

Methods

adjoint :: Int -> Int

InvolutiveMultiplication Int8

Methods

adjoint :: Int8 -> Int8

InvolutiveMultiplication Int16

Methods

adjoint :: Int16 -> Int16

InvolutiveMultiplication Int32

Methods

adjoint :: Int32 -> Int32

InvolutiveMultiplication Int64

Methods

adjoint :: Int64 -> Int64

InvolutiveMultiplication Integer

Methods

adjoint :: Integer -> Integer

InvolutiveMultiplication Natural

Methods

adjoint :: Natural -> Natural

InvolutiveMultiplication Word

Methods

adjoint :: Word -> Word

InvolutiveMultiplication Word8

Methods

adjoint :: Word8 -> Word8

InvolutiveMultiplication Word16

Methods

adjoint :: Word16 -> Word16

InvolutiveMultiplication Word32

Methods

adjoint :: Word32 -> Word32

InvolutiveMultiplication Word64

Methods

adjoint :: Word64 -> Word64

InvolutiveMultiplication ()

Methods

adjoint :: () -> ()

InvolutiveMultiplication Euclidean

Methods

adjoint :: Euclidean -> Euclidean

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

Methods

adjoint :: Trig r -> Trig r

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

Methods

adjoint :: Quaternion' r -> Quaternion' r

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

Methods

adjoint :: Hyper r -> Hyper r

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

Methods

adjoint :: Dual' r -> Dual' r

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

Methods

adjoint :: Quaternion r -> Quaternion r

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

Methods

adjoint :: Hyper' r -> Hyper' r

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

Methods

adjoint :: Dual r -> Dual r

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

Methods

adjoint :: Complex r -> Complex r

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

Methods

adjoint :: (h -> r) -> h -> r

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

Methods

adjoint :: (a, b) -> (a, b)

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

Methods

adjoint :: (a, b, c) -> (a, b, c)

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

Methods

adjoint :: (a, b, c, d) -> (a, b, c, d)

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

Methods

adjoint :: (a, b, c, d, e) -> (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 Natural

InvolutiveSemiring Word

InvolutiveSemiring Word8

InvolutiveSemiring Word16

InvolutiveSemiring Word32

InvolutiveSemiring Word64

InvolutiveSemiring ()

InvolutiveSemiring Euclidean

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

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

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

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

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

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

Involutive Algebras

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

Minimal complete definition

inv

Methods

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

Instances

InvolutiveSemiring r => InvolutiveAlgebra r ()

Methods

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

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

Methods

inv :: (TrigBasis -> k) -> TrigBasis -> k

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

Methods

inv :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r

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

Methods

inv :: (HyperBasis -> k) -> HyperBasis -> k

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

Methods

inv :: (DualBasis' -> k) -> DualBasis' -> k

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

Methods

inv :: (QuaternionBasis -> r) -> QuaternionBasis -> r

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

Methods

inv :: (HyperBasis' -> k) -> HyperBasis' -> k

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

Methods

inv :: (DualBasis -> k) -> DualBasis -> k

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

Methods

inv :: (ComplexBasis -> k) -> ComplexBasis -> k

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

Methods

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

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

Methods

inv :: ((a, b, c) -> r) -> (a, b, c) -> r

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

Methods

inv :: ((a, b, c, d) -> r) -> (a, b, c, d) -> r

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

Methods

inv :: ((a, b, c, d, e) -> r) -> (a, b, c, d, e) -> r

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

Minimal complete definition

coinv

Methods

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

Instances

InvolutiveSemiring r => InvolutiveCoalgebra r ()

Methods

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

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

Methods

coinv :: (TrigBasis -> k) -> TrigBasis -> k

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

Methods

coinv :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r

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

Methods

coinv :: (HyperBasis -> k) -> HyperBasis -> k

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

Methods

coinv :: (DualBasis' -> k) -> DualBasis' -> k

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

Methods

coinv :: (QuaternionBasis -> r) -> QuaternionBasis -> r

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

Methods

coinv :: (HyperBasis' -> k) -> HyperBasis' -> k

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

Methods

coinv :: (DualBasis -> k) -> DualBasis -> k

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

Methods

coinv :: (ComplexBasis -> k) -> ComplexBasis -> k

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

Methods

coinv :: ((a, b) -> r) -> (a, b) -> r

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

Methods

coinv :: ((a, b, c) -> r) -> (a, b, c) -> r

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

Methods

coinv :: ((a, b, c, d) -> r) -> (a, b, c, d) -> r

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

Methods

coinv :: ((a, b, c, d, e) -> r) -> (a, b, c, d, e) -> r

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

Trivial Involution

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 Natural

TriviallyInvolutive Word

TriviallyInvolutive Word8

TriviallyInvolutive Word16

TriviallyInvolutive Word32

TriviallyInvolutive Word64

TriviallyInvolutive ()

TriviallyInvolutive Euclidean

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

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

(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)

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