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Language	Haskell98

Numeric.Algebra.Involutive

Contents

Involution
Involutive Algebras
Trivial Involution

Synopsis

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 #