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Numeric.Algebra.Involutive

Contents

Involution
Involutive Algebras
Trivial Involution

Synopsis

Involution

class Multiplicative r => InvolutiveMultiplication r whereSource

An semigroup with involution

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

Methods

adjoint :: r -> rSource

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

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)

Involutive Algebras

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

Methods

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

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 whereSource

Methods

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

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 Source

Instances

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

Trivial Involution

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

 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)
(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 Source

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 Source

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 Source

Instances

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