Numeric.Algebra.Involutive

Involution

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(Semiring (a, b, c, d, e), InvolutiveMultiplication (a, b, c, d, e), 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

Methods

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

Instances

(Algebra r (), InvolutiveSemiring r) => InvolutiveAlgebra r ()

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

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

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

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

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

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

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

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

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

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

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

(InvolutiveSemiring r, Algebra r (a, b, c, d, e), 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

(Coalgebra r (), InvolutiveSemiring r) => InvolutiveCoalgebra r ()

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

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

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

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

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

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

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

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

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

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

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

(InvolutiveSemiring r, Coalgebra r (a, b, c, d, e), 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

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 Word

TriviallyInvolutive Word8

TriviallyInvolutive Word16

TriviallyInvolutive Word32

TriviallyInvolutive Word64

TriviallyInvolutive ()

TriviallyInvolutive Natural

TriviallyInvolutive Euclidean

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

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

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

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

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

(Commutative (a, b, c, d, e), InvolutiveMultiplication (a, b, c, d, e), 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

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

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

(CommutativeAlgebra r (a, b, c), TriviallyInvolutive r, InvolutiveAlgebra r (a, b, c), TriviallyInvolutiveAlgebra r a, TriviallyInvolutiveAlgebra r b, TriviallyInvolutiveAlgebra r c) => TriviallyInvolutiveAlgebra r (a, b, c)

(CommutativeAlgebra r (a, b, c, d), TriviallyInvolutive r, InvolutiveAlgebra r (a, b, c, d), TriviallyInvolutiveAlgebra r a, TriviallyInvolutiveAlgebra r b, TriviallyInvolutiveAlgebra r c, TriviallyInvolutiveAlgebra r d) => TriviallyInvolutiveAlgebra r (a, b, c, d)

(CommutativeAlgebra r (a, b, c, d, e), TriviallyInvolutive r, InvolutiveAlgebra r (a, b, c, d, e), 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

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

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

(CocommutativeCoalgebra r (a, b, c), TriviallyInvolutive r, InvolutiveCoalgebra r (a, b, c), TriviallyInvolutiveCoalgebra r a, TriviallyInvolutiveCoalgebra r b, TriviallyInvolutiveCoalgebra r c) => TriviallyInvolutiveCoalgebra r (a, b, c)

(CocommutativeCoalgebra r (a, b, c, d), TriviallyInvolutive r, InvolutiveCoalgebra r (a, b, c, d), TriviallyInvolutiveCoalgebra r a, TriviallyInvolutiveCoalgebra r b, TriviallyInvolutiveCoalgebra r c, TriviallyInvolutiveCoalgebra r d) => TriviallyInvolutiveCoalgebra r (a, b, c, d)

(CocommutativeCoalgebra r (a, b, c, d, e), TriviallyInvolutive r, InvolutiveCoalgebra r (a, b, c, d, e), 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