algebra-4.3: Constructive abstract algebra

Numeric.Algebra.Involutive

Synopsis

# Involution

class Multiplicative r => InvolutiveMultiplication r where Source #

An semigroup with involution

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

Minimal complete definition

Methods

adjoint :: r -> r Source #

Instances

Instances

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

# Involutive Algebras

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

Minimal complete definition

inv

Methods

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

Instances

 Source # Methodsinv :: (() -> r) -> () -> r Source # Source # Methodsinv :: (ComplexBasis -> k) -> ComplexBasis -> k Source # Source # Methodsinv :: (DualBasis -> k) -> DualBasis -> k Source # Source # Methodsinv :: (HyperBasis' -> k) -> HyperBasis' -> k Source # Source # Methodsinv :: (QuaternionBasis -> r) -> QuaternionBasis -> r Source # Source # Methodsinv :: (DualBasis' -> k) -> DualBasis' -> k Source # Source # Methodsinv :: (HyperBasis -> k) -> HyperBasis -> k Source # Source # Methodsinv :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r Source # Source # Methodsinv :: (TrigBasis -> k) -> TrigBasis -> k Source # (InvolutiveAlgebra r a, InvolutiveAlgebra r b) => InvolutiveAlgebra r (a, b) Source # Methodsinv :: ((a, b) -> r) -> (a, b) -> r Source # (InvolutiveAlgebra r a, InvolutiveAlgebra r b, InvolutiveAlgebra r c) => InvolutiveAlgebra r (a, b, c) Source # Methodsinv :: ((a, b, c) -> r) -> (a, b, c) -> r Source # (InvolutiveAlgebra r a, InvolutiveAlgebra r b, InvolutiveAlgebra r c, InvolutiveAlgebra r d) => InvolutiveAlgebra r (a, b, c, d) Source # Methodsinv :: ((a, b, c, d) -> r) -> (a, b, c, d) -> r Source # (InvolutiveAlgebra r a, InvolutiveAlgebra r b, InvolutiveAlgebra r c, InvolutiveAlgebra r d, InvolutiveAlgebra r e) => InvolutiveAlgebra r (a, b, c, d, e) Source # Methodsinv :: ((a, b, c, d, e) -> r) -> (a, b, c, d, e) -> r Source #

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

Minimal complete definition

coinv

Methods

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

Instances

 Source # Methodscoinv :: (() -> r) -> () -> r Source # Source # Methodscoinv :: (ComplexBasis -> k) -> ComplexBasis -> k Source # Source # Methodscoinv :: (DualBasis -> k) -> DualBasis -> k Source # Source # Methodscoinv :: (HyperBasis' -> k) -> HyperBasis' -> k Source # Source # Methodscoinv :: (QuaternionBasis -> r) -> QuaternionBasis -> r Source # Source # Methodscoinv :: (DualBasis' -> k) -> DualBasis' -> k Source # Source # Methodscoinv :: (HyperBasis -> k) -> HyperBasis -> k Source # Source # Methodscoinv :: (QuaternionBasis' -> r) -> QuaternionBasis' -> r Source # Source # Methodscoinv :: (TrigBasis -> k) -> TrigBasis -> k Source # (InvolutiveCoalgebra r a, InvolutiveCoalgebra r b) => InvolutiveCoalgebra r (a, b) Source # Methodscoinv :: ((a, b) -> r) -> (a, b) -> r Source # (InvolutiveCoalgebra r a, InvolutiveCoalgebra r b, InvolutiveCoalgebra r c) => InvolutiveCoalgebra r (a, b, c) Source # Methodscoinv :: ((a, b, c) -> r) -> (a, b, c) -> r Source # (InvolutiveCoalgebra r a, InvolutiveCoalgebra r b, InvolutiveCoalgebra r c, InvolutiveCoalgebra r d) => InvolutiveCoalgebra r (a, b, c, d) Source # Methodscoinv :: ((a, b, c, d) -> r) -> (a, b, c, d) -> r Source # (InvolutiveCoalgebra r a, InvolutiveCoalgebra r b, InvolutiveCoalgebra r c, InvolutiveCoalgebra r d, InvolutiveCoalgebra r e) => InvolutiveCoalgebra r (a, b, c, d, e) Source # Methodscoinv :: ((a, b, c, d, e) -> r) -> (a, b, c, d, e) -> r Source #

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 Source #

# Trivial Involution

adjoint = id

Instances

 Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # (TriviallyInvolutive r, TriviallyInvolutiveAlgebra r a) => TriviallyInvolutive (a -> r) Source # Source # (TriviallyInvolutive a, TriviallyInvolutive b, TriviallyInvolutive c) => TriviallyInvolutive (a, b, c) Source # (TriviallyInvolutive a, TriviallyInvolutive b, TriviallyInvolutive c, TriviallyInvolutive d) => TriviallyInvolutive (a, b, c, d) Source # (TriviallyInvolutive a, TriviallyInvolutive b, TriviallyInvolutive c, TriviallyInvolutive d, TriviallyInvolutive e) => TriviallyInvolutive (a, b, c, d, e) Source #

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

Instances

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

Instances

 Source # Source # Source # Source # Source #

Instances

 Source #