Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	None
Language	Haskell2010

OAlg.Structure.Algebraic.Definition

Contents

Description

definition of algebraic structures, i.e. Distributive structures with a suitable Vectorial structure.

Synopsis

class (Distributive a, Vectorial a) => Algebraic a
data Alg k
class (TransformableDst (s k), TransformableVec k s, Transformable (s k) (Alg k)) => TransformableAlg k (s :: Type -> Type)
class (Semiring r, Commutative r, Algebraic r, Scalar r ~ r) => AlgebraicSemiring r
type AlgebraicRing r = (AlgebraicSemiring r, Ring r)
type AlgebraicField r = (AlgebraicSemiring r, Field r)

Algebraic

class (Distributive a, Vectorial a) => Algebraic a Source #

Distributive structures with a suitable Vectorial structure.

Property Let a be a Algebraic structure, then holds:

For all r in Scalar a and a, b in a with start a == end b holds: r!(a*b) == (r!a)*b and r!(a*b) == a*(r!b).

Instances

Instances details

Algebraic N Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
Algebraic Q Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
Algebraic Z Source #
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Algebraic Integer Source #
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Algebraic () Source #
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Algebraic Int Source #
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Algebraic x => Algebraic (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
Algebraic a => Algebraic (Op a) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
Entity p => Algebraic (Orientation p) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
(Semiring r, Commutative r, Multiplicative m, FibredOriented m, Ord m) => Algebraic (Sum r m) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
(Algebraic x, Typeable t, Typeable n) => Algebraic (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero
(Algebraic a, Typeable t, Typeable n, Typeable m) => Algebraic (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation

data Alg k Source #

type representing the class of k-Algebraic structures.

Instances

Instances details

TransformableAlg k Alg Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableVec k Alg Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableG Op (Alg k) (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tauG :: Struct (Alg k) x -> Struct (Alg k) (Op x) Source #
TransformableAdd (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableTyp (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableDst (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableFbr (Alg k) Source #
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TransformableFbrOrt (Alg k) Source #
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TransformableMlt (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableOrt (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
Transformable (Alg k) Ent Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Ent x Source #
Transformable (Alg k) Add Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Add x Source #
Transformable (Alg k) Typ Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Typ x Source #
Transformable (Alg k) Dst Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Dst x Source #
Transformable (Alg k) Fbr Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Fbr x Source #
Transformable (Alg k) FbrOrt Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct FbrOrt x Source #
Transformable (Alg k) Mlt Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Mlt x Source #
Transformable (Alg k) Ort Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Ort x Source #
Transformable (Alg k) (Vec k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct (Vec k) x Source #
type Structure (Alg k) x Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition type Structure (Alg k) x = (Algebraic x, k ~ Scalar x)

class (TransformableDst (s k), TransformableVec k s, Transformable (s k) (Alg k)) => TransformableAlg k (s :: Type -> Type) Source #

helper class to avoid undecidable instances.

Instances

Instances details

TransformableAlg k Alg Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition

Algebraic Semiring

class (Semiring r, Commutative r, Algebraic r, Scalar r ~ r) => AlgebraicSemiring r Source #

Commutative Semirings with a sound Algebraic structure.

Property Let r be an instance of AlgebraicSemiring, then holds:

For all x and y in r holds: x ! y == x * y.

Note

The purpose of this structure is on the one hand to summarize the somewhat lengthy constraints and on the other hand to ensure that the scalar multiplication (!) is compatible with the Multiplicative structure.
The property 1. for a Algebraic structure forces the Semiring to be Commutative.

Instances

Instances details

AlgebraicSemiring N Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
AlgebraicSemiring Q Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
AlgebraicSemiring Z Source #
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AlgebraicSemiring Integer Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
AlgebraicSemiring Int Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition

type AlgebraicRing r = (AlgebraicSemiring r, Ring r) Source #

algebraic rings.

type AlgebraicField r = (AlgebraicSemiring r, Field r) Source #

algebraic fields.