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

OAlg.Structure.Multiplicative.Definition

Contents

Multiplicative
Transposable
Commutative
Invertible
Cayleyan
X

Description

multiplicative structures, i.e. structures with a partially defined multiplication (*).

Synopsis

class Oriented c => Multiplicative c where
- one :: Point c -> c
- (*) :: c -> c -> c
- npower :: c -> N -> c
one' :: Multiplicative c => p c -> Point c -> c
isOne :: Multiplicative c => c -> Bool
data Mlt
class (TransformableOrt s, Transformable s Mlt) => TransformableMlt s
tauMlt :: Transformable s Mlt => Struct s x -> Struct Mlt x
class (TransposableOriented c, Multiplicative c) => TransposableMultiplicative c
class Multiplicative c => Commutative c
class Multiplicative c => Invertible c where
- tryToInvert :: c -> Solver c
- invert :: c -> c
- isInvertible :: c -> Bool
- zpower :: c -> Z -> c
data Inv c = Inv c c
class Invertible c => Cayleyan c
xosPathAt :: forall c (s :: Site). Multiplicative c => XOrtSite s c -> N -> Point c -> X (Path c)
xosPath :: forall c (s :: Site). Multiplicative c => XOrtSite s c -> N -> X (Path c)
xosXOrtSitePath :: forall c (s :: Site). Multiplicative c => XOrtSite s c -> N -> XOrtSite s (Path c)

Multiplicative

class Oriented c => Multiplicative c where Source #

Oriented structures with a partially defined multiplication and having one as the neutral element of the multiplication. An entity of a Multiplicative structure will be called a factor.

Properties Let c be a type instance of the class Multiplicative, then holds:

For all p in Point c holds: orientation (one p) == p :> p.
For all f and g in c holds:
1. if start f == end g then f * g is valid and start (f * g) == start g and end (f * g) == end f.
2. if start f /= end g then f * g is not valid and its evaluation will end up in a NotMultiplicable exception.
For all f in c holds: one (end f) * f == f and f * one (start f) == f.
For all f, g and h in c with start g == end h and start f == end g holds: (f * g) * h == f * (g * h).
For all f in c holds:
1. npower f 1 == f.
2. If f is a endo than npower f 0 == one (start f) and For all n in N holds: npower f (succ n) == f * npower f n.

Such a c will be called a multiplicative structure and an entity f of c will be called factor. The associated factor one p to a p in Point c will be called the one at p.

Note If the types c and Point c are interpreted as sets M and O and * as a partially defined function from M x M -> M then this forms a small category with objects in O and morphisms in M.

Minimal complete definition

one, (*)

Methods

one :: Point c -> c Source #

the neutral element associated to each point. If there is no ambiguity for one p we will briefly denote it by 1 r or just 1.

(*) :: c -> c -> c infixl 7 Source #

the multiplication of two factors.

npower :: c -> N -> c Source #

n times the multiplication of a given factor f.

Instances

Instances details

Multiplicative N Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point N -> N Source # (*) :: N -> N -> N Source # npower :: N -> N -> N Source #
Multiplicative Q Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point Q -> Q Source # (*) :: Q -> Q -> Q Source # npower :: Q -> N -> Q Source #
Multiplicative Z Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point Z -> Z Source # (*) :: Z -> Z -> Z Source # npower :: Z -> N -> Z Source #
Multiplicative Variant Source #
Instance details Defined in OAlg.Data.Variant Methods one :: Point Variant -> Variant Source # (*) :: Variant -> Variant -> Variant Source # npower :: Variant -> N -> Variant Source #
Multiplicative N' Source #
Instance details Defined in OAlg.Entity.Natural Methods one :: Point N' -> N' Source # (*) :: N' -> N' -> N' Source # npower :: N' -> N -> N' Source #
Multiplicative W' Source #
Instance details Defined in OAlg.Entity.Natural Methods one :: Point W' -> W' Source # (*) :: W' -> W' -> W' Source # npower :: W' -> N -> W' Source #
Multiplicative Integer Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point Integer -> Integer Source # (*) :: Integer -> Integer -> Integer Source # npower :: Integer -> N -> Integer Source #
Multiplicative () Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point () -> () Source # (*) :: () -> () -> () Source # npower :: () -> N -> () Source #
Multiplicative Int Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point Int -> Int Source # (*) :: Int -> Int -> Int Source # npower :: Int -> N -> Int Source #
Multiplicative c => Multiplicative (Id c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Id c) -> Id c Source # (*) :: Id c -> Id c -> Id c Source # npower :: Id c -> N -> Id c Source #
Distributive x => Multiplicative (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods one :: Point (Matrix x) -> Matrix x Source # (*) :: Matrix x -> Matrix x -> Matrix x Source # npower :: Matrix x -> N -> Matrix x Source #
Galoisian x => Multiplicative (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods one :: Point (GL2 x) -> GL2 x Source # (*) :: GL2 x -> GL2 x -> GL2 x Source # npower :: GL2 x -> N -> GL2 x Source #
Oriented x => Multiplicative (GLT x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods one :: Point (GLT x) -> GLT x Source # (*) :: GLT x -> GLT x -> GLT x Source # npower :: GLT x -> N -> GLT x Source #
Oriented x => Multiplicative (ColTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods one :: Point (ColTrafo x) -> ColTrafo x Source # (*) :: ColTrafo x -> ColTrafo x -> ColTrafo x Source # npower :: ColTrafo x -> N -> ColTrafo x Source #
Oriented x => Multiplicative (RowTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods one :: Point (RowTrafo x) -> RowTrafo x Source # (*) :: RowTrafo x -> RowTrafo x -> RowTrafo x Source # npower :: RowTrafo x -> N -> RowTrafo x Source #
(Show x, Validable x, Eq x, Typeable x) => Multiplicative (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods one :: Point (ProductSymbol x) -> ProductSymbol x Source # (*) :: ProductSymbol x -> ProductSymbol x -> ProductSymbol x Source # npower :: ProductSymbol x -> N -> ProductSymbol x Source #
(Entity i, Ord i) => Multiplicative (Permutation i) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation Methods one :: Point (Permutation i) -> Permutation i Source # (*) :: Permutation i -> Permutation i -> Permutation i Source # npower :: Permutation i -> N -> Permutation i Source #
Fibred f => Multiplicative (Sheaf f) Source #	`'Data.List.(++)'` is not commutative!
Instance details Defined in OAlg.Structure.Fibred.Definition Methods one :: Point (Sheaf f) -> Sheaf f Source # (*) :: Sheaf f -> Sheaf f -> Sheaf f Source # npower :: Sheaf f -> N -> Sheaf f Source #
Multiplicative c => Multiplicative (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Inv c) -> Inv c Source # (*) :: Inv c -> Inv c -> Inv c Source # npower :: Inv c -> N -> Inv c Source #
Multiplicative c => Multiplicative (Op c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Op c) -> Op c Source # (*) :: Op c -> Op c -> Op c Source # npower :: Op c -> N -> Op c Source #
Entity p => Multiplicative (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Orientation p) -> Orientation p Source # (*) :: Orientation p -> Orientation p -> Orientation p Source # npower :: Orientation p -> N -> Orientation p Source #
Oriented q => Multiplicative (Path q) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Path q) -> Path q Source # (*) :: Path q -> Path q -> Path q Source # npower :: Path q -> N -> Path q Source #
(Oriented x, Typeable p, p ~ Point x) => Multiplicative (Dim x p) Source #
Instance details Defined in OAlg.Entity.Matrix.Dim Methods one :: Point (Dim x p) -> Dim x p Source # (*) :: Dim x p -> Dim x p -> Dim x p Source # npower :: Dim x p -> N -> Dim x p Source #
(Oriented a, Integral r) => Multiplicative (Product r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods one :: Point (Product r a) -> Product r a Source # (*) :: Product r a -> Product r a -> Product r a Source # npower :: Product r a -> N -> Product r a Source #
(Semiring r, Commutative r, Multiplicative m, FibredOriented m, Ord m) => Multiplicative (Sum r m) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods one :: Point (Sum r m) -> Sum r m Source # (*) :: Sum r m -> Sum r m -> Sum r m Source # npower :: Sum r m -> N -> Sum r m Source #
(Multiplicative x, Sliced i x, Typeable s) => Multiplicative (SliceFactor s i x) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods one :: Point (SliceFactor s i x) -> SliceFactor s i x Source # (*) :: SliceFactor s i x -> SliceFactor s i x -> SliceFactor s i x Source # npower :: SliceFactor s i x -> N -> SliceFactor s i x Source #
(Distributive x, Typeable t, Typeable n) => Multiplicative (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero Methods one :: Point (ConsecutiveZeroHom t n x) -> ConsecutiveZeroHom t n x Source # (*) :: ConsecutiveZeroHom t n x -> ConsecutiveZeroHom t n x -> ConsecutiveZeroHom t n x Source # npower :: ConsecutiveZeroHom t n x -> N -> ConsecutiveZeroHom t n x Source #
(Multiplicative a, Typeable t, Typeable n, Typeable m) => Multiplicative (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods one :: Point (DiagramTrafo t n m a) -> DiagramTrafo t n m a Source # (*) :: DiagramTrafo t n m a -> DiagramTrafo t n m a -> DiagramTrafo t n m a Source # npower :: DiagramTrafo t n m a -> N -> DiagramTrafo t n m a Source #

one' :: Multiplicative c => p c -> Point c -> c Source #

the one to a given point. The type p c serves only as proxy and one' is lazy in it.

Note As Point may be a non-injective type family, the type checker needs some times a little bit more information to pic the right one.

isOne :: Multiplicative c => c -> Bool Source #

check for being equal to one.

data Mlt Source #

type representing the class of Multiplicative structures.

Instances

Instances details

TransformableTyp Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransformableType Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransformableMlt Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransformableOrt Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransformableOp Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Transformable Dst Mlt Source #
Instance details Defined in OAlg.Structure.Distributive.Definition Methods tau :: Struct Dst x -> Struct Mlt x Source #
Transformable DstX Mlt Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition Methods tau :: Struct DstX x -> Struct Mlt x Source #
Transformable Mlt Ent Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tau :: Struct Mlt x -> Struct Ent x Source #
Transformable Mlt Typ Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tau :: Struct Mlt x -> Struct Typ x Source #
Transformable Mlt Ort Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tau :: Struct Mlt x -> Struct Ort x Source #
Transformable Mlt Type Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tau :: Struct Mlt x -> Struct Type x Source #
Transformable MltX Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods tau :: Struct MltX x -> Struct Mlt x Source #
TransformableGRefl Op Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransformableG Op Mlt Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tauG :: Struct Mlt x -> Struct Mlt (Op x) Source #
(HomMultiplicativeDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => NaturalConic h Cone Mlt p d t n m Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => NaturalConic h ConeZeroHead Mlt p d t n m Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
Conic c => XStandardEligibleConeFactorG c Mlt 'Injective d 'Discrete n m (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods xStandardEligibleConeFactorG :: XEligibleConeFactorG c Mlt 'Injective d 'Discrete n m (Matrix Z) Source #
Conic c => XStandardEligibleConeFactorG c Mlt 'Projective d 'Discrete n m (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods xStandardEligibleConeFactorG :: XEligibleConeFactorG c Mlt 'Projective d 'Discrete n m (Matrix Z) Source #
(Conic c, Diagrammatic d) => XStandardEligibleConeG c Mlt p d 'Discrete n m (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods xStandardEligibleConeG :: XEligibleConeG c Mlt p d 'Discrete n m (Matrix Z) Source #
TransformableGRefl Op (Mlt, Sld i) Source #
Instance details Defined in OAlg.Entity.Slice.Sliced
TransformableG Op (Mlt, t) Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tauG :: Struct (Mlt, t) x -> Struct Mlt (Op x) Source #
(HomMultiplicativeDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => NaturalTransformable h (->) (SDualBi (ConeG Cone Mlt p d t n m)) (SDualBi (ConeG Cone Mlt p d t n m)) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality
(HomDistributiveDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => NaturalTransformable h (->) (SDualBi (ConeG ConeZeroHead Mlt p d t n m)) (SDualBi (ConeG Cone Mlt p d t n m)) Source #
Instance details Defined in OAlg.Limes.Cone.ZeroHead.Duality
HomOrientedSlicedFree (Inv2 (HomFree Mlt)) Source #
Instance details Defined in OAlg.Entity.Slice.Free
Transformable (Alg k) Mlt Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Mlt x Source #
(HomMultiplicativeDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p), t ~ Dual (Dual t)) => ApplicativeG (SDualBi (ConeG Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality Methods amapG :: h x y -> SDualBi (ConeG Cone Mlt p d t n m) x -> SDualBi (ConeG Cone Mlt p d t n m) y Source #
(HomMultiplicativeDisjunctive h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => ApplicativeG (SDualBi (Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Duality Methods amapG :: h x y -> SDualBi (Cone Mlt p d t n m) x -> SDualBi (Cone Mlt p d t n m) y Source #
(HomMultiplicativeDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p), t ~ Dual (Dual t)) => FunctorialG (SDualBi (ConeG Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Conic.Duality
(HomMultiplicativeDisjunctive h, FunctorialOriented h, NaturalDiagrammaticBi h d t n m, p ~ Dual (Dual p)) => FunctorialG (SDualBi (Cone Mlt p d t n m)) h (->) Source #
Instance details Defined in OAlg.Limes.Cone.Duality
p ~ Dual (Dual p) => ApplicativeG (SDualBi (LiftableFree p)) (Inv2 (HomFree Mlt)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods amapG :: Inv2 (HomFree Mlt) x y -> SDualBi (LiftableFree p) x -> SDualBi (LiftableFree p) y Source #
p ~ Dual (Dual p) => FunctorialG (SDualBi (LiftableFree p)) (Inv2 (HomFree Mlt)) (->) Source #
Instance details Defined in OAlg.Entity.Slice.Free
FunctorialOriented (Sub (Mlt, SldFr) (HomDisjEmpty Mlt Op)) Source #
Instance details Defined in OAlg.Entity.Slice.Free
TransformableOp (Mlt, Sld i) Source #
Instance details Defined in OAlg.Entity.Slice.Sliced
Transformable s Mlt => Transformable (s, SldFr) Mlt Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods tau :: Struct (s, SldFr) x -> Struct Mlt x Source #
Transformable s Mlt => Transformable (s, Sld i) Mlt Source #
Instance details Defined in OAlg.Entity.Slice.Sliced Methods tau :: Struct (s, Sld i) x -> Struct Mlt x Source #
TransformableObjectClass (Mlt, SldFr) (HomDisj Mlt Op (HomEmpty Mlt)) Source #
Instance details Defined in OAlg.Entity.Slice.Free
(Entity p, Diagrammatic d, XStandard p, XStandard (d t n m (Orientation p))) => XStandard (Cone Mlt 'Injective d t n m (Orientation p)) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Methods xStandard :: X (Cone Mlt 'Injective d t n m (Orientation p)) Source #
(Entity p, Diagrammatic d, XStandard p, XStandard (d t n m (Orientation p))) => XStandard (Cone Mlt 'Projective d t n m (Orientation p)) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Methods xStandard :: X (Cone Mlt 'Projective d t n m (Orientation p)) Source #
type Hom Mlt h Source #
Instance details Defined in OAlg.Hom.Multiplicative type Hom Mlt h = HomMultiplicative h
type HomD Mlt h Source #
Instance details Defined in OAlg.Hom.Multiplicative type HomD Mlt h = HomMultiplicativeDisjunctive h
type Structure Mlt x Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition type Structure Mlt x = Multiplicative x

class (TransformableOrt s, Transformable s Mlt) => TransformableMlt s Source #

helper class to avoid undecidable instances.

Instances

Instances details

TransformableMlt Dst Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
TransformableMlt DstX Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition
TransformableMlt Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransformableMlt MltX Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition
TransformableMlt (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableMlt s => TransformableMlt (s, SldFr) Source #
Instance details Defined in OAlg.Entity.Slice.Free
TransformableMlt s => TransformableMlt (s, Sld i) Source #
Instance details Defined in OAlg.Entity.Slice.Sliced

tauMlt :: Transformable s Mlt => Struct s x -> Struct Mlt x Source #

tau for Mlt.

Transposable

class (TransposableOriented c, Multiplicative c) => TransposableMultiplicative c Source #

transposable Multiplicative structures.

Property Let c be a TransposableMultiplicative structure, then holds:

For all p in Point c holds: transpose (one p) = one p.
For all f, g in c with start f == end g holds: transpose (f * g) == transpose g * transpose f.

Instances

Instances details

TransposableMultiplicative N Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransposableMultiplicative Q Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
TransposableMultiplicative Z Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
(Distributive x, TransposableDistributive x) => TransposableMultiplicative (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
(Galoisian x, TransposableDistributive x) => TransposableMultiplicative (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
TransposableMultiplicative c => TransposableMultiplicative (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Entity p => TransposableMultiplicative (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition

Commutative

class Multiplicative c => Commutative c Source #

commutative multiplicative structures.

Property Let c be a Commutative structure, then holds: For all f and g in c with start f == end f, start g == end g and start f == end g holds: f * g == g * f.

Instances

Instances details

Commutative N Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Commutative Q Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Commutative Z Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Commutative N' Source #
Instance details Defined in OAlg.Entity.Natural
Commutative W' Source #
Instance details Defined in OAlg.Entity.Natural
Commutative Integer Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Commutative () Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Commutative Int Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Commutative c => Commutative (Op c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition

Invertible

class Multiplicative c => Invertible c where Source #

multiplicative structures having a multiplicative inverse.

Definition Let f and g be two factors in a Multiplicative structure _c then we call g a multiplicative inverse to f (or short inverse) if and only if the following hold:

start g == end f and end g == start f.
f * g = one (end f) and g * f == one (start f).

Properties For all f in a Invertible structure c holds:

isInvertible f is equivalent to solvable (tryToInvert f).
if isInvertible f holds, then invert f is valid and it is the multiplicative inverse of f. Furthermore invert f == solve (tryToInvert m).
if not isInvertible f holds, then invert f is not valid and evaluating it will end up in a NotInvertible-exception.

Note

It is not required that every factor has a multiplicative inverse (see Cayleyan for such structures).
This structure is intended for multiplicative structures having a known algorithm to evaluate for every invertible f its inverse.

Minimal complete definition

tryToInvert

Methods

tryToInvert :: c -> Solver c Source #

solver to evaluate the multiplicative inverse - if it exists.

invert :: c -> c Source #

the inverse.

isInvertible :: c -> Bool Source #

check for being invertible.

zpower :: c -> Z -> c Source #

if 0 <= z then n times the multiplication for the given factor else prj z times the multiplication of the inverse of the given factor.

Instances

Instances details

Invertible N Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: N -> Solver N Source # invert :: N -> N Source # isInvertible :: N -> Bool Source # zpower :: N -> Z -> N Source #
Invertible Q Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Q -> Solver Q Source # invert :: Q -> Q Source # isInvertible :: Q -> Bool Source # zpower :: Q -> Z -> Q Source #
Invertible Z Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Z -> Solver Z Source # invert :: Z -> Z Source # isInvertible :: Z -> Bool Source # zpower :: Z -> Z -> Z Source #
Invertible Integer Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Integer -> Solver Integer Source # invert :: Integer -> Integer Source # isInvertible :: Integer -> Bool Source # zpower :: Integer -> Z -> Integer Source #
Invertible () Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: () -> Solver () Source # invert :: () -> () Source # isInvertible :: () -> Bool Source # zpower :: () -> Z -> () Source #
Invertible Int Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Int -> Solver Int Source # invert :: Int -> Int Source # isInvertible :: Int -> Bool Source # zpower :: Int -> Z -> Int Source #
Galoisian x => Invertible (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods tryToInvert :: GL2 x -> Solver (GL2 x) Source # invert :: GL2 x -> GL2 x Source # isInvertible :: GL2 x -> Bool Source # zpower :: GL2 x -> Z -> GL2 x Source #
Oriented x => Invertible (GLT x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Methods tryToInvert :: GLT x -> Solver (GLT x) Source # invert :: GLT x -> GLT x Source # isInvertible :: GLT x -> Bool Source # zpower :: GLT x -> Z -> GLT x Source #
Oriented x => Invertible (ColTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods tryToInvert :: ColTrafo x -> Solver (ColTrafo x) Source # invert :: ColTrafo x -> ColTrafo x Source # isInvertible :: ColTrafo x -> Bool Source # zpower :: ColTrafo x -> Z -> ColTrafo x Source #
Oriented x => Invertible (RowTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods tryToInvert :: RowTrafo x -> Solver (RowTrafo x) Source # invert :: RowTrafo x -> RowTrafo x Source # isInvertible :: RowTrafo x -> Bool Source # zpower :: RowTrafo x -> Z -> RowTrafo x Source #
(Entity i, Ord i) => Invertible (Permutation i) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation Methods tryToInvert :: Permutation i -> Solver (Permutation i) Source # invert :: Permutation i -> Permutation i Source # isInvertible :: Permutation i -> Bool Source # zpower :: Permutation i -> Z -> Permutation i Source #
Multiplicative c => Invertible (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Inv c -> Solver (Inv c) Source # invert :: Inv c -> Inv c Source # isInvertible :: Inv c -> Bool Source # zpower :: Inv c -> Z -> Inv c Source #
Invertible c => Invertible (Op c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Op c -> Solver (Op c) Source # invert :: Op c -> Op c Source # isInvertible :: Op c -> Bool Source # zpower :: Op c -> Z -> Op c Source #
Entity p => Invertible (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Orientation p -> Solver (Orientation p) Source # invert :: Orientation p -> Orientation p Source # isInvertible :: Orientation p -> Bool Source # zpower :: Orientation p -> Z -> Orientation p Source #
(Oriented a, Integral r, Ring r) => Invertible (Product r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods tryToInvert :: Product r a -> Solver (Product r a) Source # invert :: Product r a -> Product r a Source # isInvertible :: Product r a -> Bool Source # zpower :: Product r a -> Z -> Product r a Source #

data Inv c Source #

invertible factors within a Multiplicative structures c, which forms a sub Multiplicative structure on c, given by the canonical inclusion inj which is given by \Inv f _ -> f.

Property Let Inv f f' be in Inv c where c is a Multiplicative structure, then holds:

orientation f' == opposite (orientation f).
f' * f == one (start f).
f * f' == one (end f).

Note The canonical inclusion is obviously not injective on the set of all values of type Inv c to c. But restricted to the valid ones it is injective, because the inverses of a f in c are uniquely determined by f.

Constructors

Inv c c

Instances

Instances details

Show c => Show (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

showsPrec :: Int -> Inv c -> ShowS #

show :: Inv c -> String #

showList :: [Inv c] -> ShowS #

Eq c => Eq (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

(==) :: Inv c -> Inv c -> Bool #

(/=) :: Inv c -> Inv c -> Bool #

TransposableMultiplicative c => Transposable (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

transpose :: Inv c -> Inv c Source #

Multiplicative c => Validable (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

valid :: Inv c -> Statement Source #

Multiplicative c => Exponential (Inv c) Source #

Instance details

Defined in OAlg.Structure.Exponential

Associated Types

type Exponent (Inv c)
Instance details Defined in OAlg.Structure.Exponential type Exponent (Inv c) = Z

Methods

(^) :: Inv c -> Exponent (Inv c) -> Inv c Source #

Multiplicative c => Cayleyan (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Multiplicative c => Invertible (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

tryToInvert :: Inv c -> Solver (Inv c) Source #

invert :: Inv c -> Inv c Source #

isInvertible :: Inv c -> Bool Source #

zpower :: Inv c -> Z -> Inv c Source #

Multiplicative c => Multiplicative (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

one :: Point (Inv c) -> Inv c Source #

(*) :: Inv c -> Inv c -> Inv c Source #

npower :: Inv c -> N -> Inv c Source #

TransposableMultiplicative c => TransposableMultiplicative (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Multiplicative c => Oriented (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

orientation :: Inv c -> Orientation (Point (Inv c)) Source #

start :: Inv c -> Point (Inv c) Source #

end :: Inv c -> Point (Inv c) Source #

TransposableMultiplicative c => TransposableOriented (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

EqPoint c => EqPoint (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

ShowPoint c => ShowPoint (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

TypeablePoint c => TypeablePoint (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

ValidablePoint c => ValidablePoint (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Embeddable (Inv c) c Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

Methods

inj :: Inv c -> c Source #

type Exponent (Inv c) Source #

Instance details

Defined in OAlg.Structure.Exponential

type Exponent (Inv c) = Z

type Point (Inv c) Source #

Instance details

Defined in OAlg.Structure.Multiplicative.Definition

type Point (Inv c) = Point c

Cayleyan

class Invertible c => Cayleyan c Source #

Invertible structures where every element is invertible.

Property Let c be a Cayleyan structure, then holds: For all f in c holds: isInvertible f == True.

Note

If the type Point c is singleton, then the mathematical interpretation of c is a group.
The name of this structures is given by Arthur Cayley who introduced the concept (and the name) of an abstract group in 1854 (https://en.wikipedia.org/wiki/Arthur_Cayley).
Usually in mathematics such a structure is called a groupoid.

Instances

Instances details

Cayleyan () Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Galoisian x => Cayleyan (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
Oriented x => Cayleyan (GLT x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
Oriented x => Cayleyan (ColTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation
Oriented x => Cayleyan (RowTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation
(Entity i, Ord i) => Cayleyan (Permutation i) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation
Multiplicative c => Cayleyan (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Cayleyan c => Cayleyan (Op c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Entity p => Cayleyan (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
(Oriented a, Integral r, Ring r) => Cayleyan (Product r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition

X

xosPathAt :: forall c (s :: Site). Multiplicative c => XOrtSite s c -> N -> Point c -> X (Path c) Source #

random variable of paths at the given point and the given length (see xosPathMaxAt and as c is Multiplicative, the underlying random variable for factors for a given point is not empty).

xosPath :: forall c (s :: Site). Multiplicative c => XOrtSite s c -> N -> X (Path c) Source #

random variable of paths with the given length.

xosXOrtSitePath :: forall c (s :: Site). Multiplicative c => XOrtSite s c -> N -> XOrtSite s (Path c) Source #

the induced random variable for paths.