Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

OAlg.Structure.Additive.Definition

Contents

Additive
Abelian
Orphan instances

Description

additive structures, i.e. structures with a partially defined addition (+).

Synopsis

class Fibred a => Additive a where
- zero :: Root a -> a
- (+) :: a -> a -> a
- ntimes :: N -> a -> a
zero' :: Additive a => p a -> Root a -> a
data Add
class (ForgetfulFbr s, Transformable s Add) => ForgetfulAdd s
class Additive a => Abelian a where
- negate :: a -> a
- (-) :: a -> a -> a
- ztimes :: Z -> a -> a
isZero :: Additive a => a -> Bool
data Abl
class (ForgetfulFbr s, ForgetfulAdd s, Transformable s Abl) => ForgetfulAbl s

Additive

class Fibred a => Additive a where Source #

Fibred structures with a partialy defined addition and having zero as the neutral element of the summation. An entity of a Additive structure will be called a summand.

Properties Let a be a Additive structure, then holds:

For all r in Root a holds: root (zero r) == r.
For all f and g in a holds:
1. If root f == root g then f + g is valid and root (f + g) == root f.
2. If root f /= root g then f + g is not valid and its evaluation will end up in a NotAddable-exception.
For all f, g in a with root f == root g holds: f + g == g + f.
For all f in a holds: f + zero (root f) == f
For all f, g, h in a with root f == root g == root h holds: (f + g) + h == f + (g + h).
For all f in a and n in N holds: ntimes 0 f == zero (root f) and ntimes (n + 1) f == f + ntimes n f.

Minimal complete definition

zero, (+)

Methods

zero :: Root a -> a Source #

the neutral element associated to each root. If there is no ambiguity for zero r we will briefly denote it by 0 r or just 0.

(+) :: a -> a -> a infixl 6 Source #

the addition for two summands.

ntimes :: N -> a -> a Source #

n times of a summand.

Instances

Instances details

Additive N Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root N -> N Source # (+) :: N -> N -> N Source # ntimes :: N -> N -> N Source #
Additive Q Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root Q -> Q Source # (+) :: Q -> Q -> Q Source # ntimes :: N -> Q -> Q Source #
Additive Z Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root Z -> Z Source # (+) :: Z -> Z -> Z Source # ntimes :: N -> Z -> Z Source #
Additive N' Source #
Instance details Defined in OAlg.Entity.Natural Methods zero :: Root N' -> N' Source # (+) :: N' -> N' -> N' Source # ntimes :: N -> N' -> N' Source #
Additive W' Source #
Instance details Defined in OAlg.Entity.Natural Methods zero :: Root W' -> W' Source # (+) :: W' -> W' -> W' Source # ntimes :: N -> W' -> W' Source #
Additive Integer Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root Integer -> Integer Source # (+) :: Integer -> Integer -> Integer Source # ntimes :: N -> Integer -> Integer Source #
Additive () Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root () -> () Source # (+) :: () -> () -> () Source # ntimes :: N -> () -> () Source #
Additive Int Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root Int -> Int Source # (+) :: Int -> Int -> Int Source # ntimes :: N -> Int -> Int Source #
(Additive a, FibredOriented a) => Additive (Op a) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root (Op a) -> Op a Source # (+) :: Op a -> Op a -> Op a Source # ntimes :: N -> Op a -> Op a Source #
(Additive x, FibredOriented x) => Additive (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods zero :: Root (Matrix x) -> Matrix x Source # (+) :: Matrix x -> Matrix x -> Matrix x Source # ntimes :: N -> Matrix x -> Matrix x Source #
Semiring r => Additive (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Methods zero :: Root (Vector r) -> Vector r Source # (+) :: Vector r -> Vector r -> Vector r Source # ntimes :: N -> Vector r -> Vector r Source #
Entity p => Additive (Orientation p) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root (Orientation p) -> Orientation p Source # (+) :: Orientation p -> Orientation p -> Orientation p Source # ntimes :: N -> Orientation p -> Orientation p Source #
(Fibred a, Ord a, Semiring r, Commutative r) => Additive (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods zero :: Root (Sum r a) -> Sum r a Source # (+) :: Sum r a -> Sum r a -> Sum r a Source # ntimes :: N -> Sum r a -> Sum r a Source #
(Distributive r, Total r, Commutative r, Ord a, Entity a) => Additive (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods zero :: Root (SumSymbol r a) -> SumSymbol r a Source # (+) :: SumSymbol r a -> SumSymbol r a -> SumSymbol r a Source # ntimes :: N -> SumSymbol r a -> SumSymbol r a Source #
(Distributive a, Typeable t, Typeable n, Typeable m) => Additive (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods zero :: Root (Transformation t n m a) -> Transformation t n m a Source # (+) :: Transformation t n m a -> Transformation t n m a -> Transformation t n m a Source # ntimes :: N -> Transformation t n m a -> Transformation t n m a Source #

zero' :: Additive a => p a -> Root a -> a Source #

the zero to a given root. The type p c serves only as proxy and zero' 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 zero.

data Add Source #

type representing the class of Additive structures.

Instances

Instances details

ForgetfulAdd Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulTyp Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulFbr Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition
Transformable Abl Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Add x Source #
Transformable Add Ent Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Add x -> Struct Ent x Source #
Transformable Add Typ Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Add x -> Struct Typ x Source #
Transformable Add Fbr Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Add x -> Struct Fbr x Source #
Transformable Dst Add Source #
Instance details Defined in OAlg.Structure.Distributive.Definition Methods tau :: Struct Dst x -> Struct Add x Source #
(Semiring r, Commutative r) => EmbeddableMorphism (HomSymbol r) Add Source #
Instance details Defined in OAlg.Entity.Matrix.Vector
EmbeddableMorphism h Add => EmbeddableMorphism (OpHom h) Add Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
Transformable (Alg k) Add Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Add x Source #
Transformable (Vec k) Add Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition Methods tau :: Struct (Vec k) x -> Struct Add x Source #
type Hom Add h Source #
Instance details Defined in OAlg.Hom.Additive type Hom Add h = HomAdditive h
type Structure Add x Source #
Instance details Defined in OAlg.Structure.Additive.Definition type Structure Add x = Additive x

class (ForgetfulFbr s, Transformable s Add) => ForgetfulAdd s Source #

transformable to Additive structure.

Instances

Instances details

ForgetfulAdd Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulAdd Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulAdd Dst Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
ForgetfulAdd (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
ForgetfulAdd (Vec k) Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition

Abelian

class Additive a => Abelian a where Source #

Additive structures having for each summand an additve inverse.

Properties Let a be a Additive structure, then holds:

For all f in a holds: root (negate f) == root f.
For all f in a holds: f + negate f == zero (root f).
For all f and g in a holds:
1. If root f == root g then f - g is valid and root (f - g) == root f.
2. If root f /= root g then f - g is not valid and its evaluation will end up in a NotAddable-exception.
For f and g in a with root f == root g holds: f - g == f + negate g.
For all z in Z and f in a holds:
1. If 0 <= z then ztimes z f == ntimes (prj z) f.
2. If z < 0 then ztimes z f == negate (ntimes (prj z) f).

Minimal complete definition

negate | (-)

Methods

negate :: a -> a Source #

negation of a summand.

(-) :: a -> a -> a infixl 6 Source #

subtraction of two summands.

Properties

ztimes :: Z -> a -> a Source #

z times of a sumand.

Instances

Instances details

Abelian Q Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Q -> Q Source # (-) :: Q -> Q -> Q Source # ztimes :: Z -> Q -> Q Source #
Abelian Z Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Z -> Z Source # (-) :: Z -> Z -> Z Source # ztimes :: Z -> Z -> Z Source #
Abelian Integer Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Integer -> Integer Source # (-) :: Integer -> Integer -> Integer Source # ztimes :: Z -> Integer -> Integer Source #
Abelian () Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: () -> () Source # (-) :: () -> () -> () Source # ztimes :: Z -> () -> () Source #
Abelian Int Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Int -> Int Source # (-) :: Int -> Int -> Int Source # ztimes :: Z -> Int -> Int Source #
(Abelian a, FibredOriented a) => Abelian (Op a) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Op a -> Op a Source # (-) :: Op a -> Op a -> Op a Source # ztimes :: Z -> Op a -> Op a Source #
(Abelian x, FibredOriented x) => Abelian (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods negate :: Matrix x -> Matrix x Source # (-) :: Matrix x -> Matrix x -> Matrix x Source # ztimes :: Z -> Matrix x -> Matrix x Source #
Ring r => Abelian (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Methods negate :: Vector r -> Vector r Source # (-) :: Vector r -> Vector r -> Vector r Source # ztimes :: Z -> Vector r -> Vector r Source #
Entity p => Abelian (Orientation p) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Orientation p -> Orientation p Source # (-) :: Orientation p -> Orientation p -> Orientation p Source # ztimes :: Z -> Orientation p -> Orientation p Source #
(Fibred a, Ord a, Ring r, Commutative r) => Abelian (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods negate :: Sum r a -> Sum r a Source # (-) :: Sum r a -> Sum r a -> Sum r a Source # ztimes :: Z -> Sum r a -> Sum r a Source #
(Distributive r, Total r, Commutative r, Ord a, Abelian r, Entity a) => Abelian (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods negate :: SumSymbol r a -> SumSymbol r a Source # (-) :: SumSymbol r a -> SumSymbol r a -> SumSymbol r a Source # ztimes :: Z -> SumSymbol r a -> SumSymbol r a Source #
(Distributive a, Abelian a, Typeable t, Typeable n, Typeable m) => Abelian (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods negate :: Transformation t n m a -> Transformation t n m a Source # (-) :: Transformation t n m a -> Transformation t n m a -> Transformation t n m a Source # ztimes :: Z -> Transformation t n m a -> Transformation t n m a Source #

isZero :: Additive a => a -> Bool Source #

check for beeing zero.

data Abl Source #

type representing the class of Abelian structures.

Instances

Instances details

ForgetfulAbl Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulAdd Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulTyp Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
ForgetfulFbr Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
Transformable Abl Ent Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Ent x Source #
Transformable Abl Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Add x Source #
Transformable Abl Typ Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Typ x Source #
Transformable Abl Fbr Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Fbr x Source #
type Structure Abl x Source #
Instance details Defined in OAlg.Structure.Additive.Definition type Structure Abl x = Abelian x

class (ForgetfulFbr s, ForgetfulAdd s, Transformable s Abl) => ForgetfulAbl s Source #

transformable to Abelian structure.

Instances

Instances details

ForgetfulAbl Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition

Orphan instances

Additive a => Projectible a (Sheaf a) Source #
Instance details Methods prj :: Sheaf a -> a Source #