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

OAlg.AbelianGroup.ZMod

Contents

Cyclic Group
Homomorphism
X
Proposition
Exception

Description

Homomorphisms between cyclic groups.

A group is called cyclic if it is generated by one element. Such a group G is obviously abelian and isomorphic to Z/nZ for some n :: N, which is called its order. The group homomorphisms between cyclic groups form a Z-algebraic structure which is presented here.

Synopsis

newtype ZMod = ZMod N
zmOrd :: ZMod -> N
data ZModHom
toZ :: ZModHom -> Z
fromZ :: Z -> ZModHom
zmh :: Orientation ZMod -> Z -> ZModHom
zmhEligible :: ZMod -> ZMod -> Z -> Bool
zmhGenOrd :: Orientation ZMod -> (ZModHom, N)
zmhGenerator :: Orientation ZMod -> ZModHom
xZModHom :: X Z -> Orientation ZMod -> X ZModHom
xZModHomTo :: N -> X Z -> X ZMod -> XOrtSite To ZModHom
xZModHomFrom :: N -> X Z -> X ZMod -> XOrtSite From ZModHom
xZModTo :: N -> ZMod -> X ZMod
xZModFrom :: N -> ZMod -> X ZMod
prpZModHom :: Statement
data ZModException = NotEligible String

Cyclic Group

newtype ZMod Source #

cyclic group Z/n, i.e. the quotient group of Z divided by nZ.

Constructors

ZMod N

Instances

Instances details

Show ZMod Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods showsPrec :: Int -> ZMod -> ShowS # show :: ZMod -> String # showList :: [ZMod] -> ShowS #
Eq ZMod Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods (==) :: ZMod -> ZMod -> Bool # (/=) :: ZMod -> ZMod -> Bool #
Ord ZMod Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods compare :: ZMod -> ZMod -> Ordering # (<) :: ZMod -> ZMod -> Bool # (<=) :: ZMod -> ZMod -> Bool # (>) :: ZMod -> ZMod -> Bool # (>=) :: ZMod -> ZMod -> Bool # max :: ZMod -> ZMod -> ZMod # min :: ZMod -> ZMod -> ZMod #
Validable ZMod Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods valid :: ZMod -> Statement #
XStandard ZMod Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods xStandard :: X ZMod #
Entity ZMod Source #
Instance details Defined in OAlg.AbelianGroup.ZMod

zmOrd :: ZMod -> N Source #

order of the cyclic group.

Homomorphism

data ZModHom Source #

additive homomorphisms between cyclic groups ZMod constructable via zmh.

Note The homomorphisms between two cyclic groups a and b form again a cyclic group which will be denoted by ZModHom (a:>b) (see zmhGenOrd).

Instances

Instances details

Show ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods showsPrec :: Int -> ZModHom -> ShowS # show :: ZModHom -> String # showList :: [ZModHom] -> ShowS #
Eq ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods (==) :: ZModHom -> ZModHom -> Bool # (/=) :: ZModHom -> ZModHom -> Bool #
Ord ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods compare :: ZModHom -> ZModHom -> Ordering # (<) :: ZModHom -> ZModHom -> Bool # (<=) :: ZModHom -> ZModHom -> Bool # (>) :: ZModHom -> ZModHom -> Bool # (>=) :: ZModHom -> ZModHom -> Bool # max :: ZModHom -> ZModHom -> ZModHom # min :: ZModHom -> ZModHom -> ZModHom #
Validable ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods valid :: ZModHom -> Statement #
Entity ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod
Abelian ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods negate :: ZModHom -> ZModHom # (-) :: ZModHom -> ZModHom -> ZModHom # ztimes :: Z -> ZModHom -> ZModHom #
Additive ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods zero :: Root ZModHom -> ZModHom # (+) :: ZModHom -> ZModHom -> ZModHom # ntimes :: N -> ZModHom -> ZModHom #
Algebraic ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod
Distributive ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod
Fibred ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Associated Types type Root ZModHom # Methods root :: ZModHom -> Root ZModHom #
FibredOriented ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod
Commutative ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod
Invertible ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods tryToInvert :: ZModHom -> Solver ZModHom # invert :: ZModHom -> ZModHom # isInvertible :: ZModHom -> Bool # zpower :: ZModHom -> Z -> ZModHom #
Multiplicative ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods one :: Point ZModHom -> ZModHom # (*) :: ZModHom -> ZModHom -> ZModHom # npower :: ZModHom -> N -> ZModHom #
Oriented ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Associated Types type Point ZModHom # Methods orientation :: ZModHom -> Orientation (Point ZModHom) # start :: ZModHom -> Point ZModHom # end :: ZModHom -> Point ZModHom #
Vectorial ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Associated Types type Scalar ZModHom # Methods (!) :: Scalar ZModHom -> ZModHom -> ZModHom #
XStandardOrtSite 'From ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods xStandardOrtSite :: XOrtSite 'From ZModHom #
XStandardOrtSite 'To ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods xStandardOrtSite :: XOrtSite 'To ZModHom #
type Root ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod type Root ZModHom = Orientation ZMod
type Point ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod type Point ZModHom = ZMod
type Scalar ZModHom Source #
Instance details Defined in OAlg.AbelianGroup.ZMod type Scalar ZModHom = Z

toZ :: ZModHom -> Z Source #

the underlying integer.

fromZ :: Z -> ZModHom Source #

the endomorphism in ZMod 0 given by the integer, i.e. the inverse of toZ.

zmh :: Orientation ZMod -> Z -> ZModHom Source #

the induced homomorphism.

Property Let a, b be in ZMod and r in Z then holds:

zmh a b r is valid if and only if zmhEligible a b r is true.
If zmhEligible a b r is false then the evaluation of zmh a b r will end up by throwing a NotEligible-exception.

zmhEligible :: ZMod -> ZMod -> Z -> Bool Source #

predicate to determine the eligibility of a value in Z to induce a homomorphism between the given cyclic groups.

Property Let a = ZMod a', b = ZMod b' be in ZMod and r in Z, then holds: zmhEligible a b r is true if and only if mod0 (r * inj a') b' == 0.

zmhGenOrd :: Orientation ZMod -> (ZModHom, N) Source #

zmhGenOrd (a:>b) = (r,o) where r is a generator for ZModHom(a:>b) with order o.

Note It follows that inj o ! r == zero (a:>b) and ZModHom(a:>b) is isomorphic to Z/o which is represented by ZMod o.

zmhGenerator :: Orientation ZMod -> ZModHom Source #

zmhGenerator (a:>b) = h is a generator for the abelian group ZModHom(a:>b).

X

Since the algebraic structure of ZModHom is not balanced - which means that between any two cyclic groups there may be no nontrivial homomorphisms - it is appropriate to work with XOrtSite instead of XOrtOrientation for random variables with values in ZModHom.

xZModHom :: X Z -> Orientation ZMod -> X ZModHom Source #

random variable for homomorphisms for the given orientation of ZMod.

xZModHomTo :: N -> X Z -> X ZMod -> XOrtSite To ZModHom Source #

random variable for homomorphisms based on xZModTo.

xZModHomFrom :: N -> X Z -> X ZMod -> XOrtSite From ZModHom Source #

random variable for homomorphisms based on xZModFrom.

xZModTo :: N -> ZMod -> X ZMod Source #

random variable of cyclic groups admitting nontrivial homomorphisms to the given group.

xZModFrom :: N -> ZMod -> X ZMod Source #

random variable of cyclic groups admitting non trivial homomorphisms from the given one.

Proposition

prpZModHom :: Statement Source #

validity of the Z-algebraic structure of ZModHom.

Exception

data ZModException Source #

exceptions for cyclic groups which are sub exceptions from SomeOAlgException.

Constructors

NotEligible String

Instances

Instances details

Exception ZModException Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods toException :: ZModException -> SomeException # fromException :: SomeException -> Maybe ZModException # displayException :: ZModException -> String #
Show ZModException Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods showsPrec :: Int -> ZModException -> ShowS # show :: ZModException -> String # showList :: [ZModException] -> ShowS #
Eq ZModException Source #
Instance details Defined in OAlg.AbelianGroup.ZMod Methods (==) :: ZModException -> ZModException -> Bool # (/=) :: ZModException -> ZModException -> Bool #