OAlg.AbelianGroup.Definition

homomorphisms between finitely generated abelian groups.

Abelian Group

newtype AbGroup

finitely generate abelian group, i.e. the cartesian product of cyclic groups Z/n and are represented as a formal product with symbols in ZMod.

Definition Let g be in AbGroup. We call g smith normal if and only if there exists a sequence n 0, n 1 .. n (k-1) in N with length k and a exponent r in N such that:

1. 2 <= n i for all 0 <= i < k.
2. n (i + 1) mod n i == 0 for all i in 0 <= i < k-1.
3. g == abg (n 0) * abg (n 1) * .. * abg (n (k-1)) * abg 0 ^ r.

Theorem Every finitely generated abelian group is isomorphic to a group in smith normal form. This isomorphism is given by isoSmithNormal.

Examples Finitely generated abelian groups constructed via abg and its multiplicative structure:

>>> abg 12
AbGroup[Z/12]

represents the cyclic group Z/12.

>>> abg 2 * abg 3
AbGroup[Z/2*Z/3]

represents the cartesian product of the groups Z/2 and Z/3.

>>> abg 6 * abg 4 * abg 4
AbGroup[Z/6*Z/4^2]

represents the cartesian product of the groups Z/6, Z/4 and Z/4.

>>> abg 0 ^ 6
AbGroup[Z^6]

represents the free abelian group Z ^ 6 of dimension 6.

>>> one () :: AbGroup
AbGroup[]

represents the cartesian product of zero cyclic groups and

>>> one () * abg 4 * abg 6 == abg 4 * abg 6
True

Examples Checking for smith normal via isSmithNormal:

>>> isSmithNormal (abg 4)
True

>>> isSmithNormal (abg 2 * abg 2)
True

>>> isSmithNormal (abg 17 * abg 51)
True

>>> isSmithNormal (abg 2 * abg 4 * abg 0 ^ 3)
True

>>> isSmithNormal (abg 5 * abg 3)
False

>>> isSmithNormal (abg 0 * abg 3 * abg 6)
False

>>> isSmithNormal (abg 1 * abg 4)
False

>>> isSmithNormal (one ())
True

Examples The associated isomorphism in AbHom of a finitely generated abelian group given by isoSmithNormal.

>>> end (isoSmithNormal (abg 3 * abg 5))
AbGroup[Z/15]

>>> end (isoSmithNormal (abg 2 * abg 4 * abg 2))
AbGroup[Z/2^2*Z/4]

>>> end (isoSmithNormal (abg 4 * abg 6))
AbGroup[Z/2*Z/12]

>>> end (isoSmithNormal (abg 1))
AbGroup[]

Constructors

AbGroup (ProductSymbol ZMod)

Instances

Show AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

showsPrec :: Int -> AbGroup -> ShowS

show :: AbGroup -> String

showList :: [AbGroup] -> ShowS

Eq AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

(==) :: AbGroup -> AbGroup -> Bool

(/=) :: AbGroup -> AbGroup -> Bool

Ord AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

compare :: AbGroup -> AbGroup -> Ordering

(<) :: AbGroup -> AbGroup -> Bool

(<=) :: AbGroup -> AbGroup -> Bool

(>) :: AbGroup -> AbGroup -> Bool

(>=) :: AbGroup -> AbGroup -> Bool

max :: AbGroup -> AbGroup -> AbGroup

min :: AbGroup -> AbGroup -> AbGroup

LengthN AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

lengthN :: AbGroup -> N

Validable AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

valid :: AbGroup -> Statement

XStandard AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

xStandard :: X AbGroup

Entity AbGroup

Defined in OAlg.AbelianGroup.Definition

Exponential AbGroup

Defined in OAlg.AbelianGroup.Definition

Associated Types

type Exponent AbGroup

Methods

(^) :: AbGroup -> Exponent AbGroup -> AbGroup

Multiplicative AbGroup

Defined in OAlg.AbelianGroup.Definition

Methods

one :: Point AbGroup -> AbGroup

(*) :: AbGroup -> AbGroup -> AbGroup

npower :: AbGroup -> N -> AbGroup

Oriented AbGroup

Defined in OAlg.AbelianGroup.Definition

Associated Types

type Point AbGroup

Methods

orientation :: AbGroup -> Orientation (Point AbGroup)

start :: AbGroup -> Point AbGroup

end :: AbGroup -> Point AbGroup

type Exponent AbGroup

Defined in OAlg.AbelianGroup.Definition

type Exponent AbGroup = N

type Point AbGroup

Defined in OAlg.AbelianGroup.Definition

type Point AbGroup = ()

abg :: N -> AbGroup

the cyclic group of the given order as a finitely generated abelian group.

isSmithNormal :: AbGroup -> Bool

checks if the given group is smith normal (see definition AbGroup).

abgDim :: AbGroup -> Dim' ZModHom

the associated dimension for matrices of ZModHom.

Homomorphism

newtype AbHom

additive homomorphism between finitely generated abelian groups which are represented by matrices over ZModHom.

Constructors

AbHom (Matrix ZModHom)

Instances

Show AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

showsPrec :: Int -> AbHom -> ShowS

show :: AbHom -> String

showList :: [AbHom] -> ShowS

Eq AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

(==) :: AbHom -> AbHom -> Bool

(/=) :: AbHom -> AbHom -> Bool

XStandardSomeFreeSliceFromLiftable AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

xStandardSomeFreeSliceFromLiftable :: XSomeFreeSliceFromLiftable AbHom

Validable AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

valid :: AbHom -> Statement

XStandard AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

xStandard :: X AbHom

Entity AbHom

Defined in OAlg.AbelianGroup.Definition

Abelian AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

negate :: AbHom -> AbHom

(-) :: AbHom -> AbHom -> AbHom

ztimes :: Z -> AbHom -> AbHom

Additive AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

zero :: Root AbHom -> AbHom

(+) :: AbHom -> AbHom -> AbHom

ntimes :: N -> AbHom -> AbHom

Algebraic AbHom

Defined in OAlg.AbelianGroup.Definition

Distributive AbHom

Defined in OAlg.AbelianGroup.Definition

Fibred AbHom

Defined in OAlg.AbelianGroup.Definition

Associated Types

type Root AbHom

Methods

root :: AbHom -> Root AbHom

FibredOriented AbHom

Defined in OAlg.AbelianGroup.Definition

Multiplicative AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

one :: Point AbHom -> AbHom

(*) :: AbHom -> AbHom -> AbHom

npower :: AbHom -> N -> AbHom

Oriented AbHom

Defined in OAlg.AbelianGroup.Definition

Associated Types

type Point AbHom

Methods

orientation :: AbHom -> Orientation (Point AbHom)

start :: AbHom -> Point AbHom

end :: AbHom -> Point AbHom

XStandardOrtOrientation AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

xStandardOrtOrientation :: XOrtOrientation AbHom

XStandardOrtSiteFrom AbHom

Defined in OAlg.AbelianGroup.Definition

XStandardOrtSiteTo AbHom

Defined in OAlg.AbelianGroup.Definition

XStandardPoint AbHom

Defined in OAlg.AbelianGroup.Definition

Vectorial AbHom

Defined in OAlg.AbelianGroup.Definition

Associated Types

type Scalar AbHom

Methods

(!) :: Scalar AbHom -> AbHom -> AbHom

XStandardOrtSite 'From AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

xStandardOrtSite :: XOrtSite 'From AbHom

XStandardOrtSite 'To AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

xStandardOrtSite :: XOrtSite 'To AbHom

Attestable k => SliceCokernelTo (Free k) AbHom

Defined in OAlg.AbelianGroup.KernelsAndCokernels

Methods

sliceCokernelTo :: Slice 'To (Free k) AbHom -> Cokernel N1 AbHom

Attestable k => SliceKernelFrom (Free k) AbHom

Defined in OAlg.AbelianGroup.KernelsAndCokernels

Methods

sliceKernelFrom :: Slice 'From (Free k) AbHom -> Kernel N1 AbHom

Attestable k => Sliced (Free k) AbHom

Defined in OAlg.AbelianGroup.Definition

Methods

slicePoint :: Free k AbHom -> Point AbHom

type Root AbHom

Defined in OAlg.AbelianGroup.Definition

type Root AbHom = Orientation AbGroup

type Point AbHom

Defined in OAlg.AbelianGroup.Definition

type Point AbHom = AbGroup

type Scalar AbHom

Defined in OAlg.AbelianGroup.Definition

type Scalar AbHom = Z

abh :: Orientation AbGroup -> [(ZModHom, N, N)] -> AbHom

the additive homomorphism with the given orientation and ZModHom-entries.

abh' :: Orientation AbGroup -> [(Z, N, N)] -> AbHom

the additive homomorphism with the given orientation and Z-entries.

abhz :: AbHom -> Matrix Z

the underlying Z-matrix.

zabh :: Matrix Z -> AbHom

the associated homomorphism between products of abg 0 given by the column - respectively row - length.

Adjunction

abhFreeAdjunction :: Adjunction AbHomFree (Matrix Z) AbHom

the projection AbHomFree as left adjoint.

data AbHomFree x y where

projection homomorphisms to Matrix Z.

Constructors

AbHomFree :: AbHomFree AbHom (Matrix Z)

FreeAbHom :: AbHomFree (Matrix Z) AbHom

Instances

Applicative AbHomFree

Defined in OAlg.AbelianGroup.Definition

Methods

amap :: AbHomFree a b -> a -> b

EmbeddableMorphismTyp AbHomFree

Defined in OAlg.AbelianGroup.Definition

Morphism AbHomFree

Defined in OAlg.AbelianGroup.Definition

Associated Types

type ObjectClass AbHomFree

Methods

homomorphous :: AbHomFree x y -> Homomorphous (ObjectClass AbHomFree) x y

domain :: AbHomFree x y -> Struct (ObjectClass AbHomFree) x

range :: AbHomFree x y -> Struct (ObjectClass AbHomFree) y

Eq2 AbHomFree

Defined in OAlg.AbelianGroup.Definition

Methods

eq2 :: AbHomFree x y -> AbHomFree x y -> Bool

Show2 AbHomFree

Defined in OAlg.AbelianGroup.Definition

Methods

show2 :: AbHomFree a b -> String

Validable2 AbHomFree

Defined in OAlg.AbelianGroup.Definition

Methods

valid2 :: AbHomFree x y -> Statement

Entity2 AbHomFree

Defined in OAlg.AbelianGroup.Definition

HomAdditive AbHomFree

Defined in OAlg.AbelianGroup.Definition

HomDistributive AbHomFree

Defined in OAlg.AbelianGroup.Definition

HomFibred AbHomFree

Defined in OAlg.AbelianGroup.Definition

Methods

rmap :: AbHomFree a b -> Root a -> Root b

HomFibredOriented AbHomFree

Defined in OAlg.AbelianGroup.Definition

HomMultiplicative AbHomFree

Defined in OAlg.AbelianGroup.Definition

HomOriented AbHomFree

Defined in OAlg.AbelianGroup.Definition

Methods

pmap :: AbHomFree a b -> Point a -> Point b

EmbeddableMorphism AbHomFree Add

Defined in OAlg.AbelianGroup.Definition

EmbeddableMorphism AbHomFree Typ

Defined in OAlg.AbelianGroup.Definition

EmbeddableMorphism AbHomFree Dst

Defined in OAlg.AbelianGroup.Definition

EmbeddableMorphism AbHomFree Fbr

Defined in OAlg.AbelianGroup.Definition

EmbeddableMorphism AbHomFree FbrOrt

Defined in OAlg.AbelianGroup.Definition

EmbeddableMorphism AbHomFree Mlt

Defined in OAlg.AbelianGroup.Definition

EmbeddableMorphism AbHomFree Ort

Defined in OAlg.AbelianGroup.Definition

Show (AbHomFree x y)

Defined in OAlg.AbelianGroup.Definition

Methods

showsPrec :: Int -> AbHomFree x y -> ShowS

show :: AbHomFree x y -> String

showList :: [AbHomFree x y] -> ShowS

Eq (AbHomFree x y)

Defined in OAlg.AbelianGroup.Definition

Methods

(==) :: AbHomFree x y -> AbHomFree x y -> Bool

(/=) :: AbHomFree x y -> AbHomFree x y -> Bool

Validable (AbHomFree x y)

Defined in OAlg.AbelianGroup.Definition

Methods

valid :: AbHomFree x y -> Statement

(Typeable x, Typeable y) => Entity (AbHomFree x y)

Defined in OAlg.AbelianGroup.Definition

type ObjectClass AbHomFree

Defined in OAlg.AbelianGroup.Definition

type ObjectClass AbHomFree = Alg Z

Limes

abhProducts :: Products n AbHom

products for AbHom.

abhSums :: Sums n AbHom

sums for AbHom.

Generator

abgGeneratorTo :: AbGroup -> Generator To AbHom

the generator for a finitely generated abelian group.

Property Let a be in AbGroup, then holds a == g where Generator (DiagramChainTo g _) _ _ _ _ = abgGeneratorTo a.

X

xAbHom :: Q -> Orientation AbGroup -> X AbHom

random variable for AbHom given by a density and an orientation.

xAbHomTo :: Q -> N -> N -> N -> AbGroup -> X AbHom

random variable of homomorphisms between abelian groups with end equal to the given one.

   r s t
  [f    ] a
  [     ] b
  [g h  ] c

xAbHomFrom :: Q -> N -> N -> N -> AbGroup -> X AbHom

random variable of homomorphisms between abelian groups with start equal to the given one.

   a b c
  [f    ] r
  [g   l] s
  [h    ] t

stdMaxDim :: N

the maximal length of abelian groups for the standard random variable of type X AbGroup.

Property 1 <= stdMaxDim.

Proposition

prpAbHom :: Statement

validity of the algebraic structure of AbHom.