HAPprime 
  
  
                           Datatypes reference manual 
  
  
                                 Version 0.3.2
  
  
                                13 February 2009
  
  
                                   Paul Smith
  
  
  
  Paul Smith
      Email:    mailto:paul.smith@nuigalway.ie
      Homepage: http://www.maths.nuigalway.ie/~pas
      Address:  Department of Mathematics,
                National University of Ireland, Galway
                Galway,
                Ireland.
  
  
  
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  Copyright
  © 2006-2009 Paul Smith
  
  HAPprime  is  released under the GNU General Public License (GPL). This file
  is  part  of  HAPprime, though as documentation it is released under the GNU
  Free                Documentation                License                (see
  http://www.gnu.org/licenses/licenses.html#FDL).
  
  HAPprime  is  free  software; you can redistribute it and/or modify it under
  the  terms  of  the  GNU  General  Public  License  as published by the Free
  Software  Foundation;  either  version 2 of the License, or (at your option)
  any later version.
  
  HAPprime  is distributed in the hope that it will be useful, but WITHOUT ANY
  WARRANTY;  without  even  the implied warranty of MERCHANTABILITY or FITNESS
  FOR  A  PARTICULAR  PURPOSE.  See  the  GNU  General Public License for more
  details.
  
  You should have received a copy of the GNU General Public License along with
  HAPprime;  if  not,  write  to the Free Software Foundation, Inc., 59 Temple
  Place, Suite 330, Boston, MA 02111-1307 USA
  
  For more details, see http://www.fsf.org/licenses/gpl.html.
  
  
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  Acknowledgements
  HAPprime  is supported by a Marie Curie Transfer of Knowledge grant based at
  the Department of Mathematics, NUI Galway (MTKD-CT-2006-042685)
  
  
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  Contents (HAPprime Datatypes)
  
  1 Introduction
  2 Resolutions
    2.1 The HAPResolution datatype in HAPprime
    2.2 Implementation: Constructing resolutions
    2.3 Resolution construction functions
      2.3-1 LengthOneResolutionPrimePowerGroup
      2.3-2 LengthZeroResolutionPrimePowerGroup
    2.4 Resolution data access functions
      2.4-1 ResolutionLength
      2.4-2 ResolutionGroup
      2.4-3 ResolutionFpGModuleGF
      2.4-4 ResolutionModuleRank
      2.4-5 ResolutionModuleRanks
      2.4-6 BoundaryFpGModuleHomomorphismGF
      2.4-7 ResolutionsAreEqual
    2.5 Example: Computing and working with resolutions
    2.6 Miscellaneous resolution functions
      2.6-1 BestCentralSubgroupForResolutionFiniteExtension
  3 Graded algebras
    3.1 Graded algebras in HAP (and HAPprime)
    3.2 Data access functions
      3.2-1 ModPRingGeneratorDegrees
      3.2-2 ModPRingNiceBasis
      3.2-3 ModPRingNiceBasisAsPolynomials
      3.2-4 ModPRingBasisAsPolynomials
    3.3 Other functions
      3.3-1 PresentationOfGradedStructureConstantAlgebra
    3.4 Example: Graded algebras and mod-p cohomology rings
  4 Presentations of graded algebras
    4.1 The GradedAlgebraPresentation datatype
    4.2 Construction function
      4.2-1 GradedAlgebraPresentation construction functions
    4.3 Data access functions
      4.3-1 BaseRing
      4.3-2 CoefficientsRing
      4.3-3 IndeterminatesOfGradedAlgebraPresentation
      4.3-4 GeneratorsOfPresentationIdeal
      4.3-5 PresentationIdeal
      4.3-6 IndeterminateDegrees
      4.3-7 Example: Constructing and accessing data of a
      GradedAlgebraPresentation
    4.4 Other functions
      4.4-1 TensorProduct
      4.4-2 IsIsomorphicGradedAlgebra
      4.4-3 IsAssociatedGradedRing
      4.4-4 DegreeOfRepresentative
      4.4-5 MaximumDegreeForPresentation
      4.4-6 SubspaceDimensionDegree
      4.4-7 SubspaceBasisRepsByDegree
      4.4-8 CoefficientsOfPoincareSeries
      4.4-9 HilbertPoincareSeries
      4.4-10 LHSSpectralSequence
    4.5 Example: Computing the Lyndon-Hoschild-Serre spectral sequence and
    mod-p cohomology ring for a small p-group
  5 FG-modules
    5.1 The FpGModuleGF datatype
    5.2 Implementation details: Block echelon form
      5.2-1 Generating vectors and their block structure
      5.2-2 Matrix echelon reduction and head elements
      5.2-3 Echelon block structure and minimal generators
      5.2-4 Intersection of two modules
    5.3 Construction functions
      5.3-1 FpGModuleGF construction functions
      5.3-2 FpGModuleFromFpGModuleGF
      5.3-3 MutableCopyModule
      5.3-4 CanonicalAction
      5.3-5 Example: Constructing a FpGModuleGF
    5.4 Data access functions
      5.4-1 ModuleGroup
      5.4-2 ModuleGroupOrder
      5.4-3 ModuleAction
      5.4-4 ModuleActionBlockSize
      5.4-5 ModuleGroupAndAction
      5.4-6 ModuleCharacteristic
      5.4-7 ModuleField
      5.4-8 ModuleAmbientDimension
      5.4-9 AmbientModuleDimension
      5.4-10 DisplayBlocks
      5.4-11 Example: Accessing data about a FpGModuleGF
    5.5 Generator and vector space functions
      5.5-1 ModuleGenerators
      5.5-2 ModuleGeneratorsAreMinimal
      5.5-3 ModuleGeneratorsAreEchelonForm
      5.5-4 ModuleIsFullCanonical
      5.5-5 ModuleGeneratorsForm
      5.5-6 ModuleRank
      5.5-7 ModuleVectorSpaceBasis
      5.5-8 ModuleVectorSpaceDimension
      5.5-9 MinimalGeneratorsModule
      5.5-10 RadicalOfModule
      5.5-11 Example: Generators and basis vectors of a FpGModuleGF
    5.6 Block echelon functions
      5.6-1 EchelonModuleGenerators
      5.6-2 ReverseEchelonModuleGenerators
      5.6-3 Example: Converting a FpGModuleGF to block echelon form
    5.7 Sum and intersection functions
      5.7-1 DirectSumOfModules
      5.7-2 DirectDecompositionOfModule
      5.7-3 IntersectionModules
      5.7-4 SumModules
      5.7-5 Example: Sum and intersection of FpGModuleGFs
    5.8 Miscellaneous functions
      5.8-1 =
      5.8-2 IsModuleElement
      5.8-3 IsSubModule
      5.8-4 RandomElement
      5.8-5 Random Submodule
  6 FG-module homomorphisms
    6.1 The FpGModuleHomomorphismGF datatype
    6.2 Calculating the kernel of a FG-module homorphism by splitting into two
    homomorphisms
    6.3 Calculating the kernel of a FG-module homorphism by column reduction
    and partitioning
    6.4 Construction functions
      6.4-1 FpGModuleHomomorphismGF construction functions
      6.4-2 Example: Constructing a FpGModuleHomomorphismGF
    6.5 Data access functions
      6.5-1 SourceModule
      6.5-2 TargetModule
      6.5-3 ModuleHomomorphismGeneratorMatrix
      6.5-4 DisplayBlocks
      6.5-5 DisplayModuleHomomorphismGeneratorMatrix
      6.5-6 DisplayModuleHomomorphismGeneratorMatrixBlocks
      6.5-7 Example: Accessing data about a FpGModuleHomomorphismGF
    6.6 Image and kernel functions
      6.6-1 ImageOfModuleHomomorphism
      6.6-2 PreImageRepresentativeOfModuleHomomorphism
      6.6-3 KernelOfModuleHomomorphism
      6.6-4 Example: Kernel and Image of a FpGModuleHomomorphismGF
  7 Ring homomorphisms
    7.1 The HAPRingHomomorphism datatype
      7.1-1 Implementation details
      7.1-2 Elimination orderings
    7.2 Construction functions
      7.2-1 HAPRingToSubringHomomorphism
      7.2-2 HAPSubringToRingHomomorphism
      7.2-3 HAPRingHomomorphismByIndeterminateMap
      7.2-4 HAPRingReductionHomomorphism
      7.2-5 PartialCompositionRingHomomorphism
      7.2-6 HAPZeroRingHomomorphism
      7.2-7 InverseRingHomomorphism
      7.2-8 CompositionRingHomomorphism
    7.3 Data access functions
      7.3-1 SourceGenerators
      7.3-2 SourceRelations
      7.3-3 SourcePolynomialRing
      7.3-4 ImageGenerators
      7.3-5 ImageRelations
      7.3-6 ImagePolynomialRing
    7.4 General functions
      7.4-1 ImageOfRingHomomorphism
      7.4-2 PreimageOfRingHomomorphism
    7.5 Example: Constructing and using a HAPRingHomomorphism
  8 Derivations
    8.1 The HAPDerivation datatype
    8.2 Computing the kernel and homology of a derivation
    8.3 Construction function
      8.3-1 HAPDerivation construction functions
    8.4 Data access function
      8.4-1 DerivationRing
      8.4-2 DerivationImages
      8.4-3 DerivationRelations
      8.4-4 Example: Constructing and accessing data of a HAPDerivation
    8.5 Image, kernel and homology functions
      8.5-1 ImageOfDerivation
      8.5-2 KernelOfDerivation
      8.5-3 HomologyOfDerivation
      8.5-4 Example: Homology of a HAPDerivation
  9 Poincaré series
    9.1 Computing the Poincaré series using spectral sequences
    9.2 Computing the Poincaré series using a minimal resolution
      9.2-1 PoincareSeriesAutoMem
    9.3 Example Poincaré series computations
    9.4 The Poincaré series of groups of order 64 and 128
  10 General Functions
    10.1 Matrices
      10.1-1 SumIntersectionMatDestructive
      10.1-2 SolutionMat
      10.1-3 IsSameSubspace
      10.1-4 PrintDimensionsMat
      10.1-5 Example: matrices and vector spaces
    10.2 Polynomials
      10.2-1 TermsOfPolynomial
      10.2-2 IsMonomial
      10.2-3 UnivariateMonomialsOfMonomial
      10.2-4 IndeterminateAndExponentOfUnivariateMonomial
      10.2-5 IndeterminatesOfPolynomial
      10.2-6 ReduceIdeal
      10.2-7 ReducedPolynomialRingPresentation
      10.2-8 Example: monomials, polynomials and ring presentations
    10.3 Singular
      10.3-1 SingularSetNormalFormIdeal
      10.3-2 SingularPolynomialNormalForm
      10.3-3 SingularGroebnerBasis
      10.3-4 SingularReducedGroebnerBasis
    10.4 Groups
      10.4-1 HallSeniorNumber
  
  
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