chap0.txt

  
  
                               MatricesForHomalg
  
  
                        Matrices for the homalg project
  
  
                               Version 2018.01.07
  
  
                                   June 2015
  
  
                                Mohamed Barakat
  
                             Markus Lange-Hegermann
  
                                   Vinay Wagh
  
  
  
            (this manual is still under construction)
            This  manual  is  best  viewed  as an HTML document. The
            latest version is available online at:
            http://homalg.math.rwth-aachen.de/~barakat/homalg-project/MatricesForHomalg/chap0.html
            An   offline   version   should   be   included  in  the
            documentation  subfolder of the package. This package is
            part of the homalg-project:
            http://homalg.math.rwth-aachen.de/index.php/core-packages/matricesforhomalg
  
  
  
  Mohamed Barakat
      Email:    mailto:[email protected]
      Homepage: http://www.mathematik.uni-kl.de/~barakat/
      Address:  Department of Mathematics,
                University of Kaiserslautern,
                67653 Kaiserslautern,
                Germany
  
  
  Markus Lange-Hegermann
      Email:    mailto:[email protected]
      Homepage: http://wwwb.math.rwth-aachen.de/~markus/
      Address:  Lehrstuhl  B  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52056 Aachen, Germany
  
  
  Vinay Wagh
      Email:    mailto:[email protected]
      Homepage: http://www.iitg.ernet.in/vinay.wagh/
      Address:  E-102, Department of Mathematics,
                Indian Institute of Technology Guwahati,
                Guwahati, Assam, India.
                PIN: 781 039.
  
  
  
  -------------------------------------------------------
  Copyright
  © 2007-2015 by Mohamed Barakat and Markus Lange-Hegermann
  
  This  package  may  be distributed under the terms and conditions of the GNU
  Public License Version 2.
  
  
  -------------------------------------------------------
  
  
  Contents (MatricesForHomalg)
  
  1 Introduction
    1.1 What is the role of the MatricesForHomalg package in the homalg
    project?
      1.1-1 MatricesForHomalg provides ...
      1.1-2 homalg delegates ...
      1.1-3 The black box concept
    1.2 This manual
  2 Installation of the MatricesForHomalg Package
  3 Rings
    3.1 Rings: Category and Representations
      3.1-1 IsHomalgRing
      3.1-2 IsPreHomalgRing
      3.1-3 IsHomalgRingElement
      3.1-4 IsHomalgInternalRingRep
    3.2 Rings: Constructors
      3.2-1 HomalgRingOfIntegers
      3.2-2 HomalgFieldOfRationals
      3.2-3 \/
    3.3 Rings: Properties
      3.3-1 IsZero
      3.3-2 ContainsAField
      3.3-3 IsRationalsForHomalg
      3.3-4 IsFieldForHomalg
      3.3-5 IsDivisionRingForHomalg
      3.3-6 IsIntegersForHomalg
      3.3-7 IsResidueClassRingOfTheIntegers
      3.3-8 IsBezoutRing
      3.3-9 IsIntegrallyClosedDomain
      3.3-10 IsUniqueFactorizationDomain
      3.3-11 IsKaplanskyHermite
      3.3-12 IsDedekindDomain
      3.3-13 IsDiscreteValuationRing
      3.3-14 IsFreePolynomialRing
      3.3-15 IsWeylRing
      3.3-16 IsLocalizedWeylRing
      3.3-17 IsGlobalDimensionFinite
      3.3-18 IsLeftGlobalDimensionFinite
      3.3-19 IsRightGlobalDimensionFinite
      3.3-20 HasInvariantBasisProperty
      3.3-21 HasLeftInvariantBasisProperty
      3.3-22 HasRightInvariantBasisProperty
      3.3-23 IsLocal
      3.3-24 IsSemiLocalRing
      3.3-25 IsIntegralDomain
      3.3-26 IsHereditary
      3.3-27 IsLeftHereditary
      3.3-28 IsRightHereditary
      3.3-29 IsHermite
      3.3-30 IsLeftHermite
      3.3-31 IsRightHermite
      3.3-32 IsNoetherian
      3.3-33 IsLeftNoetherian
      3.3-34 IsRightNoetherian
      3.3-35 IsCohenMacaulay
      3.3-36 IsGorenstein
      3.3-37 IsKoszul
      3.3-38 IsArtinian
      3.3-39 IsLeftArtinian
      3.3-40 IsRightArtinian
      3.3-41 IsOreDomain
      3.3-42 IsLeftOreDomain
      3.3-43 IsRightOreDomain
      3.3-44 IsPrincipalIdealRing
      3.3-45 IsLeftPrincipalIdealRing
      3.3-46 IsRightPrincipalIdealRing
      3.3-47 IsRegular
      3.3-48 IsFiniteFreePresentationRing
      3.3-49 IsLeftFiniteFreePresentationRing
      3.3-50 IsRightFiniteFreePresentationRing
      3.3-51 IsSimpleRing
      3.3-52 IsSemiSimpleRing
      3.3-53 IsSuperCommutative
      3.3-54 BasisAlgorithmRespectsPrincipalIdeals
      3.3-55 AreUnitsCentral
      3.3-56 IsMinusOne
      3.3-57 IsMonic
      3.3-58 IsMonicUptoUnit
      3.3-59 IsLeftRegular
      3.3-60 IsRightRegular
      3.3-61 IsRegular
    3.4 Rings: Attributes
      3.4-1 Inverse
      3.4-2 homalgTable
      3.4-3 RingElementConstructor
      3.4-4 TypeOfHomalgMatrix
      3.4-5 ConstructorForHomalgMatrices
      3.4-6 Zero
      3.4-7 One
      3.4-8 MinusOne
      3.4-9 ProductOfIndeterminates
      3.4-10 RationalParameters
      3.4-11 IndeterminatesOfPolynomialRing
      3.4-12 RelativeIndeterminatesOfPolynomialRing
      3.4-13 IndeterminateCoordinatesOfRingOfDerivations
      3.4-14 RelativeIndeterminateCoordinatesOfRingOfDerivations
      3.4-15 IndeterminateDerivationsOfRingOfDerivations
      3.4-16 RelativeIndeterminateDerivationsOfRingOfDerivations
      3.4-17 IndeterminateAntiCommutingVariablesOfExteriorRing
      3.4-18 RelativeIndeterminateAntiCommutingVariablesOfExteriorRing
      3.4-19 IndeterminatesOfExteriorRing
      3.4-20 CoefficientsRing
      3.4-21 KrullDimension
      3.4-22 LeftGlobalDimension
      3.4-23 RightGlobalDimension
      3.4-24 GlobalDimension
      3.4-25 GeneralLinearRank
      3.4-26 ElementaryRank
      3.4-27 StableRank
      3.4-28 AssociatedGradedRing
    3.5 Rings: Operations and Functions
  4 Ring Maps
    4.1 Ring Maps: Category and Representations
      4.1-1 IsHomalgRingMap
      4.1-2 IsHomalgRingSelfMap
      4.1-3 IsHomalgRingMapRep
    4.2 Ring Maps: Constructors
      4.2-1 RingMap
    4.3 Ring Maps: Properties
      4.3-1 IsMorphism
      4.3-2 IsIdentityMorphism
      4.3-3 IsMonomorphism
      4.3-4 IsEpimorphism
      4.3-5 IsIsomorphism
      4.3-6 IsAutomorphism
    4.4 Ring Maps: Attributes
      4.4-1 Source
      4.4-2 Range
      4.4-3 DegreeOfMorphism
      4.4-4 CoordinateRingOfGraph
    4.5 Ring Maps: Operations and Functions
  5 Matrices
    5.1 Matrices: Category and Representations
      5.1-1 IsHomalgMatrix
      5.1-2 IsHomalgInternalMatrixRep
    5.2 Matrices: Constructors
      5.2-1 HomalgInitialMatrix
      5.2-2 HomalgInitialIdentityMatrix
      5.2-3 HomalgZeroMatrix
      5.2-4 HomalgIdentityMatrix
      5.2-5 HomalgVoidMatrix
      5.2-6 HomalgMatrix
      5.2-7 HomalgDiagonalMatrix
      5.2-8 \*
    5.3 Matrices: Properties
      5.3-1 IsZero
      5.3-2 IsOne
      5.3-3 IsUnitFree
      5.3-4 IsPermutationMatrix
      5.3-5 IsSpecialSubidentityMatrix
      5.3-6 IsSubidentityMatrix
      5.3-7 IsLeftRegular
      5.3-8 IsRightRegular
      5.3-9 IsInvertibleMatrix
      5.3-10 IsLeftInvertibleMatrix
      5.3-11 IsRightInvertibleMatrix
      5.3-12 IsEmptyMatrix
      5.3-13 IsDiagonalMatrix
      5.3-14 IsScalarlMatrix
      5.3-15 IsUpperTriangularMatrix
      5.3-16 IsLowerTriangularMatrix
      5.3-17 IsStrictUpperTriangularMatrix
      5.3-18 IsStrictLowerTriangularMatrix
      5.3-19 IsUpperStairCaseMatrix
      5.3-20 IsLowerStairCaseMatrix
      5.3-21 IsTriangularMatrix
      5.3-22 IsBasisOfRowsMatrix
      5.3-23 IsBasisOfColumnsMatrix
      5.3-24 IsReducedBasisOfRowsMatrix
      5.3-25 IsReducedBasisOfColumnsMatrix
      5.3-26 IsInitialMatrix
      5.3-27 IsInitialIdentityMatrix
      5.3-28 IsVoidMatrix
    5.4 Matrices: Attributes
      5.4-1 NrRows
      5.4-2 NrColumns
      5.4-3 DeterminantMat
      5.4-4 ZeroRows
      5.4-5 ZeroColumns
      5.4-6 NonZeroRows
      5.4-7 NonZeroColumns
      5.4-8 PositionOfFirstNonZeroEntryPerRow
      5.4-9 PositionOfFirstNonZeroEntryPerColumn
      5.4-10 RowRankOfMatrix
      5.4-11 ColumnRankOfMatrix
      5.4-12 LeftInverse
      5.4-13 RightInverse
      5.4-14 CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries
      5.4-15 CoefficientsOfNumeratorOfHilbertPoincareSeries
      5.4-16 UnreducedNumeratorOfHilbertPoincareSeries
      5.4-17 NumeratorOfHilbertPoincareSeries
      5.4-18 HilbertPoincareSeries
      5.4-19 HilbertPolynomial
      5.4-20 AffineDimension
      5.4-21 AffineDegree
      5.4-22 ProjectiveDegree
      5.4-23 ConstantTermOfHilbertPolynomialn
      5.4-24 MatrixOfSymbols
    5.5 Matrices: Operations and Functions
      5.5-1 HomalgRing
      5.5-2 LeftInverse
      5.5-3 RightInverse
      5.5-4 LeftInverseLazy
      5.5-5 RightInverseLazy
      5.5-6 Involution
      5.5-7 CertainRows
      5.5-8 CertainColumns
      5.5-9 UnionOfRows
      5.5-10 UnionOfColumns
      5.5-11 DiagMat
      5.5-12 KroneckerMat
      5.5-13 \*
      5.5-14 \+
      5.5-15 \-
      5.5-16 \*
      5.5-17 \=
      5.5-18 GetColumnIndependentUnitPositions
      5.5-19 GetRowIndependentUnitPositions
      5.5-20 GetUnitPosition
      5.5-21 Eliminate
      5.5-22 BasisOfRowModule
      5.5-23 BasisOfColumnModule
      5.5-24 DecideZeroRows
      5.5-25 DecideZeroColumns
      5.5-26 SyzygiesGeneratorsOfRows
      5.5-27 SyzygiesGeneratorsOfColumns
      5.5-28 SyzygiesGeneratorsOfRows
      5.5-29 SyzygiesGeneratorsOfColumns
      5.5-30 ReducedBasisOfRowModule
      5.5-31 ReducedBasisOfColumnModule
      5.5-32 ReducedSyzygiesGeneratorsOfRows
      5.5-33 ReducedSyzygiesGeneratorsOfColumns
      5.5-34 BasisOfRowsCoeff
      5.5-35 BasisOfColumnsCoeff
      5.5-36 DecideZeroRowsEffectively
      5.5-37 DecideZeroColumnsEffectively
      5.5-38 BasisOfRows
      5.5-39 BasisOfColumns
      5.5-40 DecideZero
      5.5-41 SyzygiesOfRows
      5.5-42 SyzygiesOfColumns
      5.5-43 ReducedSyzygiesOfRows
      5.5-44 ReducedSyzygiesOfColumns
      5.5-45 RightDivide
      5.5-46 LeftDivide
      5.5-47 RightDivide
      5.5-48 LeftDivide
      5.5-49 GenerateSameRowModule
      5.5-50 GenerateSameColumnModule
  6 Ring Relations
    6.1 Ring Relations: Categories and Representations
      6.1-1 IsHomalgRingRelations
      6.1-2 IsHomalgRingRelationsAsGeneratorsOfLeftIdeal
      6.1-3 IsHomalgRingRelationsAsGeneratorsOfRightIdeal
      6.1-4 IsRingRelationsRep
    6.2 Ring Relations: Constructors
    6.3 Ring Relations: Properties
      6.3-1 CanBeUsedToDecideZero
      6.3-2 IsInjectivePresentation
    6.4 Ring Relations: Attributes
    6.5 Ring Relations: Operations and Functions
  A The Basic Matrix Operations
    A.1 Main
    A.2 Effective
    A.3 Relative
    A.4 Reduced
  B The Matrix Tool Operations
    B.1 The Tool Operations without a Fallback Method
      B.1-1 InitialMatrix
      B.1-2 InitialIdentityMatrix
      B.1-3 ZeroMatrix
      B.1-4 IdentityMatrix
      B.1-5 Involution
      B.1-6 CertainRows
      B.1-7 CertainColumns
      B.1-8 UnionOfRows
      B.1-9 UnionOfColumns
      B.1-10 DiagMat
      B.1-11 KroneckerMat
      B.1-12 MulMat
      B.1-13 AddMat
      B.1-14 SubMat
      B.1-15 Compose
      B.1-16 IsZeroMatrix
      B.1-17 NrRows
      B.1-18 NrColumns
      B.1-19 Determinant
    B.2 The Tool Operations with a Fallback Method
      B.2-1 AreEqualMatrices
      B.2-2 IsIdentityMatrix
      B.2-3 IsDiagonalMatrix
      B.2-4 ZeroRows
      B.2-5 ZeroColumns
      B.2-6 GetColumnIndependentUnitPositions
      B.2-7 GetRowIndependentUnitPositions
      B.2-8 GetUnitPosition
      B.2-9 PositionOfFirstNonZeroEntryPerRow
      B.2-10 PositionOfFirstNonZeroEntryPerColumn
  C Logic Subpackages
    C.1 LIRNG: Logical Implications for Rings
    C.2 LIMAP: Logical Implications for Ring Maps
    C.3 LIMAT: Logical Implications for Matrices
    C.4 COLEM: Clever Operations for Lazy Evaluated Matrices
      C.4-1 Eval
      C.4-2 Eval
      C.4-3 Eval
      C.4-4 Eval
      C.4-5 Eval
      C.4-6 Eval
      C.4-7 Eval
      C.4-8 Eval
      C.4-9 Eval
      C.4-10 Eval
      C.4-11 Eval
      C.4-12 Eval
      C.4-13 Eval
      C.4-14 Eval
      C.4-15 Eval
      C.4-16 Eval
      C.4-17 Eval
  D The subpackage ResidueClassRingForHomalg as a sample ring package
    D.1 The Mandatory Basic Operations
      D.1-1 BasisOfRowModule
      D.1-2 BasisOfColumnModule
      D.1-3 DecideZeroRows
      D.1-4 DecideZeroColumns
      D.1-5 SyzygiesGeneratorsOfRows
      D.1-6 SyzygiesGeneratorsOfColumns
      D.1-7 BasisOfRowsCoeff
      D.1-8 BasisOfColumnsCoeff
      D.1-9 DecideZeroRowsEffectively
      D.1-10 DecideZeroColumnsEffectively
      D.1-11 RelativeSyzygiesGeneratorsOfRows
      D.1-12 RelativeSyzygiesGeneratorsOfColumns
    D.2 The Mandatory Tool Operations
      D.2-1 InitialMatrix
      D.2-2 InitialIdentityMatrix
      D.2-3 ZeroMatrix
      D.2-4 IdentityMatrix
      D.2-5 Involution
      D.2-6 CertainRows
      D.2-7 CertainColumns
      D.2-8 UnionOfRows
      D.2-9 UnionOfColumns
      D.2-10 DiagMat
      D.2-11 KroneckerMat
      D.2-12 MulMat
      D.2-13 AddMat
      D.2-14 SubMat
      D.2-15 Compose
      D.2-16 IsZeroMatrix
      D.2-17 NrRows
      D.2-18 NrColumns
      D.2-19 Determinant
    D.3 Some of the Recommended Tool Operations
      D.3-1 AreEqualMatrices
      D.3-2 IsOne
      D.3-3 IsDiagonalMatrix
      D.3-4 ZeroRows
      D.3-5 ZeroColumns
  E Debugging MatricesForHomalg
    E.1 Increase the assertion level
    E.2 Using homalgMode
      E.2-1 homalgMode
  F Overview of the MatricesForHomalg Package Source Code
    F.1 Rings, Ring Maps, Matrices, Ring Relations
    F.2 The Low Level Algorithms
    F.3 Logical Implications for MatricesForHomalg Objects
    F.4 The subpackage ResidueClassRingForHomalg
    F.5 The homalgTable for GAP4 built-in rings