  
  
                              [1X MatricesForHomalg [101X
  
  
                       [1X Matrices for the homalg project [101X
  
  
                                   2022.08-02
  
  
                                 17 August 2022
  
  
                                Mohamed Barakat
  
                             Markus Lange-Hegermann
  
                                 Martin Leuner
  
                                   Vinay Wagh
  
  
  
  Mohamed Barakat
      Email:    [7Xmailto:mohamed.barakat@uni-siegen.de[107X
      Homepage: [7Xhttps://mohamed-barakat.github.io[107X
      Address:  [33X[0;14YWalter-Flex-Str. 3[133X
                [33X[0;14Y57072 Siegen[133X
                [33X[0;14YGermany[133X
  
  
  Markus Lange-Hegermann
      Email:    [7Xmailto:markus.lange-hegermann@hs-owl.de[107X
      Homepage: [7Xhttps://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/[107X
      Address:  [33X[0;14YMarkus Lange-Hegermann[133X
                [33X[0;14YHochschule Ostwestfalen-Lippe[133X
                [33X[0;14YLiebigstraße 87[133X
                [33X[0;14Y32657 Lemgo[133X
                [33X[0;14YGermany[133X
  
  
  Martin Leuner
      Email:    [7Xmailto:leuner@momo.math.rwth-aachen.de[107X
      Homepage: [7Xhttp://wwwb.math.rwth-aachen.de/Mitarbeiter/leuner.php[107X
      Address:  [33X[0;14YMartin Leuner[133X
                [33X[0;14YLehrstuhl B fuer Mathematik, RWTH Aachen[133X
                [33X[0;14YTemplergraben 64[133X
                [33X[0;14Y52062 Aachen[133X
                [33X[0;14YGermany[133X
  
  
  Vinay Wagh
      Email:    [7Xmailto:waghoba@gmail.com[107X
      Homepage: [7Xhttp://www.iitg.ernet.in/vinay.wagh/[107X
      Address:  [33X[0;14YE-102, Department of Mathematics,[133X
                [33X[0;14YIndian Institute of Technology Guwahati,[133X
                [33X[0;14YGuwahati, Assam, India.[133X
                [33X[0;14YPIN: 781 039.[133X
                [33X[0;14YIndia[133X
  
  
  
  -------------------------------------------------------
  
  
  [1XContents (MatricesForHomalg)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YWhat is the role of the [5XMatricesForHomalg[105X package in the [5Xhomalg[105X
    project?[133X
      1.1-1 [33X[0;0Y[5XMatricesForHomalg[105X provides ...[133X
      1.1-2 [33X[0;0Y[5Xhomalg[105X delegates ...[133X
      1.1-3 [33X[0;0YThe black box concept[133X
    1.2 [33X[0;0YThis manual[133X
  2 [33X[0;0YInstallation of the [5XMatricesForHomalg[105X Package[133X
  3 [33X[0;0YRings[133X
    3.1 [33X[0;0YRings: Category and Representations[133X
      3.1-1 IsHomalgRing
      3.1-2 IsPreHomalgRing
      3.1-3 IsHomalgRingElement
      3.1-4 IsHomalgInternalRingRep
    3.2 [33X[0;0YRings: Constructors[133X
      3.2-1 HomalgRingOfIntegers
      3.2-2 \/
    3.3 [33X[0;0YRings: Properties[133X
      3.3-1 IsZero
      3.3-2 IsNonZeroRing
      3.3-3 ContainsAField
      3.3-4 IsRationalsForHomalg
      3.3-5 IsFieldForHomalg
      3.3-6 IsDivisionRingForHomalg
      3.3-7 IsIntegersForHomalg
      3.3-8 IsResidueClassRingOfTheIntegers
      3.3-9 IsBezoutRing
      3.3-10 IsIntegrallyClosedDomain
      3.3-11 IsUniqueFactorizationDomain
      3.3-12 IsKaplanskyHermite
      3.3-13 IsDedekindDomain
      3.3-14 IsDiscreteValuationRing
      3.3-15 IsFreePolynomialRing
      3.3-16 IsWeylRing
      3.3-17 IsLocalizedWeylRing
      3.3-18 IsGlobalDimensionFinite
      3.3-19 IsLeftGlobalDimensionFinite
      3.3-20 IsRightGlobalDimensionFinite
      3.3-21 HasInvariantBasisProperty
      3.3-22 IsLocal
      3.3-23 IsSemiLocalRing
      3.3-24 IsIntegralDomain
      3.3-25 IsHereditary
      3.3-26 IsLeftHereditary
      3.3-27 IsRightHereditary
      3.3-28 IsHermite
      3.3-29 IsLeftHermite
      3.3-30 IsRightHermite
      3.3-31 IsNoetherian
      3.3-32 IsLeftNoetherian
      3.3-33 IsRightNoetherian
      3.3-34 IsCohenMacaulay
      3.3-35 IsGorenstein
      3.3-36 IsKoszul
      3.3-37 IsArtinian
      3.3-38 IsLeftArtinian
      3.3-39 IsRightArtinian
      3.3-40 IsOreDomain
      3.3-41 IsLeftOreDomain
      3.3-42 IsRightOreDomain
      3.3-43 IsPrincipalIdealRing
      3.3-44 IsLeftPrincipalIdealRing
      3.3-45 IsRightPrincipalIdealRing
      3.3-46 IsRegular
      3.3-47 IsFiniteFreePresentationRing
      3.3-48 IsLeftFiniteFreePresentationRing
      3.3-49 IsRightFiniteFreePresentationRing
      3.3-50 IsSimpleRing
      3.3-51 IsSemiSimpleRing
      3.3-52 IsSuperCommutative
      3.3-53 BasisAlgorithmRespectsPrincipalIdeals
      3.3-54 AreUnitsCentral
      3.3-55 IsMinusOne
      3.3-56 IsMonic
      3.3-57 IsMonicUptoUnit
      3.3-58 IsLeftRegular
      3.3-59 IsRightRegular
      3.3-60 IsRegular
    3.4 [33X[0;0YRings: Attributes[133X
      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 [33X[0;0YRings: Operations and Functions[133X
  4 [33X[0;0YRing Maps[133X
    4.1 [33X[0;0YRing Maps: Category and Representations[133X
      4.1-1 IsHomalgRingMap
      4.1-2 IsHomalgRingSelfMap
      4.1-3 IsHomalgRingMapRep
    4.2 [33X[0;0YRing Maps: Constructors[133X
      4.2-1 RingMap
    4.3 [33X[0;0YRing Maps: Properties[133X
      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 [33X[0;0YRing Maps: Attributes[133X
      4.4-1 Source
      4.4-2 Range
      4.4-3 DegreeOfMorphism
      4.4-4 CoordinateRingOfGraph
    4.5 [33X[0;0YRing Maps: Operations and Functions[133X
  5 [33X[0;0YMatrices[133X
    5.1 [33X[0;0YMatrices: Category and Representations[133X
      5.1-1 IsHomalgMatrix
      5.1-2 IsHomalgInternalMatrixRep
    5.2 [33X[0;0YMatrices: Constructors[133X
      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 HomalgMatrixListList
      5.2-8 HomalgDiagonalMatrix
      5.2-9 \*
      5.2-10 CoercedMatrix
    5.3 [33X[0;0YMatrices: Properties[133X
      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 [33X[0;0YMatrices: Attributes[133X
      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 [33X[0;0YMatrices: Operations and Functions[133X
      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 TransposedMatrix
      5.5-8 CertainRows
      5.5-9 CertainColumns
      5.5-10 UnionOfRows
      5.5-11 UnionOfColumns
      5.5-12 DiagMat
      5.5-13 KroneckerMat
      5.5-14 DualKroneckerMat
      5.5-15 \*
      5.5-16 \+
      5.5-17 \-
      5.5-18 \*
      5.5-19 \=
      5.5-20 GetColumnIndependentUnitPositions
      5.5-21 GetRowIndependentUnitPositions
      5.5-22 GetUnitPosition
      5.5-23 Eliminate
      5.5-24 BasisOfRowModule
      5.5-25 BasisOfColumnModule
      5.5-26 DecideZeroRows
      5.5-27 DecideZeroColumns
      5.5-28 SyzygiesGeneratorsOfRows
      5.5-29 SyzygiesGeneratorsOfColumns
      5.5-30 SyzygiesGeneratorsOfRows
      5.5-31 SyzygiesGeneratorsOfColumns
      5.5-32 ReducedBasisOfRowModule
      5.5-33 ReducedBasisOfColumnModule
      5.5-34 ReducedSyzygiesGeneratorsOfRows
      5.5-35 ReducedSyzygiesGeneratorsOfColumns
      5.5-36 BasisOfRowsCoeff
      5.5-37 BasisOfColumnsCoeff
      5.5-38 DecideZeroRowsEffectively
      5.5-39 DecideZeroColumnsEffectively
      5.5-40 BasisOfRows
      5.5-41 BasisOfColumns
      5.5-42 DecideZero
      5.5-43 SyzygiesOfRows
      5.5-44 SyzygiesOfColumns
      5.5-45 ReducedSyzygiesOfRows
      5.5-46 ReducedSyzygiesOfColumns
      5.5-47 RightDivide
      5.5-48 LeftDivide
      5.5-49 RightDivide
      5.5-50 LeftDivide
      5.5-51 GenerateSameRowModule
      5.5-52 GenerateSameColumnModule
      5.5-53 SimplifyHomalgMatrixByLeftAndRightMultiplicationWithInvertibleMatrices
      5.5-54 SimplifyHomalgMatrixByLeftMultiplicationWithInvertibleMatrix
      5.5-55 SimplifyHomalgMatrixByRightMultiplicationWithInvertibleMatrix
      5.5-56 CoefficientsWithGivenMonomials
  6 [33X[0;0YRing Relations[133X
    6.1 [33X[0;0YRing Relations: Categories and Representations[133X
      6.1-1 IsHomalgRingRelations
      6.1-2 IsHomalgRingRelationsAsGeneratorsOfLeftIdeal
      6.1-3 IsHomalgRingRelationsAsGeneratorsOfRightIdeal
      6.1-4 IsRingRelationsRep
    6.2 [33X[0;0YRing Relations: Constructors[133X
    6.3 [33X[0;0YRing Relations: Properties[133X
      6.3-1 CanBeUsedToDecideZero
      6.3-2 IsInjectivePresentation
    6.4 [33X[0;0YRing Relations: Attributes[133X
    6.5 [33X[0;0YRing Relations: Operations and Functions[133X
  A [33X[0;0YThe Basic Matrix Operations[133X
    A.1 [33X[0;0YMain[133X
    A.2 [33X[0;0YEffective[133X
    A.3 [33X[0;0YRelative[133X
    A.4 [33X[0;0YReduced[133X
  B [33X[0;0YThe Matrix Tool Operations[133X
    B.1 [33X[0;0YThe Tool Operations [13Xwithout[113X a Fallback Method[133X
      B.1-1 InitialMatrix
      B.1-2 InitialIdentityMatrix
      B.1-3 ZeroMatrix
      B.1-4 IdentityMatrix
      B.1-5 Involution
      B.1-6 TransposedMatrix
      B.1-7 CertainRows
      B.1-8 CertainColumns
      B.1-9 UnionOfRows
      B.1-10 UnionOfRowsPair
      B.1-11 UnionOfColumns
      B.1-12 UnionOfColumnsPair
      B.1-13 DiagMat
      B.1-14 KroneckerMat
      B.1-15 DualKroneckerMat
      B.1-16 MulMat
      B.1-17 AddMat
      B.1-18 SubMat
      B.1-19 Compose
      B.1-20 IsZeroMatrix
      B.1-21 NrRows
      B.1-22 NrColumns
      B.1-23 Determinant
      B.1-24 CoefficientsWithGivenMonomials
    B.2 [33X[0;0YThe Tool Operations with a Fallback Method[133X
      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 [33X[0;0YLogic Subpackages[133X
    C.1 [33X[0;0Y[5XLIRNG[105X: Logical Implications for Rings[133X
    C.2 [33X[0;0Y[5XLIMAP[105X: Logical Implications for Ring Maps[133X
    C.3 [33X[0;0Y[5XLIMAT[105X: Logical Implications for Matrices[133X
    C.4 [33X[0;0Y[5XCOLEM[105X: Clever Operations for Lazy Evaluated Matrices[133X
      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
      C.4-18 Eval
      C.4-19 Eval
      C.4-20 Eval
      C.4-21 Eval
  D [33X[0;0YThe subpackage [5XResidueClassRingForHomalg[105X as a sample ring package[133X
    D.1 [33X[0;0YThe Mandatory Basic Operations[133X
      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 [33X[0;0YThe Mandatory Tool Operations[133X
      D.2-1 InitialMatrix
      D.2-2 InitialIdentityMatrix
      D.2-3 ZeroMatrix
      D.2-4 IdentityMatrix
      D.2-5 Involution
      D.2-6 TransposedMatrix
      D.2-7 CertainRows
      D.2-8 CertainColumns
      D.2-9 UnionOfRows
      D.2-10 UnionOfColumns
      D.2-11 DiagMat
      D.2-12 KroneckerMat
      D.2-13 DualKroneckerMat
      D.2-14 MulMat
      D.2-15 AddMat
      D.2-16 SubMat
      D.2-17 Compose
      D.2-18 IsZeroMatrix
      D.2-19 NrRows
      D.2-20 NrColumns
      D.2-21 Determinant
    D.3 [33X[0;0YSome of the Recommended Tool Operations[133X
      D.3-1 AreEqualMatrices
      D.3-2 IsOne
      D.3-3 IsDiagonalMatrix
      D.3-4 ZeroRows
      D.3-5 ZeroColumns
  E [33X[0;0YDebugging [5XMatricesForHomalg[105X[133X
    E.1 [33X[0;0YIncrease the assertion level[133X
    E.2 [33X[0;0Y[10XUsing homalgMode[110X[133X
      E.2-1 homalgMode
  F [33X[0;0YOverview of the [5XMatricesForHomalg[105X Package Source Code[133X
    F.1 [33X[0;0YRings, Ring Maps, Matrices, Ring Relations[133X
    F.2 [33X[0;0YThe Low Level Algorithms[133X
    F.3 [33X[0;0YLogical Implications for [5XMatricesForHomalg[105X Objects[133X
    F.4 [33X[0;0YThe subpackage [5XResidueClassRingForHomalg[105X[133X
    F.5 [33X[0;0YThe homalgTable for [5XGAP4[105X built-in rings[133X
  
  
  [32X
