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MatricesForHomalg

Matrices for the homalg project

2024.11-02

16 November 2024

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57072 Siegen
Germany

Markus Lange-Hegermann
Email: markus.lange-hegermann@hs-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

Martin Leuner
Email: leuner@momo.math.rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/Mitarbeiter/leuner.php
Address:
Martin Leuner
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

Vinay Wagh
Email: waghoba@gmail.com
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.
India

Contents

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 \/
 3.3 Rings: Properties

  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 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 HomalgMatrixListList
  5.2-8 HomalgRowVector
  5.2-9 HomalgColumnVector
  5.2-10 HomalgDiagonalMatrix
  5.2-11 \*
  5.2-12 CoercedMatrix
 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 IsScalarMatrix
  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 NumberRows
  5.4-2 NumberColumns
  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 TransposedMatrix
  5.5-8 CertainRows
  5.5-9 CertainColumns
  5.5-10 UnionOfRows
  5.5-11 UnionOfColumns
  5.5-12 ConvertRowToMatrix
  5.5-13 ConvertColumnToMatrix
  5.5-14 ConvertMatrixToRow
  5.5-15 ConvertMatrixToColumn
  5.5-16 DiagMat
  5.5-17 KroneckerMat
  5.5-18 DualKroneckerMat
  5.5-19 \*
  5.5-20 \+
  5.5-21 \-
  5.5-22 \*
  5.5-23 \=
  5.5-24 GetColumnIndependentUnitPositions
  5.5-25 GetRowIndependentUnitPositions
  5.5-26 GetUnitPosition
  5.5-27 Eliminate
  5.5-28 BasisOfRowModule
  5.5-29 BasisOfColumnModule
  5.5-30 DecideZeroRows
  5.5-31 DecideZeroColumns
  5.5-32 SyzygiesGeneratorsOfRows
  5.5-33 SyzygiesGeneratorsOfColumns
  5.5-34 SyzygiesGeneratorsOfRows
  5.5-35 SyzygiesGeneratorsOfColumns
  5.5-36 ReducedBasisOfRowModule
  5.5-37 ReducedBasisOfColumnModule
  5.5-38 ReducedSyzygiesGeneratorsOfRows
  5.5-39 ReducedSyzygiesGeneratorsOfColumns
  5.5-40 BasisOfRowsCoeff
  5.5-41 BasisOfColumnsCoeff
  5.5-42 DecideZeroRowsEffectively
  5.5-43 DecideZeroColumnsEffectively
  5.5-44 BasisOfRows
  5.5-45 BasisOfColumns
  5.5-46 DecideZero
  5.5-47 SyzygiesOfRows
  5.5-48 SyzygiesOfColumns
  5.5-49 ReducedSyzygiesOfRows
  5.5-50 ReducedSyzygiesOfColumns
  5.5-51 RightDivide
  5.5-52 LeftDivide
  5.5-53 RightDivide
  5.5-54 LeftDivide
  5.5-55 SafeRightDivide
  5.5-56 SafeLeftDivide
  5.5-57 UniqueRightDivide
  5.5-58 UniqueLeftDivide
  5.5-59 GenerateSameRowModule
  5.5-60 GenerateSameColumnModule
  5.5-61 SimplifyHomalgMatrixByLeftAndRightMultiplicationWithInvertibleMatrices
  5.5-62 SimplifyHomalgMatrixByLeftMultiplicationWithInvertibleMatrix
  5.5-63 SimplifyHomalgMatrixByRightMultiplicationWithInvertibleMatrix
  5.5-64 CoefficientsWithGivenMonomials
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 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 NumberRows
  B.1-22 NumberColumns
  B.1-23 Determinant
  B.1-24 CoefficientsWithGivenMonomials
 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
  C.4-18 Eval
  C.4-19 Eval
  C.4-20 Eval
  C.4-21 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 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 NumberRows
  D.2-20 NumberColumns
  D.2-21 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
References
Index

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