GAP (MatricesForHomalg)

  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

5.5 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.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.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

MatricesForHomalg

Matrices for the homalg project

Copyright

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