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MatricesForHomalg

Matrices for the homalg project

2022.12-01

1 December 2022

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
Liebigstra├če 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
2 Installation of the MatricesForHomalg Package
3 Rings
 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
4 Ring Maps
5 Matrices
 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 GenerateSameRowModule

  5.5-56 GenerateSameColumnModule

  5.5-57 SimplifyHomalgMatrixByLeftAndRightMultiplicationWithInvertibleMatrices

  5.5-58 SimplifyHomalgMatrixByLeftMultiplicationWithInvertibleMatrix

  5.5-59 SimplifyHomalgMatrixByRightMultiplicationWithInvertibleMatrix

  5.5-60 CoefficientsWithGivenMonomials
6 Ring Relations
A The Basic Matrix Operations
B The Matrix Tool Operations
C Logic Subpackages
D The subpackage ResidueClassRingForHomalg as a sample ring package
E Debugging MatricesForHomalg
F Overview of the MatricesForHomalg Package Source Code
References
Index

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