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MatricesForHomalg

Matrices for the homalg project

Version 2020.10-04

June 2015

(this manual is still under construction)

This manual is best viewed as an HTML document. The latest version is available online at:

https://homalg-project.github.io/homalg_project/MatricesForHomalg/doc/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: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Department of Mathematics,
University of Siegen,
57072 Siegen,
Germany

Markus Lange-Hegermann
Email: markus.lange.hegermann@rwth-aachen.de
Homepage: http://wwwb.math.rwth-aachen.de/~markus/
Address:
Lehrstuhl B für Mathematik, RWTH Aachen, Templergraben 64, 52056 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.

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 or (at your option) any later version.

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 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 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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