MatroidGeneration

Generate low-rank matroids

Documentation	Latest Release	Build Status	Code Coverage

This GAP package was used to create the publicly available read-only ArangoDB collection matroids_split_public.

Construct a matroid from the database and find the database key of a matroid

gap> LoadPackage( "MatroidGeneration" );
true
gap> db := AttachMatroidsDatabase();
#I  Connecting to http+ssl://matroid.mathematik.uni-siegen.de:443
[object ArangoDatabase "MatroidsDB"]
gap> sha := "ef53049d834fba1b21f36c365d7f1d46d7fce2f2";;
gap> d := db.matroids_split_public.document( sha );
[ArangoDocument]
gap> matroid := MatroidByCoatomsNC( d );
<A matroid>
gap> coatoms := MinimalListOfCoatoms( matroid );
[ [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ], [ 1, 10, 11, 12 ], [ 2, 6, 10, 13 ],
  [ 2, 7, 11, 14 ], [ 3, 6, 12, 14 ], [ 3, 8, 11, 13 ], [ 4, 9, 10, 14 ],
  [ 4, 7, 13 ], [ 5, 7, 12 ], [ 5, 8, 10 ], [ 5, 9, 11 ], [ 5, 13, 14 ],
  [ 9, 12, 13 ], [ 1, 13 ], [ 1, 14 ], [ 2, 8 ], [ 2, 9 ], [ 2, 12 ],
  [ 3, 7 ], [ 3, 9 ], [ 3, 10 ], [ 4, 6 ], [ 4, 8 ], [ 4, 11 ],
  [ 4, 12 ], [ 5, 6 ], [ 6, 11 ], [ 7, 10 ], [ 8, 12 ], [ 8, 14 ] ]
gap> ShaSum( String( coatoms ) ) = sha;
true

Accessing the collection through the web interface

The collection can be accessed through the publicly available ArangoDB web interface:

	credentials
server	https://matroid.mathematik.uni-siegen.de/
username	matroid
password	matroid
database	MatroidsDB
collection	matroids_split_public

Accessing the collection through GAP

The package can also be used to access the collection directly.

First start gap and make sure you can

gap> LoadPackage( "MatroidGeneration" );
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Loading  ferret 1.0.2 (Backtrack Search in Permutation Groups)
by Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/).
Homepage: https://gap-packages.github.io/ferret/
Report issues at https://github.com/gap-packages/ferret/issues
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Loading  Gauss 2020.06.27 (Extended Gauss functionality for GAP)
by Simon Görtzen (https://www.linkedin.com/in/simongoertzen/).
maintained by:
   Mohamed Barakat (https://mohamed-barakat.github.io) and
   Sebastian Gutsche (https://sebasguts.github.io).
Homepage: https://homalg-project.github.io/homalg_project/Gauss/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  json 2.0.1 (Reading and Writing JSON)
by Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/).
Homepage: https://gap-packages.github.io/json/
Report issues at https://github.com/gap-packages/json/issues
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Loading  ToolsForHomalg 2020.05.12 (Provides special methods and knowledge propagation tools)
by Mohamed Barakat (https://mohamed-barakat.github.io),
   Sebastian Gutsche (https://sebasguts.github.io), and
   Markus Lange-Hegermann (http://wwwb.math.rwth-aachen.de/~markus/).
Homepage: https://homalg-project.github.io/homalg_project/ToolsForHomalg/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  images 1.3.0 (Minimal and Canonical images)
by Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/),
   Markus Pfeiffer (http://www.morphism.de/~markusp/),
   Rebecca Waldecker (http://conway1.mathematik.uni-halle.de/~waldecker/), and
   Eliza Jonauskyte (ej31@st-andrews.ac.uk).
maintained by:
   Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/).
Homepage: https://gap-packages.github.io/images/
Report issues at https://github.com/gap-packages/images/issues
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Loading  MatricesForHomalg 2020.06.30 (Lazy evaluated matrices with clever operations for the homalg project)
by Mohamed Barakat (https://mohamed-barakat.github.io),
   Markus Lange-Hegermann (https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/),
   Martin Leuner (http://wwwb.math.rwth-aachen.de/Mitarbeiter/leuner.php), and
   Vinay Wagh (http://www.iitg.ernet.in/vinay.wagh/).
maintained by:
   Mohamed Barakat (https://mohamed-barakat.github.io).
Homepage: https://homalg-project.github.io/homalg_project/MatricesForHomalg/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  ParallelizedIterators 2019.06.06 (Parallely evaluate recursive iterators)
by Mohamed Barakat (https://mohamed-barakat.github.io),
   Reimer Behrends (https://github.com/rbehrends), and
   Lukas Kühne (https://github.com/lukaskuehne).
Homepage: https://homalg-project.github.io/ParallelizedIterators
Report issues at https://github.com/homalg-project/ParallelizedIterators/issues
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Loading  GaussForHomalg 2020.06.27 (Gauss functionality for the homalg project)
by Simon Görtzen (https://www.linkedin.com/in/simongoertzen/).
maintained by:
   Mohamed Barakat (https://mohamed-barakat.github.io).
Homepage: https://homalg-project.github.io/homalg_project/GaussForHomalg/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  HomalgToCAS 2020.06.27 (A window to the outer world)
by Thomas Bächler (http://wwwb.math.rwth-aachen.de/~thomas/),
   Mohamed Barakat (https://mohamed-barakat.github.io),
   Simon Görtzen (https://www.linkedin.com/in/simongoertzen/),
   Sebastian Gutsche (https://sebasguts.github.io), and
   Vinay Wagh (http://www.iitg.ernet.in/vinay.wagh/).
with contributions by:
   Thomas Breuer (http://www.math.rwth-aachen.de/~Thomas.Breuer/) and
   Frank Lübeck (http://www.math.rwth-aachen.de/~Frank.Luebeck/).
maintained by:
   Mohamed Barakat (https://mohamed-barakat.github.io).
Homepage: https://homalg-project.github.io/homalg_project/HomalgToCAS/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  IO_ForHomalg 2020.04.30 (IO capabilities for the homalg project)
by Thomas Bächler (http://wwwb.math.rwth-aachen.de/~thomas/),
   Mohamed Barakat (https://mohamed-barakat.github.io),
   Sebastian Gutsche (https://algebra.mathematik.uni-siegen.de/gutsche/),
   Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef/), and
   Daniel Robertz (http://wwwb.math.rwth-aachen.de/~daniel/).
maintained by:
   Mohamed Barakat (https://mohamed-barakat.github.io).
Homepage: https://homalg-project.github.io/homalg_project/IO_ForHomalg/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  RingsForHomalg 2020.06.30 (Dictionaries of external rings for the homalg project)
by Mohamed Barakat (https://mohamed-barakat.github.io),
   Simon Görtzen (https://www.linkedin.com/in/simongoertzen/),
   Markus Kirschmer (http://www.math.rwth-aachen.de/~Markus.Kirschmer/),
   Markus Lange-Hegermann (https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/),
   Oleksandr Motsak (http://www.mathematik.uni-kl.de/~motsak/),
   Daniel Robertz (http://wwwb.math.rwth-aachen.de/~daniel/),
   Hans Schönemann (http://www.mathematik.uni-kl.de/~hannes/),
   Andreas Steenpaß (steenpass@mathematik.uni-kl.de), and
   Vinay Wagh (http://www.iitg.ernet.in/vinay.wagh/).
with contributions by:
   Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef/).
maintained by:
   Mohamed Barakat (https://mohamed-barakat.github.io).
Homepage: https://homalg-project.github.io/homalg_project/RingsForHomalg/
Report issues at https://github.com/homalg-project/homalg_project/issues
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Loading  ArangoDBInterface 2020.04.16 (An interface to ArangoDB)
by Mohamed Barakat (https://mohamed-barakat.github.io/).
Homepage: https://homalg-project.github.io/ArangoDBInterface
Report issues at https://github.com/homalg-project/ArangoDBInterface/issues
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Loading  alcove 2019-09-01 (alcove - Objects in Algebraic Combinatorics)
by Martin Leuner (http://wwwb.math.rwth-aachen.de/~leuner/).
Homepage: https://github.com/martin-leuner/alcove
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Loading  MatroidGeneration 2019.06.16 (Generate low-rank matroids)
by Mohamed Barakat (https://mohamed-barakat.github.io/) and
   Lukas Kühne (https://github.com/lukaskuehne).
Homepage: https://homalg-project.github.io/MatroidGeneration
Report issues at https://github.com/homalg-project/MatroidGeneration/issues
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true

MatroidGeneration recursively depends on the above packages, which can all be downloaded from https://github.com/homalg-project/.

Then attach the database MatroidsDB:

gap> db := AttachMatroidsDatabase();
#I  Connecting to http+ssl://matroid.mathematik.uni-siegen.de:443
================================================================

                                       _
  __ _ _ __ __ _ _ __   __ _  ___  ___| |__
 / _` | '__/ _` | '_ \ / _` |/ _ \/ __| '_ \
| (_| | | | (_| | | | | (_| | (_) \__ \ | | |
 \__,_|_|  \__,_|_| |_|\__, |\___/|___/_| |_|
                       |___/

arangosh (ArangoDB 3.4.5 [darwin] 64bit, using jemalloc, build , VPack 0.1.33, RocksDB 5.16.0, ICU 58.1, V8 5.7.492.77, OpenSSL 1.0.2r  26 Feb 2019)
Copyright (c) ArangoDB GmbH

Command-line history will be persisted when the shell is exited.
Connected to ArangoDB 'http+ssl://matroid.mathematik.uni-siegen.de:443' version: 3.3.19 [UNKNOWN, server], database: 'MatroidsDB', username: 'matroid'

Type 'tutorial' for a tutorial or 'help' to see common examples

================================================================
[object ArangoDatabase "MatroidsDB"]
gap> ?QueryDatabase

Content of the collection

The database collection matroids_split_public contains the values of the following keys:

Rank
NumberOfAtoms
Characteristic
Exponents
AutomorphismGroup
AutomorphismGroupSize
AutomorphismGroupSizeFactors
CharacteristicSet
CoordinateRingOfModuliSpace
DimensionOverZ
DressWenzelCondition
InequalitiesOfModuliSpace
IsBalanced
IsCharPolyIntegrallySplitting
IsDivisionallyFree
IsEssentiallyUniquelyRepresentableOverZ
IsFactored
IsInductivelyFree
IsRepresentable
IsSimplicial
IsSupersolvable
IsTutteUnique
IsUniquelyRepresentableOverZ
IsStronglyBalanced
ListOfCoatoms
MatrixOfVectorMatroid
MultiplicityVector
TuttePolynomial

Verifying Terao's freeness conjecture for rank 3 matroids with 14 atoms

The first intended use case is to verify Terao's freeness conjecture for rank 3 matroids with 14 atoms. This can be either done by a query in the web interface or in GAP.

For details see our paper "On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture".

in the web interface using filters

After choosing the collection COLLECTION > matroids_split_public press filter button and enter the following filter values (which will be combined with the AND connective)

Rank == 3
NumberOfAtoms == 14
IsRepresentable == true
IsInductivelyFree == false
IsStronglyBalanced == true
IsUniquelyRepresentableOverZ == false

and then press ENTER. The result will be No documents.

in the web interface using a query

There should be a predefined query QUERIES > Queries > Check_Terao_14 which upon execution returns an empty list. Here is its content, also in case it was accidentally deleted or modified by another user:

FOR d IN matroids_split_public
  FILTER d.Rank == 3 &&
         d.NumberOfAtoms == 14 &&
         d.IsRepresentable == true &&
         d.IsInductivelyFree == false &&
         d.IsStronglyBalanced == true &&
         d.IsUniquelyRepresentableOverZ == false
  SORT d.NumberOfAtoms, d.Characteristic, d.DimensionOverZ DESC
  RETURN { NumberOfAtoms : d.NumberOfAtoms,
           Characteristic: d.Characteristic,
           DimensionOverZ : d.DimensionOverZ,
           CoordinateRingOfModuliSpace : d.CoordinateRingOfModuliSpace,
           InequalitiesOfModuliSpace: d.InequalitiesOfModuliSpace,
           Exponents: d.Exponents,
           MultiplicityVector : d.MultiplicityVector,
          _key : d._key }

in GAP

gap> LoadPackage( "MatroidGeneration" );
true
gap> db := AttachMatroidsDatabase();
#I  Connecting to http+ssl://matroid.mathematik.uni-siegen.de:443
[object ArangoDatabase "MatroidsDB"]
gap> q := QueryDatabase( rec(
>      	  Rank := 3,
>      	  NumberOfAtoms := 14,
>      	  IsRepresentable := true,
>	  IsInductivelyFree := false,
>	  IsStronglyBalanced := true,
>	  IsUniquelyRepresentableOverZ := false ),
> db.matroids_split_public );
[ArangoQueryCursor in [object ArangoDatabase "MatroidsDB"]]
gap> q.count();
0

Dependencies

To obtain current versions of all dependencies, git clone (or git pull to update) the following repositories:

	Repository	git URL
1.	homalg_project	https://github.com/homalg-project/homalg_project.git
2.	ArangoDBInterface	https://github.com/homalg-project/ArangoDBInterface.git
3.	ParallelizedIterators	https://github.com/homalg-project/ParallelizedIterators.git
4.	alcove	https://github.com/homalg-project/alcove.git

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README.md

README.md

MatroidGeneration

Generate low-rank matroids

Table of contents

Construct a matroid from the database and find the database key of a matroid

Accessing the collection through the web interface

Accessing the collection through GAP

Content of the collection

Verifying Terao's freeness conjecture for rank 3 matroids with 14 atoms

in the web interface using filters

in the web interface using a query

in GAP

Dependencies

Files

README.md

Latest commit

History

README.md

File metadata and controls

MatroidGeneration

Generate low-rank matroids

Table of contents

Construct a matroid from the database and find the database key of a matroid

Accessing the collection through the web interface

Accessing the collection through GAP

Content of the collection

Verifying Terao's freeness conjecture for rank 3 matroids with 14 atoms

in the web interface using filters

in the web interface using a query

in GAP

Dependencies