LOOPS: Computing with quasigroups and loops in GAP

Gábor P. Nagy (University of Szeged, Hungary)
Petr Vojtěchovský (University of Denver, USA)

This is the official distribution webpage of the LOOPS package for GAP 4. You will find here:

• brief description of the package
• installation instructions
• the most recent version of the package for download
• detailed documentation
• new features not found in previous versions
• known bugs of the most recent version
• older versions of the package
• acknowledgement

Note: The package has been accepted by GAP in May 2015 and is available with the standard distribution of GAP.

Brief description

LOOPS is a package for GAP 4 whose purpose is to provide researchers in nonassociative algebra with a powerful computational tool concerning finite loops and quasigroups; and to extend GAP toward the realm of nonassociative structures.

The package consists of three complementary parts: core methods for quasigroups and loops, specific methods for loops (mainly Moufang loops), libraries of small loops.

Installation instructions

To install LOOPS, you should have GAP version 4.8 or newer installed on your computer. Then

1. download any of the available versions (Windows, Unix) of LOOPS
2. unpack the file into the pkg directory of GAP

After this step, you will see the subfolder loops in the pkg folder.

To load LOOPS into GAP, start GAP and use LoadPackage( "loops" );.

Download

The most recent version of LOOPS is 3.4.0.

Before you download LOOPS, you might want to check the README.loops and PackageInfo.g files, and the documentation.

• Unix version: loops-3.4.0.tar.gz
• Windows version: loops-3.4.0-win.zip

Documentation

The documentation for the package is included with the distribution and is located in the loops/doc folder. You can also access the documentation directly from here.

• pdf (pdf version)
• ps (postscript version)
• html (html version)

New features

In version 3.4.0

• Expanded library of CC loops (orders 32, 64, 81, 125, 343).
• Added library of left Bruck and right Bruck loops (orders 27, 81).
• Redesigned library of commutative automorphic loops.
• Improved IsotopismLoops.
• Added QuasigroupIsomorph, LoopIsomorph in analogy to LoopIsotope.
• Renamed IsLatin to IsLatinMagma to avoid conflicts with other packages.
• Added filters for automorphic loops.
• Expanded displayed view of loops to include left Bruck, right Bruck, left automorphic, left automorphic, automorphic and commutative loops.
• Updated testing (switched from ReadTest to Test).

In version 3.3.0

• Added SmallGeneratingSet( Q ) for a quasigroup Q. It returns a small generating set of Q and it is used in isomorphism searches.
• New version of Discriminator. It is a bit slower but generally speeds up isomorphism searches because it is more powerful than the old version.
• Improved auxiliary routines for packing of multiplication tables (commutativity is exploited), and added auxiliary routines for packing of cocycles. Consequently, some library files are smaller.
• Added a library of commutative automorphic loops of order 3, 9, 27, 81 and 243 (except for certain factors for 243 as described in the manual).
• Changed how Subloop( Q, S ) and Subquasigroup( Q, S ) work. It is now safe to call Subquasigroup( Q ) in all cases; a loop will be returned iff the argument is a loop.
• IsPowerAssociative( Q ) and IsDiassociative( Q ) now work for quasigroups, too. The methods are much faster.

In version 3.2.0:

• Added CanonicalCopy, which returns an isomorphic copy of a quasigroup or a loop with a Cayley table based on 1, ..., |Q|.
• Documentation and test files updated.
• Fixed a bug in InterestingLoop(96,1).
• Much more support for quasigroups, namely:
  • Discriminator (which is used of isomorphism searches) has been updated and now works for quasigroups as well.
  • Added IsomorphismQuasigroups( Q1, Q2 ).
  • Added QuasigroupsUpToIsomorphism( ls ).
  • Added AllSubquasigroups( Q ) that returns a list of all (nonempty) subquasigroups of a quasigroup.
  • AutomorphismGroup( Q ) now works for quasigroups as well.
  • DirectProduct now works for a list of quasigroups, loops and/or groups.

In version 3.1.0:

• Completely new documentation, prepared with GAPDoc.
• Added left Bol loops and right Bol loops of order pq, for odd primes p>q.
• Changed Opposite from global function to an attribute, and added operations OppositeLoop, OppositeQuasigroup, to avoid a conflict with other GAP packages.

Click to see the log of older changes.

Known bugs in the most recent version

There are issues when both the semigroup and loops packages are loaded. In the semigroup package, semigroups are declared as DeclareSynonymAttr( "IsSemigroup", IsMagma and IsAssociative ). An associative quasigroup/loop satisfies both IsMagma and IsAssociative and it might therefore be treated as a semigroup in GAP. Unfortunately, some methods for semigroups do not anticipate loops and their representation and they might therefore crash. The semigroup package actually causes problems in GAP even without loops loaded.

RightCosets(Q,S) does not return all right cosets but only a selection of right cosets that cover Q.

Older versions of the package

Since several published papers rely on older versions of the package, we post them here so that calculations can be verified.

For Unix: version 1.0.0, 1.1.0, 1.2.0, 1.3.0, 1.4.0, 1.5.0, 1.9.0, 2.0.0, 2.1.0, 2.1.1, 2.1.2, 2.1.3, 2.2.0, 3.0.0, 3.1.0, 3.2.0, 3.3.0.

For Windows: version 1.0.0, 1.1.0, 1.2.0, 1.3.0, 1.4.0, 1.5.0, 1.9.0, 2.0.0, 2.1.0, 2.1.1, 2.1.2, 2.1.3, 2.2.0, 3.0.0, 3.1.0, 3.2.0, 3.3.0.

Acknowledgement

G. P. Nagy was supported by OTKA grants F042959 and T043758, and P. Vojtechovsky was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.

We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Ales Drapal, Graham Ellis, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck, Jonathan D.H. Smith, David Stanovsky and Glen Whitney.

The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of right Bruck loops of order 27, 81 was obtained jointly with Izabella Stuhl.

last update: October 27, 2017