# Semigroups

A package for semigroups and monoids

Version 3.0.8

This project is maintained by J. D. Mitchell

### Version 3.0.8 (released 10/11/2017)

This version contains some minor bugfixes, and updates for compatibility GAP 4.9 and Orb 4.8.0.

### Version 3.0.7 (released 02/10/2017)

This version contains some minor bugfixes, fixes some issues where some tests in the main GAP repo returned different output when Semigroups was loaded than when it was not, and updates the kernel module for version 0.5.2 of libsemigroups. The configuration option --enable-debug was added.

The following issues are resolved:

• Issue 389: the most general method for NaturalPartialOrder sometimes returned incorrect results. [Wilf A. Wilson]

• Issue 393: StructureDescription for finitely presented groups failed with an error when Semigroups was loaded. [J. D. Mitchell]

• Issue 395: GAP’s test tst/testinstall/semigrp.tst failed because of a missing method for NrEquivalenceClasses for a generic semigroup congruence. [J. D. Mitchell] and [Wilf A. Wilson]

### Version 3.0.6 (released 27/09/2017)

This version contains some minor bugfixes, improves the compatibility of Semigroups with other GAP packages, and updates the kernel module for version 0.5.0 of libsemigroups

The following issues are resolved:

• Issue 371: the identity element of some types of monoids was not added to its GeneratorsOfSemigroup. This meant that the semigroup generated by GeneratorsOfSemigroup(M) was not equal to M in some rare cases. [Wilf A. Wilson]

• Issue 377: there was a bug in the method for IsInverseSemigroup for non-acting semigroups that sometimes returned a false positive. [Wilf A. Wilson]

### Version 3.0.5 (released 23/08/2017)

This version contains some minor tweaks and the following issue is resolved:

• Issue 352 There was a name clash with some other GAP packages using RandomMatrix and IsTorsion. [J. D. Mitchell]

### Version 3.0.4 (released 16/07/2017)

Some minor issues are fixed in this release:

• Issue 342 DirectProduct for transformation semigroups returned the wrong answer when applied to semigroups satisfying IsMonoidAsSemigroup. [J. D. Mitchell]

Some documentation and tests were added. [Michael Torpey]

### Version 3.0.3 (released 21/06/2017)

Some minor issues are fixed in this release:

• Issue 336 Rees (0-)matrix semigroups over non-permutation groups sometimes resulted in an error. [J. D. Mitchell]

• A method was added for IsEUnitaryInverseSemigroup for non-inverse semigroups, which previously resulted in no method found. [Chris Russell]

• Some error messages were improved in ReadGenerators and WriteGenerators. [J. D. Mitchell]

### Version 3.0.2 (released 16/06/2017)

This is an minor release fixing some minor issues in the last release.

The following issues were resolved:

• Issue 330 InversesOfSemigroupElement some times returned an incorrect value, specifically when applied to the identity of a transformation monoid. [J. D. Mitchell]

• Issue 328 when using Linux the package compiled but failed to link pthreads and so the kernel module failed to load in GAP. [J. D. Mitchell]

There are improvements to the following:

• some missing documentation was added. [Michael Torpey]

• the subsemigroup returned by IdempotentGeneratedSubsemigroup for Rees (0-)matrix semigroup over a group has a smaller generating set than previously, and can be found more quickly. [Wilf A. Wilson]

• IsomorphismSemigroups is extended so that it can be applied to arbitrary simple, 0-simple, or monogenic semigroups. [Wilf A. Wilson]

### Version 3.0.1 (released 03/06/2017)

This is an extremely minor release fixing some minor issues in the last release.

### Version 3.0.0 (released 02/06/2017)

This is a major release that dramatically expands the scope of the package. The package now features a compiled C/C++ module which interfaces with the libsemigroups C++ library to allow high-speed computations for congruences and certain categories of semigroup. There are also several new types of semigroup and a variety of new methods which can be used with them.

### Version 2.8.0 (released 26/05/2016)

In this release there are some new features and some bug fixes. In this version, we welcome Nick Ham to the contributors to the package.

#### New Features in Version 2.8.0

The new features in this release are contributed by Nick Ham:

• ApsisMonoid
• CrossedApsisMonoid
• ModularPartitionMonoid
• PlanarModularPartitionMonoid
• PlanarPartitionMonoid
• PlanarUniformBlockBijectionMonoid
• SingularApsisMonoid
• SingularCrossedApsisMonoid
• SingularModularPartitionMonoid
• SingularPlanarModularPartitionMonoid
• SingularPlanarPartitionMonoid
• SingularPlanarUniformBlockBijectionMonoid
• SingularUniformBlockBijectionMonoid
• UniformBlockBijectionMonoid

#### Issues Resolved in Version 2.8.0

• Issue 160 IrreundantGeneratingSubset behaved incorrectly when given a semigroup whose generating set consisted of a single repeated element. [Wilf A. Wilson]
• Issue 164 MatrixEntries gave an error for Rees 0-matrix semigroups whose matrices contain 0. [Wilf A. Wilson]
• Some tests failed when GAP was compiled in 32-bit mode. [Michael Torpey]

### Version 2.7.6 (released 19/04/2016)

This is a very minor release changing the name of the README (to README.md) in the PackageInfo.g file.

### Version 2.7.5 (released 19/04/2016)

This is a minor release to fix Issue 151, and to make some changes for future compatibility with GAP. In Issue 151 when the method IsomorphismPermGroup was applied to a semigroups of non-permutation transformations the returned mapping was not an isomorphism.

### Version 2.7.4 (released 02/03/2016)

This is a minor release to fix Issue 150, and to correct the required version of GAP (from 4.8.2 to 4.8.3). In Issue 150 the function IsZeroSimpleSemigroup entered an infinite loop for some examples of semigroups of partial permutations.

### Version 2.7.3 (released 15/02/2016)

This is a minor release to fix some manual examples, to correct the package url in the PackageInfo.g file, and to fix some issues with semigroups of bipartitions. It was formerly possible to create semigroups of bipartitions where the generators had different degrees, but the created semigroups were invalid; this is fixed in version 2.7.3.

### Version 2.7.2 (released 28/01/2016)

This is a minor release to fix to remove ErrorMayQuit which was renamed ErrorNoReturn in GAP 4.8.2. This change was made by Max Horn.

### Version 2.7.1 (released 19/12/2015)

This is a minor release to fix Issue 144. This issue resulted in IsInverseSemigroup sometimes returning true for semigroups which were not inverse. This occurred when the $$\mathscr{D}$$-classes of the semigroup were computed before the method for IsInverseSemigroup was first run.

### Version 2.7 (released 27/11/2015)

This is a minor release including some changes for compatibility with GAP 4.8, and some bug fixes.

#### New Features in Version 2.7

• IsomorphismReesZeroMatrixSemigroup is introduced, and it is no longer possible to apply IsomorphismReesMatrixSemigroup to a 0-simple semigroup. This change was made for the sake of consistency, so that the Range of an IsomorphismReesMatrixSemigroup is always a Rees matrix semigroup and not sometimes a Rees 0-matrix semigroup as was formerly the case.

#### Changes for GAP 4.8

• several ViewString methods for semigroups and their elements were moved from the Semigroups package to the GAP library. Some minor changes were made in the method for ViewString for semigroups, and the tests, and manual examples were updated accordingly.
• The meaning of IsMonoidAsSemigroup was changed to be consistent with the meaning of IsGroupAsSemigroup. In earlier versions, IsMonoidAsSemigroup was false for semigroups in the category IsMonoid. From Version 2.7, IsMonoidAsSemigroup is true for monoids in the category IsMonoid and for some further semigroups.

#### Issues Resolved in Version 2.7

• Issue 136: CyclesOfPartialPermSemigroup sometimes resulted in an error due to using DegreeOfPartialPermSemigroup instead of the maximum of the degree and the codegree. [James Mitchell]
• Issue 141: PartialOrderOfDClasses sometimes resulted in an error. This bug was introduced in Semigroups 2.6 and did not effect any previous versions. [James Mitchell]

### Version 2.6 (released 22/09/2015)

This release includes some bugfixes, some minor new features, and one major new feature (efficient methods for semigroups of matrices over a finite field).

#### New Features in Version 2.6

• extensive new features for computing with semigroups, monoids, and ideals, of matrices with entries in a finite field. See Chapter 7 of the manual for more details. [Markus Pfeiffer]

• The functions RectangularBand, MonogenicSemigroup, and ZeroSemigroup now have an optional first argument to specify the category of the result; the functions LeftZeroSemigroup and RightZeroSemigroup are introduced in a similar sense. [Wilf A. Wilson]

• The new property IsSemigroupWithAdjoinedZero and attribute UnderlyingSemigroupOfSemigroupWithAdjoinedZero are introduced. [Wilf A. Wilson]

• The operations MotzkinMonoid and PartialJonesMonoid were introduced. [James Mitchell]

#### Issues Resolved in Version 2.6

• Issue 131: testing membership in a Rees 0-matrix semigroup that knows it is inverse sometimes resulted in an error. [Michael Torpey]

• Issue 132: this was a feature request to introduce the operations MotzkinMonoid and PartialJonesMonoid. [James Mitchell]

• Issue 134: the operation PartialBrauerMonoid returned the wrong answer when the argument was 1. The returned semigroup was not the partial brauer monoid of degree 1. [James Mitchell]

### Version 2.5 (released 01/06/2015)

This is a minor release including several bugfixes, lots of minor improvements in the documentation, some improvements in performance, and some new features.

#### New Features in Version 2.5

• Semigroups of partial permutations now have a polynomial time (quadratic in the degree) algorithm for computing the minimal ideal [Wilf A. Wilson]

• A more efficient IsInverseSemigroup method for Rees 0-matrix semigroups is introduced, along with new methods for Idempotents and NrIdempotents for inverse Rees 0-matrix semigroups [Wilf A. Wilson]

• The documentation for congruences has been improved and new tests have been added. [Michael Torpey]

• A UniversalSemigroupCongruence now returns a much smaller set of generating pairs. [Michael Torpey]

#### Issues Resolved in Version 2.5

Issue numbers refer to the issues on the tracker.

• Issue 126: testing membership in a Rees 0-matrix semigroup that knows it is inverse sometimes resulted in an error. [James Mitchell]

• Issue 127: the main algorithm for computing with ideals of acting semigroups which know they are regular contained a bug that resulted in incorrect results. In some cases, some $$\mathscr{D}$$-classes were counted more than once, and the returned value of Size was higher than the actual size of the ideal. [James Mitchell]

• Issue 128: in some special cases UnderlyingSemigroup, ViewObj, Size, and related methods, for Rees 0-matrix semigroups over non-groups returned an error. [James Mitchell]

• The universal congruence specified by generating pairs on a (0-)simple semigroup no longer causes an error. [Michael Torpey]

### Version 2.4.1 (released 15/05/2015)

This is a extremely minor release to change 1 character in the PackageInfo.g file (wrong package archive URL).

### Version 2.4 (released 02/04/2015)

This is a minor release including several bugfixes, and improvements in performance, and some new features.

#### New Features in Version 2.4

• The function RepresentativeOfMinimalIdeal is introduced. [Wilf Wilson]

• Transformation semigroups now have a polynomial time (cubic in the degree) algorithm for computing the minimal ideal [Wilf A. Wilson]

• The functions RectangularBand, ZeroSemigroup, and MonogenicSemigroup are introduced. [Wilf A. Wilson]

• A method for choosing a random element of a semigroup has been introduced in the case that the semigroup knows its set of elements. This new method choose elements at random with uniform probability. [Wilf A. Wilson]

• The documentation and tests for congruences has been improved. [Michael Torpey and Wilf A. Wilson]

• The functionality for Rees congruences has been rewritten and improved. [Michael Torpey]

• There is a new Enumerator method for congruence classes of a semigroup congruence. [Michael Torpey]

#### Issues Resolved in Version 2.4

Issue numbers refer to the issues on the tracker.

• Issue 88: an inefficiency in JoinIrreducibleDClasses of an inverse semigroup ideal resulted in a call to GeneratorsOfSemigroup.

• Issue 94: EquivalenceClasses of the trivial congruence (generated by 0 pairs of elements) returned an error.

• Issue 95: The class containing the zero element of a Rees 0-matrix semigroup was not returned by EquivalenceClasses of a congruence over a Rees 0-matrix semigroup.

• Issue 108: IsRegularSemigroup with argument a Rees 0-matrix semigroup returned an error.

• Issue 119: NrCongruencesClasses and related methods did not work for Rees congruences.

• Issue 121: MultiplicativeZero and IsMultiplicativeZero sometimes returned incorrect results when applied to a non-acting semigroup (i.e. a semigroup not of transformations, partial permutations, partitions, or subsemigroups of a Rees 0-matrix semigroup).

• Issue 122: A bug in the creation of Green’s classes of ideals of semigroups, which resulted in an error.

• Issue 123: IsZeroSemigroup sometimes returned a false positive when applied to a non-acting semigroup.

### Version 2.3 (released 16/03/2015)

This is a minor release including some internal refactoring, and subsequent bugfixes, and stability improvements.

• Issue 116 was resolved. In some cases when the default length of hash tables in Semigroups was set to be very small, a segmentation fault occurred. This is a bug in the Orb package (see Issue 10), but we worked around it to resolve this issue.

### Version 2.2 (released 20/01/2015)

This is a minor release including some bug fixes, performance improvements, and additional functionality.

#### New Features in Version 2.2

• The functions SmallestElementSemigroup, LargestElementSemigroup, and GeneratorsSmallest.

• Free bands are introduced.

• Error messages are more uniform.

• The function RegularDClasses was introduced to resolve Issue 102.

• The documentation and code for semigroup congruences has been improved, and is better integrated with the core GAP system.

• The functions ReadGenerators and WriteGenerators were improved.

#### Issues Resolved in Version 2.2

• Some minor corrections were made to the methods for creating the ideals of some semigroups in standard examples semigroups, such as SingularTransformationSemigroup.

• Issue 102: we introduced RegularDClasses.

• Issue 104: the performance of MaximalSubsemigroups when applied to an inverse semigroup has been improved.

• Issue 105: CyclesOfPartialPerm is now documented and does not return nonsense.

• Issue 106: MaximalSubsemigroups sometimes failed when the ResClasses package was loaded. We refactored the code so that the method from ResClasses is no longer applied.

• Issue 107: A bug in the creation of Green’s classes of an ideal of a semigroup, which sometimes caused an error, has been resolved. This issue often caused MaximalSubsemigroups to stop in an error.

• Issue 110: MaximalSubsemigroups can be applied to any class of semigroup where it is possible to find an isomorphism to a transformation semigroup.

• Issue 111: POPI(1) returned the wrong semigroup. Similar issues existed in other corner cases, and these have been resolved too.

### Version 2.1.1 (released 09/09/2014)

This is a very minor release to fix an issue caused by only loading the packages needed (but not required) by Semigroups.

### Version 2.1 (released 04/09/2014)

This is a minor release including some bug fixes and performance improvements.

#### New Features in Version 2.1

• The functions:

• AsTransformationSemigroup,
• AsPartialPermSemigroup,
• AsBipartitionSemigroup,
• AsBlockBijectionSemigroup

which are shortcuts to Range(IsomorphismXSemigroup(S)).

• A method for IsTransitive for a transformation semigroup and, optionally, a positive integer or list of positive integers. This method is based on Gabow’s algorithm for determining the strongly connected components of a directed graph.

• The functions MeetSemigroupCongruence and JoinSemigroupCongruences for finding the meet and join of a pair of congruences of a semigroup.

• There is a new method for IsSynchronizingSemigroup, suggested by Peter Cameron, with better complexity than the previous method.

#### Issues Resolved in Version 2.1

Issue numbers refer to the issues on the tracker.

• There was a bug in ReadGeneratorsFile, which meant it sometimes returned fail. The mode argument for IO_FilteredFile was not given.

• There was a bug in the \in method for a congruence of a Rees 0-matrix semigroup, which sometimes returned the wrong answer for the zero of the semigroup.

• There was a bug in IteratorFromGeneratorsFile that caused it to read only every other line in the given file, and to crash if there were an odd number of lines.

• There was no hash function for bipartitions.

• The documentation for InverseSubsemigroupByProperty did not specify the arguments of the function.

• Issue 82: it is now possible to take the quotient of a semigroup by an ideal using the / operator.

• Issue 96: IsIsomorphicSemigroup sometimes returned a false negative by incorrectly comparing the output of PartialOrderOfDClasses (up to isomorphism) rather than the transitive reflexive closure of PartialOrderOfDClasses.

• Issue 97: there was a bug in the Normalizer method, which caused GAP to crash when the argument was a monoid with 0 generators.

• Issue 98: PartitionMonoid(1) returned the wrong answer, it was missing the non-identity element.

• Issue 99: the documentation for PartialOrderOfDClasses was incorrect.

• Issue 103: under certain circumstances an error was given when trying to compute with an ideal of an inverse semigroup.

### Version 2.0 (released 17/04/2014)

This is a major release including many new features and several bug fixes.

#### New Features in Version 2.0

• extensive new features for computing with elements and subsemigroups of the partition monoid. It is now possible to compute with semigroups, monoids, inverse semigroups, inverse monoids, and ideals consisting of elements of the partition monoid. Examples of subsemigroups of this type are the Brauer monoids, Temperley-Lieb monoids, and the dual symmetric inverse monoid. See Chapter 5 of the manual for more details;

• support for ideals of transformation, partial permutation, and bipartition semigroups, and subsemigroups of Rees 0-matrix semigroups. It is now possible to compute anything about one of these ideals that could formerly only be computed about a semigroup defined by a generating set. Such ideals now use a data structure similar to that used by semigroups defined by a generating set;

• the new operations IsomorphismSemigroups, IsIsomorphicSemigroup, and SmallestMultiplicationTable. Some of the methods for this operation require the Grape package to be fully installed;

• the new operation MaximalSubsemigroups, which returns the maximal subsemigroups of an arbitrary semigroup. Some of the methods for this operation require the Grape package to be fully installed;

• the operation IsMaximalSubsemigroup;

• the new operation Normalizer for computing a subgroup of a permutation group consisting of those permutations that stabilise, under conjugation, a transformation, partial perm, or bipartition semigroup. The genss package is required for this operation in some cases;

• the new operation CharacterTableOfInverseSemigroup for finding the character table of an inverse semigroup of partial permutations;

• methods for defining and manipulating the congruences of a Rees 0-matrix semigroup;

• the properties IsCongruenceFreeSemigroup, IsEUnitaryInverseSemigroup;

• the attributes:

• ComponentRepsOfTransformationSemigroup
• ComponentsOfTransformationSemigroup
• CyclesOfTransformationSemigroup
• ComponentRepsOfPartialPermSemigroup
• ComponentsOfPartialPermSemigroup
• CyclesOfPartialPermSemigroup.
• the new function IteratorFromGeneratorsFile that returns an iterator which reads semigroup elements from a file created using WriteGenerators. This function is a convenient way of, for example, looping over a collection of generators in a file without loading every object in the file into memory. This might be useful if the file contains more information than there is available memory;

• the operation EndomorphismsPartition that returns the monoid of endomorphisms preserving a partition. This monoid is defined using the minimum possible number of generators;

• a version of the function Splash that attempts to convert a string containing a dot or tikz document into a pdf and opens this pdf. Other file formats are also supported;

• the function DotSemilatticeOfIdempotents that produces a string containing a dot document of the semilattice of idempotents of an inverse semigroup grouped by $$\mathscr{D}$$-class;

• the operation NaturalLeqInverseSemigroup, which is an umbrella operation for NaturalLeqPartialPerm, and other such functions.

#### Issues Resolved in Version 2.0

Issue numbers refer to the issues on the tracker.

• the main algorithm underlying many of the methods in Semigroups has been revised to avoid computing the same information more than once. Some further internal rearranging and cleaning up was done.

• MinimalIdeal and SingularTransformationSemigroup now returns an ideal rather than a semigroup defined by a generating set;

• to reduce the size of the package archive, the examples directory has been removed. The content of the examples directory is available on this webpage.

• several bugs in the setup for subsemigroups of Rees 0-matrix semigroups were resolved. These issues would have caused GAP to give an error in certain circumstances.

• Issue 33: an error was returned when trying to calculate the size, or multiply elements in the quotient of a semigroup by an ideal.

• Issue 36,Issue 64: the function SmallGeneratingSet was ambiguous, in the sense that it was sometimes unclear how to recreate a semigroup from its small generating set. For example, SmallGeneratingSet of a monoid could return an empty list, but this empty list could not be used to recreate the monoid in GAP. This was resolved by introducing the functions SmallSemigroupGeneratingSet, SmallMonoidGeneratingSet, SmallInverseSemigroupGeneratingSet, SmallInverseMonoidGeneratingSet. These functions can also now be applied to collections of elements, i.e. not only to semigroups.

• Issue 47: ClosureSemigroup had several bugs that could, in some cases, result in incorrect results, or semigroups with invalid data structures.

• Issue 50 and Issue 59: WriteGenerators wrote nothing to a file in the case that it was not piping through xz or gzip.

• Issue 55: DotDClasses did not work when the argument was a Rees 0-matrix semigroup (it worked as intended when the argument was a subsemigroup of such a semigroup defined by a generating set).

• Issue 56: the functions Monoid and InverseMonoid sometimes did not contain their identity element.

• Issue 57: under certain circumstances a bug in the GAP kernel function INV_KER_TRANS, that didn’t handle kernels and transformations with different length and degree properly, caused GAP to give an error.

• Issue 63: there was an error in the GAP library functions Monoid and InverseMonoid, when they were passed a monoid as an argument.

• Issue 63 (and Issue 4 in the Orb package): a bug in the Orb package meant that the log of an Orb was not properly updated if the enumeration stopped early because we found something we were looking for. This caused Semigroups to return incorrect results in some rare cases.

• Issue 72: the method for IsomorphismTransformationSemigroup applied to a binary relation monoid returned an isomorphism to a transformation semigroup which was missing the image of the identity.

• Issue 89: there was a bug in TRANS_IMG_CONJ which failed to handle transformations of unequal degrees correctly. This causes incorrect results to be returned when considering semigroups generated by transformations of unequal degrees.

### Version 1.4 (released 28/10/13)

This is a minor release containing some bug fixes. Specifically: the functionality of ReadGenerators and WriteGenerators has been improved to allow the argument to be an IO file object, and support is added to read and write directly to files compressed using xz. A minor bug relating to the creation of idempotents in transformation semigroups which was triggered by the identity transformation has been resolved. The functions IsomorphismReesMatrixSemigroup, InjectionPrincipalFactor, and IsZeroSimpleSemigroup have been revised. IsZeroSimpleSemigroup formerly returned true for the 2 element zero semigroup, which is not 0-simple. IsomorphismReesMatrixSemigroup could have returned an error if called for a semigroup which was not a semigroup of partial perms or transformations. The use of AsPermutation was changed to PermutationOfImage where appropriate following a change to the library methods for AsPermutation. The declarations of IsomorphismPermGroup and ClosureSemigroup were moved/changed to avoid warnings that their methods matched more than one declaration. These warnings were exposed by doing LoadAllPackages, but were not present when loading Semigroups by itself.

### Version 1.3 (released 11/10/13)

Version 1.3 contains many bug fixes, extensions and improvements in the documentation, and several new methods and functions. Most notably are (in no particular order):

• the methods in Semigroups have been extended to apply to arbitrary subsemigroups of regular Rees 0-matrix semigroups over groups;

• a new method for MaximalSubsemigroups of Rees matrix semigroup has been implemented;

• the function Read/WriteSemigroups have been renamed Read/WriteGenerators and their performance has been improved. It is now possible to use WriteGenerators to write to a gzipped file;

• the operation SingularSemigroup has been renamed SingularTransformationSemigroup;

• the following attributes have been introduced: MinimalDClass, MaximalDClasses, StructureDescriptionMaximalSubgroups, StructureDescriptionSchutzenbergerGroups, and IsGreensDLeq.

• the attribute/operation DotDClasses has been introduced. This allows the $$\mathscr{D}$$-class diagram of a semigroup to be viewed.

• ComponentRepsOfTransformationSemigroup is reintroduced.

### Version 1.2 (released 02/08/13)

This release includes several new methods for inverse semigroups of partial permutations and for free inverse semigroups. Most notably among the new methods for inverse semigroups of partial permutations are:

• SmallerDegreePartialPermRepresentation
• VagnerPrestonRepresentation

for changing the representation of an inverse semigroup of partial permutations. The changes in this release were the result of the University of St Andrews Research for Undergraduates Summer School in 2012, and were largely written by Wilf A. Wilson and Robert Hancock.

Free inverse semigroups, and their elements, are also introduced, this part of the package was written by Julius Jonusas (who wishes to acknowledge the support of the Carnegie Trust).

### Version 1.1 (released 11/06/13)

A minor release to fix some technical issues in PackageInfo.g, the declarations of IsGreens.Class, and a minor change in the output in one test in everyfunction.tst which was consequence of the declarations of IsGreens.Class.

### Version 1.0 (released 07/06/13)

The package has been renamed from Citrus to Semigroups. The package has been completely overhauled, the performance has been improved, and the code has been generalized so that in the future the same code can be used to compute with other types of semigroups.

# Under the name Citrus:

### Version 0.9999

This is the final release of Citrus (the package will be renamed Citrus in the next release since the scope of the package has expanded to include more types of semigroups than just those of transformations and partial permutations).

A minor release fixing several bugs relating to inverse semigroups of partial permutations pointed out by partcipants at the University of St Andrews Research for Undergraduates Summer School in July 2012. Most notably by Demi Allen, Casey Donoven, Rhiannon Dougall, Robert Hancock, and Wilf A. Wilson. More specifically, SymmetricInverseSemigroup(n) returned an incorrect answer when n=1 or n=2, \in for the empty mapping and an inverse semigroup of partial perms sometimes incorrectly returned false, some harmless compiler warnings were displayed when using more recent versions of gcc, NaturalLeqPP sometimes returned the incorrect value, there was no method for IsInverseSemigroup or IsInverseMonoid for a semigroup of partial perms.

### Version 0.999

A minor release fixing several bugs relating to partial permutations and monoids thereof, pointed out by Jack Schmidt. More specifically, MultiplicativeZero sometimes incorrectly returned fail for an inverse semigroup of partial permutations, sometimes PartialPerm incorrectly returned fail when given a dense range as an argument, sometimes the size of an inverse monoid was 1 more than the correct value, and RestrictedPP sometimes failed when it should not have.

### Version 0.99

another minor release. Specific changes were: removed the declaration of SmallGeneratingSet for IsSemigroup since it appears not to be used and caused a warning to be shown when rcwa was loaded after Citrus. Added a new abstract to the PackageInfo.g file, and the documentation, and updated the webpages, in particular so that the html version of the manual is linked to that on the GAP webpage and the links to other manuals work.

### Version 0.9

renamed the function for creating the semigroup of order-preserving transformations on an n-chain from O to OrderEndomorphisms after it was pointed out that it is not sensible to have function names with one character. Also made some minor adjustments to the manual.

### Version 0.8

minor changes due to incompatibility with Smallsemi 0.6.4 which caused some test files to fail when Nilmat was loaded after Citrus. The clashes and the failed test were caused by various properties being declared for IsTransformationSemigroup rather than IsSemigroup.

### Version 0.7

the most major change is the introduction of special methods for partial permutations and inverse semigroups. So that these methods are efficient, a GAP kernel component (in C) has also been introduced for various low-level computations with partial permutations. Essentially all functions previously available for transformation semigroups are now available for inverse semigroups of partial permutations. The manual has been expanded and reorganised, some standard examples have been included (semigroups of order preserving transformations or partial permutations, the symmetric inverse semigroup, the full matrix semigroup over a finite field), the endomorphism monoids of the non-abelian groups with at most 64 elements have been included in the catalogues of examples, the functions InjectionPrincipalFactor, IsomorphismReesMatrixSemigroup, and PrincipalFactor, and some specific properties and attributes of inverse semigroups have been introduced (such as IsFactorisableSemigroup and PrimitiveIdempotents).

### Version 0.6

fixed a bug relating to the creation of transformation semigroups using MagmaByGenerators. Also added the global variable CitrusOptionsRec containing the default values of the options used by Citrus when creating a semigroup.

### Version 0.5

major changes are: the documentation has been further revised, functions for creating semigroups and monoids with certain options have been introduced, several functions have had the word Greens' removed from their names to reduce the length, the operation ClosureSemigroup has been introduced, the functions ReadCitrus and WriteCitrus for reading and writing transformations to a file have been introduced, several catalogues of examples of transformation semigroups are now included in the examples directory, methods for creating a Green’s class inside another Green’s class have been included (such as an $$\mathscr{R}$$-class of a $$\mathscr{D}$$-class or an $$\mathscr{H}$$-class of an $$\mathscr{L}$$-class), the hash functions used for transformations etc have been improved.

Some minor bugs have been fixed, and new methods or functions with the following names have also been introduced:

AntiIsomorphismTransformationSemigroup (for a trans. semigroup), IdempotentGeneratedSubsemigp, IsomorphismTransformationSemigroup (for a perm. gp), IsomorphismTransformationMonoid (for a perm. gp), NrElementsOfRank.

### Version 0.4

major changes are: the documentation has been updated, some changes to core functions for $$\mathscr{R}$$-classes/image orbits have resulted in a performance improvement, there is a method for the operation Factorization allowing an arbitrary element of a transformation semigroup to be expressed as a product of the generators.

Some minor bugs have been fixed, and new methods or functions with the following names have also been introduced:

OrbSCC, OrbSCCLookup, OrbSCCTruthTable, ReverseSchreierTreeOfSCC, SchreierTreeOfSCC, IsomorphismTransformationSemigroup (for a perm. gp).

### Version 0.3

fixed a critical (but rare) bug in AddToOrbitsOfKernels that caused computations relating to $$\mathscr{D}$$-classes, $$\mathscr{H}$$-classes, or $$\mathscr{L}$$-classes to return incorrect answers in some cases.

### Version 0.2

updated the method for \^ for a transformation and perm so that it is more efficient than the library method, same for * for a perm and transformation. New method for StructureDescription of a Brandt semigroup, and IsSubset for a trans. semigroup and trans. coll.

fixed bugs in IndexPeriodOfTransformation (it returned incorrect results) and AsPermutation. Also reduce hash table lengths so that Citrus uses less memory. Fixed bug that triggered an infinite loop when trying to find elements of a trivial trans. semigroup.

added the functions CitrusDefaultMem, CitrusHiMem, CitrusLoMem, CitrusVeryLoMem, IsBrandtSemigroup, IsLeftSimple, IsMonogenicSemigroup, IsRightSimple, IsZeroRectangularBand, IsZeroSimpleSemigroup`.