libsemigroups is a C++14 library containing implementations of several algorithms for computing finite, and finitely presented, semigroups and monoids. Namely:

• the Froidure-Pin algorithm for computing finite semigroups;
• the Todd-Coxeter algorithm for finitely presented semigroups and monoids; see also this paper;
• the Knuth-Bendix algorithm for finitely presented semigroups and monoids;
• the Schreier-Sims algorithm for permutation groups;
• a preliminary implementation of the Konieczny and Lallement-McFadden algorithm for computing finite semigroups which act on sets;
• an implementation of the Radoszewski-Rytter algorithm for testing equivalence of words in free bands;
• an implementation of the algorithm for solving the word problem for small overlap monoids, and for computing normal forms in such monoids; see Kambites, Kambites, and Mitchell-Tsalakou
• a version of Sims low index subgroup algorithm for computing one-sided congruences of a semigroup or monoid; see this paper for details.
• a version of Stephen's procedure for finitely presented semigroups and monoids (for a given word w this procedure is for determining words equivalent to w or that are left divisors of w).

libsemigroups is partly based on Algorithms for computing finite semigroups, Expository Slides, and Semigroupe 2.01 by Jean-Eric Pin.

libsemigroups is used in the Semigroups package for GAP, and it is possible to use libsemigroups directly in Python 3 via the package libsemigroups_pybind11. The development version of libsemigroups is available on github, and some related projects are here.

The main classes in libsemigroups are named after the algorithms they implement; see, for example, libsemigroups::FroidurePin, libsemigroups::Konieczny, libsemigroups::congruence::ToddCoxeter, libsemigroups::fpsemigroup::Kambites, libsemigroups::fpsemigroup::KnuthBendix, and libsemigroups::SchreierSims, libsemigroups::Sims1, or libsemigroups::Stephen.

The implementations in libsemigroups::FroidurePin, libsemigroups::Konieczny, and libsemigroups::SchreierSims are generic and easily adapted to user-defined types.

libsemigroups uses: HPCombi which uses the SSE and AVX instruction sets for very fast manipulation of transformations, partial permutations, permutations, and boolean matrices of small size; catch for tests; fmt for reporting; and eigen for some linear algebra computations.

See the documentation https://libsemigroups.readthedocs.io/en/latest/

Installation instructions are here https://libsemigroups.readthedocs.io/en/latest/install.html

If you find any problems with libsemigroups, or have any suggestions for features that you'd like to see, please use the issue tracker.

James Mitchell ([email protected])

• Reinis Cirpons ([email protected]) contributed to IsObviouslyInfinite, to integrating eigen, and contributed an implementation of the Radoszewski-Rytter algorithm for testing equivalence of words in free bands.
• Joseph Edwards ([email protected]) contributed the container StaticTriVector2.
• Luke Elliott ([email protected]) contributed to the Schreier-Sims implementation.
• Jan Engelhardt ([email protected]) contributed some bug fixes to the build system, and a number of helpful issues.
• Ilya Finkelshteyn ([email protected]) contributed to the continuous integration in AppVeyor.
• Isuru Fernando ([email protected]) contributed to the build system.
• Florent Hivert ([email protected]) contributed many helpful ideas to libsemigroups, an allocator implementation (to be included in a future version), and HPCombi.
• Max Horn ([email protected]) contributed some fixes.
• Jerry James ([email protected]) contributed some bugfixes.
• Julius Jonušas contributed to the implementation of the Froidure-Pin algorithm.
• Alex Levine contributed to the Schreier-Sims implementation.
• Michael Orlitzky ([email protected]) contributed to the build system.
• Dima Pasechnik ([email protected]) contributed to the build system.
• Chris Russell contributed some tests for finitely presented semigroups.
• Finn Smith ([email protected]) contributed the implementation of the Konieczny and Lallement-McFadden algorithm, to the Todd-Coxeter implementation, and to BMat8s.
• Nicolas Thiéry ([email protected]) contributed to the build system, packaging libsemigroups via conda, the python bindings and many helpful conversations and suggestions.
• Maria Tsalakou ([email protected]) contributed to the Knuth-Bendix implementation, related algorithms for the class ActionDigraph, and to the implementation of the Kambites class.
• Wilf Wilson ([email protected]) contributed some fixes.
• Murray Whyte ([email protected]) contributed to the documentation and reported a number of bugs.
• Michael Young ([email protected]) contributed to the congruences code in the v0.0.1 to v0.6.7.

We acknowledge financial support from the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541) (primarily for the python bindings).

We thank the Carnegie Trust for the Universities of Scotland for funding the PhD scholarship of Julius Jonušas when he worked on this project.

We thank the Engineering and Physical Sciences Research Council (EPSRC) for funding the PhD scholarships of Michael Young and Finn Smith when they worked on this project (EP/M506631/1, EP/N509759/1).