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libsemigroups - Version 2.5.1

Documentation Status Launch Binder

C++ library for semigroups and monoids

What is libsemigroups?

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;
  • 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.

How to use it

See the documentation

Installation instructions are here


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 (



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).