Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
162 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Computing finite index congruences of finitely presented semigroups and monoids (2302.06295v3)

Published 13 Feb 2023 in math.RA and cs.DS

Abstract: In this paper we describe an algorithm for computing the left, right, or 2-sided congruences of a finitely presented semigroup or monoid with finitely many classes, and alternative algorithm when the finitely presented semigroup or monoid is finite. We compare the two algorithms presented to existing algorithms and implementations. The first algorithm is a generalization of Sims' low index subgroup algorithm for finding the congruences of a monoid. The second algorithm involves determining the distinct principal congruences, and then finding all of their possible joins. Variations of this algorithm have been suggested in numerous contexts by numerous authors. We show how to utilise the theory of relative Green's relations, and a version of Schreier's Lemma for monoids, to reduce the number of principal congruences that must be generated as the first step of this approach. Both of the algorithms described in this paper are implemented in the GAP package Semigroups, and the first algorithm is available in the C++ library libsemigroups and in its python bindings libsemigroups_pybind.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (69)
  1. “The stylic monoid” In Semigroup Forum 105.1 Springer ScienceBusiness Media LLC, 2022, pp. 1–45 DOI: 10.1007/s00233-022-10285-3
  2. João Araújo, Wolfram Bentz and Gracinda M.S. Gomes “Congruences on direct products of transformation and matrix monoids” In Semigroup Forum Springer Nature, 2018 DOI: 10.1007/s00233-018-9931-8
  3. “CREAM: a Package to Compute [Auto, Endo, Iso, Mono, Epi]-morphisms, Congruences, Divisors and More for Algebras of Type (2n,1n)superscript2𝑛superscript1𝑛(2^{n},1^{n})( 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , 1 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT )”, 2022 eprint: arXiv:2202.00613
  4. Robert E. Arthur and N. Ruškuc “Presentations for Two Extensions of the Monoid of Order-Preserving Mappings on a Finite Chain” In Southeast Asian Bulletin of Mathematics 24.1 Springer ScienceBusiness Media LLC, 2000, pp. 1–7 DOI: 10.1007/s10012-000-0001-1
  5. Alex Bailey, Martin Finn-Sell and Robert Snocken “Subsemigroup, ideal and congruence growth of free semigroups” In Israel Journal of Mathematics 215.1, 2016, pp. 459–501 DOI: 10.1007/s11856-016-1384-8
  6. Matthew David George Kenworthy Brookes “Lattices of congruence relations for inverse semigroups”, 2020 URL: https://etheses.whiterose.ac.uk/29077/
  7. Andries E. Brouwer, Jan Draisma and Bart J. Frenk “Lossy Gossip and Composition of Metrics” In Discrete & Computational Geometry 53.4 Springer ScienceBusiness Media LLC, 2015, pp. 890–913 DOI: 10.1007/s00454-015-9666-1
  8. Alan J. Cain, Robert Gray and Nik Ruškuc “Green index in semigroups: generators, presentations, and automatic structures” In Semigroup Forum 85.3 Springer ScienceBusiness Media LLC, 2012, pp. 448–476 DOI: 10.1007/s00233-012-9406-2
  9. “Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups” In Combinatorial Theory 3.3 California Digital Library (CDL), 2023 DOI: 10.5070/c63362799
  10. “The Todd-Coxeter Algorithm for Semigroups and Monoids” arXiv, 2022 DOI: 10.48550/ARXIV.2203.11148
  11. “Introduction to Algorithms”, The MIT Press London, England: MIT Press, 2009
  12. “Semigroups with finitely generated universal left congruence” In Monatshefte für Mathematik 190.4 Springer ScienceBusiness Media LLC, 2019, pp. 689–724 DOI: 10.1007/s00605-019-01274-w
  13. “The monoids of orders eight, nine & ten” In Annals of Mathematics and Artificial Intelligence 56.1 Springer ScienceBusiness Media LLC, 2009, pp. 3–21 DOI: 10.1007/s10472-009-9140-y
  14. James East and James D. Mitchell “Transformation representations of diagram monoids” in preparation
  15. “Congruence lattices of ideals in categories and (partial) semigroups”, 2020 eprint: arXiv:2001.01909
  16. “Congruences on infinite partition and partial Brauer monoids”, 2018 eprint: arXiv:1809.07427
  17. “Classification of congruences of twisted partition monoids” In Advances in Mathematics 395 Elsevier BV, 2022, pp. 108097 DOI: 10.1016/j.aim.2021.108097
  18. “Properties of congruences of twisted partition monoids and their lattices” In Journal of the London Mathematical Society 106.1 Wiley, 2022, pp. 311–357 DOI: 10.1112/jlms.12574
  19. “Computing finite semigroups” In Journal of Symbolic Computation 92 Elsevier BV, 2019, pp. 110–155 DOI: 10.1016/j.jsc.2018.01.002
  20. “Congruence lattices of finite diagram monoids” In Advances in Mathematics 333 Elsevier BV, 2018, pp. 931–1003 DOI: 10.1016/j.aim.2018.05.016
  21. Vitor Hugo Fernandes “On the cyclic inverse monoid on a finite set”, 2022 eprint: arXiv:2211.02155
  22. Ralph Freese “Computing congruences efficiently” In Algebra universalis 59.3-4 Springer ScienceBusiness Media LLC, 2008, pp. 337–343 DOI: 10.1007/s00012-008-2073-1
  23. “Algorithms for computing finite semigroups” In Foundations of computational mathematics (Rio de Janeiro, 1997) Berlin: Springer, 1997, pp. 112–126
  24. Bernard A. Galler and Michael J. Fisher “An improved equivalence algorithm” In Communications of the ACM 7.5 Association for Computing Machinery (ACM), 1964, pp. 301–303 DOI: 10.1145/364099.364331
  25. “Classical Finite Transformation Semigroups” Springer London, 2009 DOI: 10.1007/978-1-84800-281-4
  26. “GAP – Groups, Algorithms, and Programming, Version 4.12.1”, 2022 The GAP Group URL: %5Curl%7Bhttps://www.gap-system.org%7D
  27. “Green index and finiteness conditions for semigroups” In Journal of Algebra 320.8 Elsevier BV, 2008, pp. 3145–3164 DOI: 10.1016/j.jalgebra.2008.07.008
  28. Peter M. Higgins “Combinatorial results for semigroups of order-preserving mappings” In Mathematical Proceedings of the Cambridge Philosophical Society 113.2 Cambridge University Press (CUP), 1993, pp. 281–296 DOI: 10.1017/s0305004100075964
  29. “Testing for isomorphism between finitely presented groups” In Groups, Combinatorics and Geometry, London Mathematical Society Lecture Note Series Cambridge University Press, 1992, pp. 459–475
  30. John E. Hopcroft and Richard M. Karp “A Linear Algorithm for Testing Equivalence of Finite Automata.”, 1971
  31. John M. Howie “Fundamentals of semigroup theory” Oxford Science Publications 12, London Mathematical Society Monographs. New Series New York: The Clarendon Press Oxford University Press, 1995, pp. x+351
  32. Alexander Hulpke “Calculating Subgroups with GAP” In Group Theory and Computation Springer Singapore, 2018, pp. 91–106 DOI: 10.1007/978-981-13-2047-7˙5
  33. Alexander Hulpke “Representing Subgroups of Finitely Presented Groups by Quotient Subgroups” In Experimental Mathematics 10.3 A K Peters, Ltd., 2001, pp. 369–382
  34. Andrzej Jura “Coset enumeration in a finitely presented semigroup” In Canad. Math. Bull. 21.1, 1978, pp. 37–46
  35. Andrzej Jura “Determining ideals of a given finite index in a finitely presented semigroup” In Demonstratio Mathematica 11.3 De Gruyter Open, 1978, pp. 813–828
  36. Andrzej Jura “Some remarks on non-existence of an algorithm for finding all ideals of a given finite index in a finitely presented semigroup” In Demonstratio Mathematica 13 De Gruyter Open, 1980, pp. 573–578
  37. “Stallings Foldings and Subgroups of Free Groups” In Journal of Algebra 248.2, 2002, pp. 608–668 DOI: 10.1006/jabr.2001.9033
  38. Mati Kilp, Ulrich Knauer and Alexander V. Mikhalev “Monoids, Acts and Categories: With Applications to Wreath Products and Graphs. A Handbook for Students and Researchers” DE GRUYTER, 2000 DOI: 10.1515/9783110812909
  39. Donald Knuth “The Art of Computer Programming: Combinatorial Algorithms, Volume 4B” Addison-Wesley Professional, 2022
  40. Donald E. Knuth “Permutations, matrices, and generalized Young tableaux” In Pacific J. Math. 34, 1970, pp. 709–727 URL: http://projecteuclid.org.ezproxy.st-andrews.ac.uk/euclid.pjm/1102971948
  41. “Le monoïde plaxique” In Noncommutative structures in algebra and geometric combinatorics (Naples, 1978) 109, Quad. “Ricerca Sci.” CNR, Rome, 1981, pp. 129–156
  42. “Groups and actions in transformation semigroups” In Mathematische Zeitschrift 228.3 Springer ScienceBusiness Media LLC, 1998, pp. 435–450 DOI: 10.1007/pl00004628
  43. Anatolii Ivanovich Mal’tsev “Symmetric groupoids” In Matematicheskii sbornik 73.1 Russian Academy of Sciences, Steklov Mathematical Institute of Russian …, 1952, pp. 136–151
  44. Victor Maltcev “On a new approach to the dual symmetric inverse monoid ℐX*superscriptsubscriptℐ𝑋\mathscr{I}_{X}^{*}script_I start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT” In Internat. J. Algebra Comput. 17.3, 2007, pp. 567–591 DOI: 10.1142/S0218196707003792
  45. “On the minimal faithful degree of Rhodes semisimple semigroups” In Journal of Algebra 633 Elsevier BV, 2023, pp. 788–813 DOI: 10.1016/j.jalgebra.2023.06.032
  46. J.C.C. McKinsey “The Decision Problem for Some Classes of Sentences Without Quantifiers” In The Journal of Symbolic Logic 8.2 Association for Symbolic Logic, 1943, pp. 61–76 URL: http://www.jstor.org/stable/2268172
  47. John Meakin “One-sided congruences on inverse semigroups” In Transactions of the American Mathematical Society 206 American Mathematical Society (AMS), 1975, pp. 67–67 DOI: 10.1090/s0002-9947-1975-0369580-9
  48. James Mitchell and Chinmaya Nagpal and Maria Tsalakou “libsemigroups_pybind11 v0.4.3” Zenodo, 2022 DOI: 10.5281/ZENODO.7307278
  49. “libsemigroups v2.7.1” Zenodo, 2023 DOI: 10.5281/zenodo.1437752
  50. “Semigroups v5.2.1” Zenodo, 2023 DOI: 10.5281/zenodo.592893
  51. J. Neubüser “An elementary introduction to coset table methods in computational group theory” In Groups - St Andrews 1981, London Mathematical Society Lecture Notes Series Cambridge University Press, 1982, pp. 1–45 DOI: 10.1017/CBO9780511661884.004
  52. Daphne Norton “Algorithms for testing equivalence of finite automata, with a grading tool for JFLAP”, 2009
  53. OEIS Foundation Inc. “The On-Line Encyclopedia of Integer Sequences” Published electronically at http://oeis.org, 2024
  54. Rui Barradas Pereira “The CREAM GAP Package - Algebra CongRuences, Endomorphisms and AutomorphisMs”, 2022 URL: https://gitlab.com/rmbper/cream
  55. Emmanuel Rauzy “Computability of finite quotients of finitely generated groups” In Journal of Group Theory 25.2 Walter de Gruyter GmbH, 2021, pp. 217–246 DOI: 10.1515/jgth-2020-0029
  56. “Syntactic and Rees Indices of Subsemigroups” In Journal of Algebra 205.2, 1998, pp. 435–450 DOI: https://doi.org/10.1006/jabr.1997.7392
  57. Nikola Ruškuc “Semigroup presentations”, 1995
  58. Boris M. Schein “The Minimal Degree of a Finite Inverse Semigroup” In Transactions of the American Mathematical Society 333.2 American Mathematical Society, 1992, pp. 877–888 URL: http://www.jstor.org/stable/2154068
  59. Ákos Seress “Permutation group algorithms” 152, Cambridge Tracts in Mathematics Cambridge: Cambridge University Press, 2003, pp. x+264 DOI: 10.1017/CBO9780511546549
  60. Charles C. Sims “Computation with Finitely Presented Groups”, Encyclopedia of Mathematics and its Applications Cambridge University Press, 1994 DOI: 10.1017/CBO9780511574702
  61. Charles C. Sims “Computation with permutation groups” In Proceedings of the second ACM symposium on Symbolic and algebraic manipulation - SYMSAC ’71 ACM Press, 1971 DOI: 10.1145/800204.806264
  62. Charles C. Sims “Computational methods in the study of permutation groups” In Computational Problems in Abstract Algebra Elsevier, 1970, pp. 169–183 DOI: 10.1016/b978-0-08-012975-4.50020-5
  63. Michael Sipser “Introduction to the Theory of Computation” Boston, MA: Course Technology, 2013
  64. “A practical method for enumerating cosets of a finite abstract group” In Proceedings of the Edinburgh Mathematical Society 5.01 Cambridge University Press (CUP), 1936, pp. 26–34 DOI: 10.1017/s0013091500008221
  65. Michael Torpey “Semigroup congruences : computational techniques and theoretical applications” University of St Andrews, 2019 DOI: 10.17630/10023-17350
  66. E.J. Tully “Representation of a semigroup by transformations acting transitively on a set” In Amer. J. Math. 83, 1961, pp. 533–541
  67. A. Wallace “Relative ideals in semigroups, I (Faucett’s Theorem)” In Colloquium Mathematicum 9.1 Institute of Mathematics, Polish Academy of Sciences, 1962, pp. 55–61 DOI: 10.4064/cm-9-1-55-61
  68. A.D. Wallace “Relative Ideals in Semigroups. II” In Acta Mathematica Academiae Scientiarum Hungaricae 14.1-2 Springer ScienceBusiness Media LLC, 1963, pp. 137–148 DOI: 10.1007/bf01901936
  69. Anthony Williams “C++ Concurrency in Action, 2E” New York, NY: Manning Publications, 2019
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets