Resources for the Digraphs minicourse

Tue, 19 Aug 2025 00:00:00 +0000

This page contains some resources for the Digraphs minicourse at GAP days Summer 2025 in Koper, Slovenia.

Here are the slides (pdf), and the worksheet worksheet (pdf).

Some data for the problems on the worksheet can be found below.

Aubrey de Grey’s graph showing that the chromatic number of the plane is at least $5$ in graph6 format: degrey.s6.gz
representatives of prime $\mathscr{J}$-classes in the boolean matrix monoid: n = 4, n = 5, n = 6, n = 7
erdos.s6.gz and labels.txt.gz.

About

Sat, 02 May 2020 00:00:00 +0000

This is the webpage of James D. Mitchell. I am a professor of mathematics at the School of Mathematics and Statistics at the University of St Andrews. I’m currently interested in computational semigroup theory, and topological algebra.

The website accompanying the paper ‘Groups that together with any transformation generate regular semigroups or idempotent generated semigroups’ by J. Araújo, J. D. Mitchell and Csaba Schneider is available here.

Address

Mathematical Institute,
North Haugh,
St Andrews,
Fife,
KY16 9SS,
Scotland.

Publications

Sat, 02 May 2020 00:00:00 +0000

A list of publications by J. D. Mitchell on MathSciNet.

See Articles and Preprints.

Articles

(with A. Abram, F. Hivert, J.-C. Novelli, and M. Tsalakou), ‘Power quotients of plactic-like monoids’, preprint; https://arxiv.org/abs/2406.16387
(with S. Bardyla, L. Elliott, and Y. Péresse), ‘A note on intrinsic topologies of groups’, submitted; https://doi.org/10.48550/arXiv.2506.11500
(with J. Edwards, and P. Ragavan) ‘Counting finite semirings’; submitted https://arxiv.org/abs/2507.03709
(with Tom Aird and Murray T. Whyte), ‘Short presentations for transformation monoids’, submitted; https://arxiv.org/abs/2406.19294
(with Luna Elliott and Alex Levine), ‘Computing congruences of finite inverse semigroups’, submitted; https://arxiv.org/abs/2406.09281
(with R. Cirpons and J. East) ‘Transformation representations of diagram monoids’, International Mathematics Research Notices, Volume 2026, Issue 6, March 2026, rnag041 https://doi.org/10.1093/imrn/rnag041; https://arxiv.org/abs/2411.14693
(with R. Cirpons, F. Hivert, A. Mahboubi, G. Melquiond, and F. L. Smith) ‘Certifying the decidability of the word problem in monoids at large’, Proceedings of the Certified Programs and Proofs conference (CPP ‘26) (2026) 128-142 https://doi.org/10.1145/3779031.3779101
(with Serhii Bardyla, Luna Elliott, and Yann Péresse) ‘Classifying the Polish semigroup topologies on the symmetric inverse monoid’, Proc. Edinburgh Math. Soc. (2026); https://doi.org/10.1017/S0013091525101302 https://arxiv.org/abs/2405.20134
(with I. Dolinka and D. G. FitzGerald) ‘Semirigidity and the enumeration of nilpotent semigroups of index three’, Semigroup Forum 112 (2026) 24-47 https://doi.org/10.1007/s00233-025-10608-0 https://arxiv.org/abs/2411.00466
(with Luna Elliott and Alex Levine), ‘E-disjunctive inverse semigroups’, Journal of Algebra, 687 (2026) 292-344; https://doi.org/10.1016/j.jalgebra.2025.08.041 https://arxiv.org/abs/2405.19825
(with M. Anagnostopoulou-Merkouri, R. Cirpons, and M. Tsalakou) ‘Computing finite index congruences of finitely presented semigroups and monoids’, Mathematics of Computation (2025); https://doi.org/10.1090/mcom/4136 https://arxiv.org/abs/2302.06295
(with R. Cirpons and F. L. Smith) ‘Off with the head: termination provers and the word problem for 1-relation monoids’, Proceedings of the 20th International Workshop on Termination (2025), WST2025 paper 16
(with F. Hivert, F. L. Smith, and W. A. Wilson) ‘Minimal generating sets for matrix monoids’, Journal of Symbolic Computation 129 (2025) 102415 https://doi.org/10.1016/j.jsc.2024.102415 https://arxiv.org/abs/2012.10323
(with Mukherjee, S. and Vojtěchovský, P.) ‘Minimal representatives of endofunctions’ Semigroup Forum 109 (2024) 626–638; https://doi.org/10.1007/s00233-024-10472-4
(with M. Brookes, J. East, C. Miller, N. Ruskuc) ‘Heights of one- and two-sided congruence lattices of semigroups’, Pacific Journal of Mathematics 333 (2024) 17-57; https://doi.org/10.2140/pjm.2024.333.17 https://arxiv.org/abs/2310.08229
(with S. Bardyla, L. Elliott, and Y. Péresse) ‘Topological embeddings into transformation monoids’, Forum Mathematicum 36 (2024) 1537-1554; https://doi.org/10.1515/forum-2023-0230 http://arxiv.org/abs/2302.08988
(with T. D. H. Coleman, F. L. Smith, and M. Tsalakou) ‘The Todd-Coxeter Algorithm for Semigroups and Monoids’, Semigroup Forum 108 (2024) 536–593; https://doi.org/10.1007/s00233-024-10431-z https://arxiv.org/abs/2203.11148
(with M. Anagnostopoulou-Merkouri and Z. Mesyan) ‘Properties of Congruence Lattices of Graph Inverse Semigroups’, Internat. J. Algebra Comput.; 34 (2024) 371-396; https://doi.org/10.1142/S0218196724500139 https://arxiv.org/abs/2108.08277
(with P. J. Cameron, J. East, D. FitzGerald, L. Pebody, and T. Quinn-Gregson) ‘Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups’, Combinatorial Theory 3 (3) (2023) #16; https://doi.org/10.5070/C63362799 https://arxiv.org/abs/2110.09701
(with C. Clark, J. Jonušas, and A. Francis) ‘An algebraic model for inversion and deletion in bacterial genome rearrangement’; J. Math. Biol. 87, 34 (2023); https://doi.org/10.1007/s00285-023-01965-x https://arxiv.org/abs/2209.07963
(with L. Elliott, J. Jonušas, Y. Péresse, M. Pinsker) ‘Polish topologies on endomorphism monoids of relational structures’, Adv. Math. 431 (2023); https://doi.org/10.1016/j.aim.2023.109214 https://arxiv.org/abs/2203.11577;
(with L. Elliott, Z. Mesyan, J. Jonušas, M. Morayne, and Y. H. Péresse) ‘Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones’, Trans. Amer. Math. Soc. 376 (2023) 8023-8093; https://doi.org/10.1090/tran/8987 https://arxiv.org/abs/1912.07029;
(with L. Elliott and A. Levine) ‘Counting monogenic monoids and inverse monoids’, Comm. Algebra 51 (2023) 4654-4661; https://doi.org/10.1080/00927872.2023.2214821 http://arxiv.org/abs/2303.12387;
(with M. Tsalakou) ‘An explicit algorithm for normal forms in small overlap monoids’, Journal of Algebra 630 (2023) 394-433; https://doi.org/10.1016/j.jalgebra.2023.04.019 https://arxiv.org/abs/2105.12125
(with R. Cirpons) ‘Polynomial time multiplication and normal forms in free bands’, Theor. Comput. Sci. 953 (2023); https://doi.org/10.1016/j.tcs.2023.113783 https://arxiv.org/abs/2209.05334
(with J. East) ‘Generating wreath products of symmetric and alternating groups’, New Zealand Journal of Mathematics, 51 (2021), 85–93; https://doi.org/10.53733/108 https://arxiv.org/abs/2104.05378v1
(with James Hyde, Julius Jonušas, Yann H. Péresse) ‘Sets of universal sequences for the symmetric group and analogous semigroups’, Proc. Amer. Math. Soc. 148 (2020), 1917-1931; https://doi.org/10.1090/proc/14881 https://arxiv.org/abs/1803.01377
(with J. East and M. Gadouleau) ‘Structural aspects of semigroups based on digraphs’, Algebraic Combinatorics 2 (2019) 711-733; https://doi.org/10.5802/alco.56 https://arxiv.org/abs/1704.00937
(with Jillian D. McPhee and Martyn Quick) ‘Automorphism groups of linearly ordered structures and endomorphisms of the ordered set (ℚ,≤) of rational numbers’, Q. J. Math. 70 (2019) 171–194; https://doi.org/10.1093/qmath/hay043 https://arxiv.org/abs/1607.03655
(with Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin) ‘Enumeration of idempotents in planar diagram monoids’, Journal of Algebra 522 (2019) 351-385; https://doi.org/10.1016/j.jalgebra.2018.11.014 https://arxiv.org/abs/1507.04838
(with J. East, A. Egri-Nagy, and Y. Péresse) ‘Computing finite semigroups’, J. Symbolic Comput. 92 (2019) 110-155; https://doi.org/10.1016/j.jsc.2018.01.002 https://arxiv.org/abs/1510.01868
(with J. East, N. Ruskuc, and M. Torpey) ‘Congruence lattices of finite diagram monoids’, Adv. Math. 333 (2018) 931-1003; https://doi.org/10.1016/j.aim.2018.05.016 https://arxiv.org/abs/1709.00142
(with C. Donoven and W. Wilson) ‘Computing maximal subsemigroups of a finite semigroup’, Journal of Algebra 505 (2018) 559-596; https://doi.org/10.1016/j.jalgebra.2018.01.044 https://arxiv.org/abs/1606.05583
(with J. East, J. Kumar, and W. Wilson) ‘Maximal subsemigroups of finite transformation and partition monoids’, Journal of Algebra 504 (2018) 176-216; https://doi.org/10.1016/j.jalgebra.2018.01.048 https://arxiv.org/abs/1706.04967
(with J. Jonušas and M. Pfeiffer) ‘Two variants of the Froiduire-Pin Algorithm for finite semigroups’, Portugal. Math. 74 (2018) 173-200; http://dx.doi.org/10.4171/PM/2001 https://arxiv.org/abs/1704.04084
(with J. East and A. Egri-Nagy) ‘Enumerating transformation semigroups’, Semigroup Forum 95 (2017) 109-125; http://dx.doi.org/10.1007/s00233-017-9869-2 https://arxiv.org/abs/1403.0274
(with J. Jonušas) ‘Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs’, Forum Mathematicum 29 (2017) 905-940; http://dx.doi.org/10.1515/forum-2016-0056 https://arxiv.org/abs/1602.05766
(with P. Cameron, M. Gadouleau, and Y. Péresse) ‘Chains of subsemigroups’, Israel Journal of Mathematics 220 (2017) 479–508; http://dx.doi.org/10.1007/s11856-017-1523-x https://arxiv.org/abs/1501.06394
(with P. J. Cameron, A. Castillo-Ramirez, and M. Gadouleau) ‘Lengths of words in transformation semigroups generated by digraphs’, Journal of Algebraic Combinatorics 45 (2017) 149–170; http://dx.doi.org/10.1007/s10801-016-0703-9 https://arxiv.org/abs/1602.00935
(with S. Craik, R. Gray, V. Kilibada, and N. Ruskuc) ‘Ends of semigroups’, Semigroup Forum 93 (2016) 330-346; http://dx.doi.org/10.1007/s00233-016-9814-9 https://arxiv.org/abs/1409.1044
(with J. Hyde, J. Jonušas, and Y. Péresse) ‘Universal sequences for the order-automorphisms of the rationals’, J. Lond. Math. Soc. 94 (2016) 21-37; http://dx.doi.org/10.1112/jlms/jdw015 https://arxiv.org/abs/1401.7823
(with Z. Mesyan) ‘The Structure of a Graph Inverse Semigroup’, Semigroup Forum 93 (2016) 111-130; http://dx.doi.org/10.1007/s00233-016-9793-x https://arxiv.org/abs/1409.4380
(with Igor Dolinka, Robert D. Gray, Jillian D. McPhee, and Martyn Quick) ‘Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph’, Math. Proc. Cambridge Philos. Soc. 160 (2016) 437-462; http://dx.doi.org/10.1017/S030500411500078X https://arxiv.org/abs/1408.4107
(with Z. Mesyan, M. Morayne, and Y. Péresse) ‘Topological graph inverse semigroups’, Topology and its applications 208 (2016) 106-126; http://dx.doi.org/10.1016/j.topol.2016.05.012 https://arxiv.org/abs/1306.5388
(with I. Dolinka and J. East) ‘Idempotent rank in the endomoprhism monoid of a non-uniform partition’, Bull. Aust. Math. Soc. 93 (2016) 73-91; http://dx.doi.org/10.1017/s0004972715000751 https://arxiv.org/abs/1504.02520
(with Araújo, W. Bentz, and C. Schneider) ‘The rank of the semigroup of transformations stabilising a partition of a finite set’, Math. Proc. Cambridge Philos. Soc. 159 (2015) 339–353; http://dx.doi.org/10.1017/S0305004115000389 https://arxiv.org/abs/1404.1598
(with J. East and Y. Péresse) ‘Maximal subsemigroups of the semigroup of all mappings on an infinite set’, Trans. Amer. Math. Soc. 367 (2015) 1911-1944; http://dx.doi.org/10.1090/S0002-9947-2014-06110-2 https://arxiv.org/abs/1104.2011
(with N. Abu-Ghazalh, Y. Péresse, and N. Ruskuc) ‘A classification of disjoint unions of two or three copies of the free monogenic semigroup’, Semigroup Forum 91 (2015) 53-61; http://dx.doi.org/10.1007/s00233-014-9638-4 https://arxiv.org/abs/1312.5518
(with Julius Jonušas) ‘A finite interval in the subsemigroup lattice of the full transformation monoid’, Semigroup Forum 89 (2014) 183-198; http://dx.doi.org/10.1007/s00233-013-9537-0 https://arxiv.org/abs/1301.2171
(with R. Gray, V. Maltcev, and N. Ruskuc) ‘Ideals and finiteness conditions for subsemigroups’, Glasgow Math. J. 56 (2014) 65-86; http://dx.doi.org/10.1017/S0017089513000086 https://arxiv.org/abs/1204.6602
(with A. Egri-Nagy and C. L. Nehaniv) ‘SgpDec: Cascade (De)Compositions of Finite Transformation Semigroups and Permutation Groups’, in Mathematical Software – ICMS 2014 Volume 8592 of the series Lecture Notes in Computer Science pp 75-82 http://dx.doi.org/10.1007/978-3-662-44199-2_13
(with J. Araújo, P. J. Cameron, and M. Neunhoeffer) ‘A classification of normalizing groups’, Journal of Algebra 373 (2013) 481-490; http://dx.doi.org/10.1016/j.jalgebra.2012.08.033 https://arxiv.org/abs/1205.0450
(with Y. Péresse) ‘Sierpiński's rank for groups and semigroups’, Wiad. Mat. 48 (2012) 209-215; a special issue of Wiadomości Matematyczne on the occasion of the European Congress of Mathematics, Krakow, Poland, 2012.
(with A. Distler) ’ The number of nilpotent semigroups of degree 3 ‘, Electronic Journal of Combinatorics 19 (2012) P51; https://arxiv.org/abs/1201.3529
(with Z. Mesyan, M. Morayne, and Y. Péresse) ‘The Bergman-Shelah preorder on transformation semigroups’, Mathematical Logic Quarterly 58 (2012) 424-433; http://dx.doi.org/10.1002/malq.201200002 https://arxiv.org/abs/1109.2706
(with Attila Egri-Nagy and Chrystopher Nehaniv), ‘Algorithms for the Efficient Calculation of the Holonomy Decomposition’, in P. Dömösi & I. Szabolcs (Eds.): Automata and Formal Languages, 13th International Conference, AFL 2011, Debrecen, Hungary, August 17-22, 2011, Proceedings, pp. 389-390, 2011 (extended abstract), ISBN 978-615-5097-19-5.
(with J. Araújo, V. H. Fernandes, M. M. Jesus, and V. Maltcev) ‘Automorphisms of partial endomorphism semigroups’, Publ. Math. Debrecen 79 (2011) 23-39; http://dx.doi.org/10.5486/PMD.2011.4703
(with J. Araújo and C. Schneider) ‘Groups that together with any transformation generate regular semigroups or idempotent generated semigroups’, Journal of Algebra 343 (2011) 93-106; http://dx.doi.org/10.1016/j.jalgebra.2011.07.002 ; see also the paper's webpage
(with Y. Péresse) ‘Generating countable sets of surjective functions’, Fund. Math. 213 (2011) 67-93; http://dx.doi.org/10.4064/fm213-1-4
(with U. Darji) ‘Approximation of automorphisms of the rationals and the random graph’, Journal of Group Theory 14 (2011) 361-388; http://dx.doi.org/10.1515/JGT.2010.057
(with M. Morayne and Y. Péresse) ‘Generating the infinite symmetric group using a closed subgroup and the least number of other elements’, Proc. Amer. Math. Soc. 139 (2011) 401-405; http://dx.doi.org/10.1090/S0002-9939-2010-10694-9
(with J. Cichoń, M. Morayne, and Y. Péresse) ‘Relative ranks of Lipschitz mappings on countable discrete metric spaces’, Topology and its Applications 158 (2011) 412-423; http://dx.doi.org/10.1016/j.topol.2010.11.020
(with M. Morayne, Y. Péresse, and M. Quick) ‘Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances’, Annals of Pure and Applied Logic 161 (2010) 1471-1485; http://dx.doi.org/10.1016/j.apal.2010.05.001
(with V. H. Fernandes, M. M. Jesus, and V. Maltcev) ‘Endomorphisms of the semigroup of order-preserving mappings’, Semigroup Forum 81 (2010) 277-285; http://dx.doi.org/10.1007/s00233-010-9220-7
(with J. Araújo, P. von Bünau and M. Neunhoeffer) ‘Computing automorphisms of semigroups’, J. Symbolic Comput. 45 (2010) 373-392; http://dx.doi.org/10.1016/j.jsc.2009.10.001
(with V. Maltcev and N. Ruskuc) ‘The Bergman property for semigroups’, J. Lond. Math. Soc. 80 (2009) 212-232; http://dx.doi.org/10.1112/jlms/jdp025
(with S. Huczynska and C. M. Roney-Dougal eds.) ‘Surveys in Combinatorics’, 2009, LMS Lecture Note Series, 365, Cambridge University Press 2009. http://dx.doi.org/10.1017/CBO9781107325975
(with U. Darji) ‘Highly transitive subgroups of the symmetric group on the natural numbers’, Colloq. Math. 112 (2008) 163-173; http://dx.doi.org/10.4064/cm112-1-9
(with R. Gray) ‘Largest subsemigroups of the full transformation monoid’, Discrete Math. 308 (2008) 4801-4810; http://dx.doi.org/10.1016/j.disc.2007.08.075
(with Y. Péresse and M. Quick) ‘Generating sequences of functions’, Q. J. Math. 58 (2007) 71-79; http://dx.doi.org/10.1093/qmath/hal011
(with J. M. André and V. H. Fernandes) ‘Largest 2-generated subsemigroups of the symmetric inverse semigroup’, Proc. Edinburgh Math. Soc. 50 (2007) 551-561; http://dx.doi.org/10.1017/S0013091505001598
(with J. Araújo) ‘Relative ranks in the monoid of endomorphisms of independence algebras of infinite rank’, Monatsh. Math. 151 (2007) 1-10; http://dx.doi.org/10.1007/s00605-006-0433-5
(with J. Cichoń and M. Morayne) ‘Generating continuous mappings with Lipschitz mappings’, Trans. Amer. Math. Soc. 359 (2007) 2059-2074; http://dx.doi.org/10.1090/S0002-9947-06-04026-8
(with P. M. Higgins, M. Morayne and N. Ruskuc) ‘Rank properties of endomorphisms of infinite partially ordered sets’, Bull. London Math. Soc. 38 (2006) 177-191; http://dx.doi.org/10.1112/S0024609305018138
(with I. Levi) ‘On rank properties of endomorphisms of finite circular orders’, Comm. Algebra 34 (2006) 1237-1250; http://dx.doi.org/10.1080/00927870500454091
(with J. Araújo) ‘An elementary proof that every singular nxn matrix is a product of idempotents’, Amer. Math. Monthly 112 (2005) 641-645; http://dx.doi.org/10.2307/30037549
‘Turan's graph theorem and maximum independent sets in Brandt semigroups’, Proceedings of the Workshop Semigroups and Languages (Lisbon, 2002), 151-163 World Sci. Publishing, 2004; http://dx.doi.org/10.1142/9789812702616_0009
(with J. Araújo and N. Silva) ‘On generating countable sets of endomorphisms’, Algebra Universalis 50 (2003) 61-67; http://dx.doi.org/10.1007/s00012-003-1809-1
(with J. M. Howie, P. M. Higgins and N. Ruskuc) ‘Countable versus uncountable rank in infinite semigroups of transformations and relations’, Proc. Edinburgh Math. Soc. 46 (2003) 531-544; http://dx.doi.org/10.1017/S0013091502000974
(with P. M. Higgins and N. Ruskuc) ‘Generating infinite full transformation semigroups using order preserving maps’, Glasgow Math. J. 45 (2003) 557-566; http://dx.doi.org/10.1017/S0017089503001460
(with C. M. Campbell and N. Ruskuc) ‘On the efficiency and deficiency of Rees matrix semigroups’, Semigroups, algorithms, automata and languages (Coimbra, 2001), 331-339, World Sci. Publishing, 2002; http://dx.doi.org/10.1142/9789812776884_0012
(with C. M. Campbell and N. Ruskuc) ‘Comparing semigroup and monoid presentations for finite monoids’, Months. Math. 134 (2002) 287-293; http://dx.doi.org/10.1007/s605-002-8263-7
(with C. M. Campbell and N. Ruskuc) ‘On defining groups efficiently without inverses’, Math. Proc. Cambridge Philos. Soc. 133 (2002) 31-36; http://dx.doi.org/10.1017/S0305004102005959

Preprints

(with J. East and J. Jonušas) ‘Generating the monoid of 2×2 matrices over max-plus and min-plus semirings’, preprint; https://arxiv.org/abs/2009.10372
(with Z. Mesyan and Y. H. Péresse) ‘Topological transformation monoids’, preprint; https://arxiv.org/abs/1809.04590

Software

Mon, 01 Jan 0001 00:00:00 +0000

Digraphs deposited GAP package; https://doi.org/10.5281/zenodo.17211546
Semigroups deposited GAP package; http://dx.doi.org/10.5281/zenodo.592893
SgpDec - Hierarchical composition and decomposition of finite permutation groups and transformation semigroups (by A. Egri-Nagy and C. Nehaniv with some contributions by me) deposited GAP package.
(with A. Distler) Smallsemi - a library of small semigroups deposited GAP package.
(with R. Cirpons and J. Edwards) libsemigroups C++ library for semigroups and monoids; https://doi.org/10.5281/zenodo.1437752
HPCombi Fast combinatorics in C++ using SSE/AVX instruction sets, (by F. Hivert with some contributions by me).
(with R. Cirpons, J. Edwards, M. Tsalakou, and M. Whyte) libsemigroups_pybind11 python package for using libsemigroups via pybind11; https://doi.org/10.5281/zenodo.17234506.
(with M. Tsalakou)step_hen a rudimentary python implementation of Stephen’s procedure for inverse semigroups, 2021.
todd_coxeter.py reference implementation of the two main variants of the Todd-Coxeter algorithm for semigroups and monoids, 2021.

About on J. D. Mitchell