Books and articles

Books

Yves-Marie André, Essay on Beauty: Translation and annotations by Alan J. Cain.

Porto, 2010
Nine Chapters on the Semigroup Art: Lecture notes for a tour through semigroups: Porto & Lisbon, 2012–25
G. H. Hardy, An Annotated Mathematician’s Apology: Annotations and commentary by Alan J. Cain.

Lisbon, 2019
Form & Number: A History of Mathematical Beauty: Lisbon, 2024

Articles

‘Subsemigroups of virtually free groups’: (with E. F. Robertson & N. Ruškuc)

Mathematical Proceedings of the Cambridge Philosophical Society 141, no. 1 (July 2006), pp. 57–66.

DOI: 10.1017/s0305004106009236
‘Subsemigroups of groups’: (with E. F. Robertson & N. Ruškuc)

Journal of Group Theory 9, no. 3 (May 2006), pp. 397–426.

DOI: 10.1515/jgt.2006.027
‘A group-embeddable non-automatic semigroup whose universal group is automatic’: Glasgow Mathematical Journal 48, no. 2 (2006), pp. 337–342.

DOI: 10.1017/s0017089506003107
‘Cancellativity is undecidable for automatic semigroups’: The Quarterly Journal of Mathematics 57, no. 3 (September 2006), pp. 285–295.

DOI: 10.1093/qmath/hai023
‘Malcev presentations for subsemigroups of groups — a survey’: In: Groups St Andrews 2005, vol. 1, no. 339 in London Mathematical Society Lecture Note Series ( , 2007), pp. 256–268.

DOI: 10.1017/CBO9780511721212.018
‘Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions’: (with E. F. Robertson & N. Ruškuc)

Journal of the Australian Mathematical Society 84, no. 1 (February 2008), pp. 39–61.

DOI: 10.1017/s1446788708000086
‘Automatic presentations for cancellative semigroups’: (with G. Oliver, N. Ruškuc & R. M. Thomas)

In: Language and Automata Theory and Applications, no. 5196 in Lecture Notes in Computer Science ( , 2008), pp. 149–159.

DOI: 10.1007/978-3-540-88282-4_15
‘Malcev presentations for subsemigroups of direct products of coherent groups’: Journal of Pure and Applied Algebra 213, no. 6 (2009), pp. 977–990.

DOI: 10.1016/j.jpaa.2008.10.006
‘Automaton semigroups’: Theoretical Computer Science 410, no. 47-49 (2009), pp. 5022–5038.

DOI: 10.1016/j.tcs.2009.07.054
‘Automatic presentations for semigroups’: (with G. Oliver, N. Ruškuc & R. M. Thomas)

Information and Computation 207, no. 11 (2009), pp. 1156–1168.

DOI: 10.1016/j.ic.2009.02.005
‘Decision problems for finitely presented and one-relation semigroups and monoids’: (with V. Maltcev)

International Journal of Algebra and Computation 19, no. 6 (September 2009), pp. 747–770.

DOI: 10.1142/s0218196709005366
‘Monoids presented by rewriting systems and automatic structures for their submonoids’: International Journal of Algebra and Computation 19, no. 6 (September 2009), pp. 771–790.

DOI: 10.1142/s0218196709005317
‘Automatic semigroups and Bruck–Reilly extensions’: Acta Mathematica Hungarica 126, no. 1-2 (2010), pp. 1–15.

DOI: 10.1007/s10474-009-8063-8
‘Automatic presentations and semigroup constructions’: (with G. Oliver, N. Ruškuc & R. M. Thomas)

Theory of Computing Systems 47, no. 2 (2010), pp. 568–592.

DOI: 10.1007/s00224-009-9216-4
‘Deus ex machina and the aesthetics of proof’: The Mathematical Intelligencer 32, no. 3 (September 2010), pp. 7–11.

DOI: 10.1007/s00283-010-9141-z
‘Unary FA-presentable semigroups’: (with N. Ruškuc & R. M. Thomas)

International Journal of Algebra and Computation 22, no. 4 (June 2012).

DOI: 10.1142/S0218196712500385
‘Green index in semigroups’: (with R. Gray & N. Ruškuc)

Semigroup Forum 85, no. 3 (2012), pp. 448–476.

DOI: 10.1007/s00233-012-9406-2
‘Context-free rewriting systems and word-hyperbolic structures with uniqueness’: (with V. Maltcev)

International Journal of Algebra and Computation 22, no. 7 (November 2012).

DOI: 10.1142/S0218196712500610
‘Hyperbolicity of monoids presented by confluent monadic rewriting systems’: Beiträge zur Algebra und Geometrie 54, no. 2 (October 2013), pp. 593–608.

DOI: 10.1007/s13366-012-0116-4
‘Automatic structures for subsemigroups of Baumslag–Solitar semigroups’: Semigroup Forum 87, no. 3 (2013), pp. 537–552.

DOI: 10.1007/s00233-013-9490-y
‘Finitely presented monoids with linear Dehn function need not have regular cross-sections’: (with V. Maltcev)

Semigroup Forum 88, no. 2 (2014), pp. 300–315.

DOI: 10.1007/s00233-013-9531-6
‘Markov semigroups, monoids, and groups’: (with V. Maltcev)

International Journal of Algebra and Computation 24, no. 5 (August 2014), pp. 609–653.

DOI: 10.1142/S021819671450026X
‘Subalgebras of FA-presentable algebras’: (with N. Ruškuc)

Algebra Universalis 72, no. 2 (October 2014), pp. 101–123.

DOI: 10.1007/s00012-014-0293-0
‘Hopfian and co-hopfian subsemigroups and extensions’: (with V. Maltcev)

Demonstratio Mathematica 47, no. 4 (2014), pp. 791–804.

DOI: 10.2478/dema-2014-0064
‘Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids’: (with R. D. Gray & A. Malheiro)

Journal of Algebra 423 (February 2015), pp. 37–53.

DOI: 10.1016/j.jalgebra.2014.09.037
‘A simple non-bisimple congruence-free finitely presented monoid’: (with V. Maltcev)

Semigroup Forum 90, no. 1 (February 2015), pp. 184–188.

DOI: 10.1007/s00233-014-9607-y
‘Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids’: (with R. D. Gray & A. Malheiro)

International Journal of Algebra and Computation 25, no. 1-2 (February–March 2015), pp. 51–80.

DOI: 10.1142/S0218196715400044
‘Deciding conjugacy in sylvester monoids and other homogeneous monoids’: (with A. Malheiro)

International Journal of Algebra and Computation 25, no. 5 (August 2015), pp. 899–915.

DOI: 10.1142/S0218196715500241
‘A countable family of finitely presented infinite congruence-free monoids’: (with F. Al-Kharousi, V. Maltcev & A. Umar)

Acta Scientiarum Mathematicarum 81, no. 3-4 (2015), pp. 437–445.

DOI: 10.14232/actasm-013-028-z
‘Decision problems for word-hyperbolic semigroups’: (with M. Pfeiffer)

Journal of Algebra 465 (November 2016), pp. 287–321.

DOI: 10.1016/j.jalgebra.2016.07.007
‘Crystallizing the hypoplactic monoid’: (with A. Malheiro)

Journal of Algebraic Combinatorics 45, no. 2 (March 2017), pp. 475–524.

DOI: 10.1007/s10801-016-0714-6
‘Growths of endomorphisms of finitely generated semigroups’: (with V. Maltcev)

Journal of the Australian Mathematical Society 102, no. 2 (April 2017), pp. 163–184.

DOI: 10.1017/S1446788716000264
‘Automaton semigroups’: (with T. Brough)

Theoretical Computer Science 674 (April 2017), pp. 1–15.

DOI: 10.1016/j.tcs.2017.02.003
‘On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids’: (with R. D. Gray & A. Malheiro)

Information and Computation 255 (August 2017), pp. 68–93.

DOI: 10.1016/j.ic.2017.05.003
‘Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, and related monoids’: (with A. Malheiro)

In: Combinatorics on Words, no. 10432 in Lecture Notes in Computer Science ( , 2017), pp. 190–202.

DOI: 10.1007/978-3-319-66396-8_18
‘Crystals and trees’: (with A. Malheiro)

Journal of Algebra 502 (May 2018), pp. 347–381.

DOI: 10.1016/j.jalgebra.2018.01.036
‘Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids’: (with A. Malheiro)

Electronic Journal of Combinatorics 25, no. 3 (2018).

URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i3p30
‘Crystal monoids & crystal bases’: (with R. D. Gray & A. Malheiro)

Journal of Combinatorial Theory, Series A 162 (February 2019), pp. 406–466.

DOI: 10.1016/j.jcta.2018.11.010
‘Visual thinking and simplicity of proof’: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (11 March 2019), art. 20180032.

DOI: 10.1098/rsta.2018.0032
‘The monoids of the patience sorting algorithm’: (with A. Malheiro & F. Silva)

International Journal of Algebra and Computation 29, no. 1 (2019), pp. 85–125.

DOI: 10.1142/S0218196718500649
‘Combinatorics of cyclic shifts in plactic, hypoplactic, sylvester, Baxter, and related monoids’: (with A. Malheiro)

Journal of Algebra 535 (1 October 2019), pp. 159–224.

DOI: 10.1016/j.jalgebra.2019.06.025
‘Context-free word problem semigroups’: (with T. Brough & M. Pfeiffer)

In: Developments in Language Theory 2019, no. 11647 in Lecture Notes in Computer Science ( , 2019), pp. 292–305.

DOI: 10.1007/978-3-030-24886-4_22
‘Combinatorics of patience sorting monoids’: (with A. Malheiro & F. Silva)

Discrete Mathematics 342, no. 9 (September 2019), pp. 2590–2611.

DOI: 10.1016/j.disc.2019.05.022
‘Two applications of monoid actions to cross-sections’: (with T. Brough & V. Maltcev)

Communications in Algebra 48, no. 5 (May 2020), pp. 1894–1903.

DOI: 10.1080/00927872.2019.1710160
‘A language hierarchy of binary relations’: (with T. Brough)

Information and Computation 275 (December 2020).

DOI: 10.1016/j.ic.2020.104607
‘Reconstructing Young tableaux’: (with E. Lehtonen)

Journal of Combinatorial Theory, Series A 187 (April 2022), art. 105578.

DOI: 10.1016/j.jcta.2021.105578
‘Identities and bases in the sylvester and baxter monoids’: (with A. Malheiro & D. Ribeiro)

Journal of Algebraic Combinatorics 58, no. 4 (December 2023), pp. 933–966.

DOI: 10.1007/s10801-022-01202-6
‘Commutative nilpotent transformation semigroups’: (with A. Malheiro & T. Paulista)

Semigroup Forum 109, no. 1 (August 2024), pp. 60–75.

DOI: 10.1007/s00233-024-10444-8
‘marginalia at work’: TUGboat 46, no. 1 (2025), pp. 49–53.

DOI: 10.47397/tb/46-1/tb142cain-marginalia
‘Experiences in typesetting a complex book’: TUGboat 46, no. 1 (2025), pp. 40–48.

DOI: 10.47397/tb/46-1/tb142cain-setting-form

Preprints

‘Unary FA-presentable binary relations’: (with N. Ruškuc)

Preprint, March 2013.

arXiv: 1303.0214
‘For a few elements more’: (with V. Maltcev)

Preprint, July 2013.

arXiv: 1307.8259
‘A note on identities in plactic monoids and monoids of upper-triangular tropical matrices’: (with G. Klein, Ł. Kubat, A. Malheiro & J. Okniński)

Preprint, May 2017.

arXiv: 1705.04596
‘Conjugacy in patience sorting monoids’: (with A. Malheiro & F. Silva)

Preprint, March 2018.

arXiv: 1803.00361
‘Every numerical semigroup arises as an automaton monoid’: (with T. M. Brough & J. P. Wächter)

Preprint, 22 March 2023.

arXiv: 2303.12715
‘Quasi-crystals for arbitrary root systems and associated generalizations of the hypoplactic monoid’: (with R. P. Guilherme & A. Malheiro)

Preprint, 3 January 2023.

arXiv: 2301.00271
‘Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions’: (with A. Malheiro, F. Rodrigues & I. Rodrigues)

Preprint, 26 September 2023.

arXiv: 2309.14887
‘A local characterization of quasi-crystal graphs ’: (with A. Malheiro, F. Rodrigues & I. Rodrigues)

Preprint, 26 September 2023.

arXiv: 2309.14898