Publications

1. Amenability and geometry of semigroups (with M. Kambites)
Transactions of the American Mathematical Society (to appear)
2. On regularity and the word problem for free idempotent generated semigroups (with I. Dolinka and N. Ruskuc)
Proceedings of the London Mathematical Society Vol. 114, 2017, pp. 401-432.
3. Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals (with J. East)
Journal of Combinatorial Theory, Series A Vol. 146, 2017, pp. 63-128.
4. Motzkin monoids and partial Brauer monoids (with I. Dolinka and J. East)
Journal of Algebra Vol. 471, 2017, pp. 251-298.
5. Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph (with I. Dolinka, J. D. McPhee, J. D. Mitchell, and M. Quick)
Mathematical Proceedings of the Cambridge Philosophical Society Vol. 160, 2016, pp. 437-462.
6. Ends of Semigroups (with S. Craik, V. Kilibarda, J. D. Mitchell, and N. Ruskuc)
Semigroup Forum Vol. 93, 2016, pp. 330-346.
7. Rewriting systems and biautomatic structures for Chinese, hypoplactic, and Sylvester monoids (with A. J. Cain and A. Malheiro)
International Journal of Algebra and Computation Vol. 25, 2015, no. 1-2.
8. A strong geometric hyperbolicity property for directed graphs and monoids (with M. Kambites )
Journal of Algebra Vol. 420, 2014, pp. 373-401.
9. The minimal number of generators of a finite semigroup
Semigroup Forum Vol. 89, 2014, pp. 135-154.
10. Homotopy Bases and Finite Derivation Type for Subgroups of Monoids (with A. Malheiro)
Journal of Algebra Vol. 410, 2014, pp. 53-84.
11. Countable locally 2-arc-transitive bipartite graphs (with J. K. Truss)
European Journal of Combinatorics Vol. 39, 2014, pp. 122-147.
12. Maximal subgroups of free idempotent generated semigroups over the full linear monoid (with I. Dolinka)
Transactions of the American Mathematical Society Vol. 366, 2014, pp. 419-455.
13. Ideals and finiteness conditions for subsemigroups (with V. Maltcev, J. D. Mitchell and N. Ruskuc)
Glasgow Mathematical Journal Vol. 56, 2014, pp. 65-86.
14. Quasi-isometry and finite presentations for left cancellative monoids (with M. Kambites)
International Journal of Algebra and Computation Vol. 23, 2013, pp. 1099-1114.
15. Groups Acting on Semimetric Spaces and Quasi-isometries of Monoids (with M. Kambites)
Transactions of the American Mathematical Society Vol. 365, 2013, pp. 555-578.
16. Homotopy bases and finite derivation type for Schützenberger groups of monoids (with A. Malheiro and S. J. Pride)
Journal of Symbolic Computation Vol. 50, 2013, pp. 50-78.
17. Maximal subgroups of free idempotent generated semigroups over the full transformation monoid (with N. Ruskuc)
Proceedings of the London Mathematical Society Vol. 104, 2012, pp. 997-1018.
18. Set-homogeneous Directed Graphs (with D. Macpherson, C. Praeger and G. Royle)
Journal of Combinatorial Theory, Series B Vol. 102, 2012, pp. 474--520.
19. On Maximal Subgroups of Free Idempotent Generated Semigroups (with N. Ruskuc)
Israel Journal of Mathematics Vol. 189, 2012, pp. 147-176.
20. A Svarc-Milnor Lemma for Monoids Acting by Isometric Embeddings (with M. Kambites)
International Journal of Algebra and Computation Vol. 21, 2011, pp. 1135-1147.
21. Presentations of Inverse Semigroups their Kernels and Extensions (with C. Carvalho and N. Ruskuc)
Journal of the Australian Mathematical Society Vol. 90, 2011, pp. 289-316.
22. Homological Finiteness Properties of Monoids, their Ideals and Maximal Subgroups (with S. J. Pride)
Journal of Pure and Applied Algebra Vol. 215, 2011, pp. 3005-3024.
23. On properties not inherited by monoids from their Schützenberger groups (with A. Malheiro and S. J. Pride)
Information and Computation Vol. 209, 2011, pp. 1120-1134.
24. Generators and Relations for Subsemigroups via Boundaries in Cayley Graphs (with N. Ruskuc)
Journal of Pure and Applied Algebra Vol. 215, 2011, pp. 2761-2779.
25. Locally-finite Connected-homogeneous Digraphs (with R. Moller)
Discrete Mathematics Vol. 311, 2011, pp. 1497-1517.
26. Finite Complete Rewriting Systems for Regular Semigroups (with A. Malheiro)
Theoretical Computer Science Vol. 412, 2011, pp. 654-661.
27. Countable connected-homogeneous graphs (with D. Macpherson)
Journal of Combinatorial Theory, Series B Vol. 100, 2010, pp. 97-118.
28. Cycle-free Partial Orders and Ends of Graphs (with J. K. Truss)
Mathematical Proceedings of the Cambridge Philosophical Society Vol. 146, 2009, pp. 535-550.
29. On Residual Finiteness of Direct Products of Algebraic Systems (with N. Ruskuc)
Monatshefte fur Mathematik Vol. 158, 2009, pp. 63-69.
30. k-CS-transitive Infinite Graphs
Journal of Combinatorial Theory, Series B Vol. 99, 2009, pp. 378-398.
31. Green Index and Finiteness Conditions for Semigroups (with N. Ruskuc)
Journal of Algebra Vol. 320, 2008, pp. 3145-3164.
32. Construction of Some Countable One-arc Transitive Bipartite Graphs (with J. K. Truss)
Discrete Mathematics Vol. 308, 2008, pp. 6392-6405.
33. Hall's Condition and Idempotent Rank of Ideals of Endomorphism Monoids
Proceedings of the Edinburgh Mathematical Society Vol. 51, 2008, pp. 1-16.
34. Largest Subsemigroups of the Full Transformation Monoid (with J. D. Mitchell)
Discrete Mathematics Vol. 308, 2008, pp. 4801-4810.
35. Idempotent Rank in Endomorphism Monoids of Finite Independence Algebras
Proceedings of the Royal Society of Edinburgh: Section A Mathematics Vol. 137A, 2007, pp. 303-331.
36. Generating Sets of Completely 0-Simple Semigroups (with N. Ruskuc)
Communications in Algebra Vol. 33, 2005, pp. 4657-4678.
Submitted papers
Thesis
[Preprint from arXiv]
Abstract
The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. We prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$ (corresponding to the $q$-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time.