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
[Full thesis]
Abstract
The use of graph theory has become widespread in the algebraic theory of semigroups.  In this context, the graph is mainly used as a visual aid to make presentation clearer and the problems more manageable. Central to such approaches is the Cayley graph of a semigroup. There are also many variations on the idea of the Cayley graph, usually special kinds of subgraph or factor graph, that have become important in their own right. Examples include Schutzenberger graphs, Schreier coset graphs and Van Kampen diagrams (for groups), Munn trees, Adian graphs, Squier complexes, semigroup diagrams, and Graham-Houghton graphs of completely 0-simple semigroups. Also, the representation of elements in finite transformation semigroups as digraphs has proved a useful tool. This thesis consists of several problems in the theory of semigroups with the common feature that they are all best attacked using graph theory. The thesis has two parts. In the first part combinatorial questions for finite semigroups and monoids are considered. In particular, we look at the problem of finding minimal generating sets for various endomorphism monoids and their ideals. This is achieved by detailed analysis of the generating sets of completely 0-simple semigroups. This investigation is carried out using the Graham-Houghton bipartite graph representation. The second part of the thesis is about infinite semigroup theory, and in particular some problems in the theory of semigroup presentations. We consider the general problem of finding presentations for subsemigroups of finitely presented semigroups. Sufficient conditions are introduced that force such a subsemigroup to be finitely presented. These conditions are given in terms of the position of the subsemigroup in the parent semigroup's left and right Cayley graphs.