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] [Final published journal version]
Abstract
We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham--Houghton graphs. We show that each proper ideal of the partition monoid $P_n$ is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of $P_n$, which coincides with the singular part of $P_n$. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley--Lieb algebras), allow one to recover formulae for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulae for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of $\{0,1\}$-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.