Preprints

  1. Mosaics. In these notes, I show how to construct all strongly E*-unitary inverse semigroups from groups acting on multiplicative graphs. This work was motivated by my own work with Johannes Kellendonk on tiling semigroups and by some unpublished notes by Ben Steinberg. A crucial step in the proof was provided by John Fountain. At this stage, I have not submitted this work; it really needs to be developed further. pdf-file.
  2. Constructing ordered groupoids.  This paper is equivalent to (in some sense) the theory developed in my paper `Constructing ordered groupoids from category actions'; it can be regarded as a co-ordinate-free version of that construction. I prove that every ordered groupoid can be constructed from a category acting suitably on a (combinatorial) groupoid. The action by the category defines a preorder on the groupoid that leads to an equivalence relation on the groupoid and an order on the quotient. Appeared in the Cahiers. pdf-file.
  3. A correspondence between balanced varieties and inverse monoids.  The construction in preprint (2) arose from trying to understand Dehornoy's construction of the geometric or structural monoid associated with a balanced variety. In this paper, I repay the debt to Dehornoy's work by showing that it is part of a more general framework generalising some ideas by Meakin and Sapir. With each balanced variety of $\Omega$-algebras, I associate an inverse monoid which is also an $\Omega$-algebra in a nice way: what I call an `inverse $\Omega$-algebra`. Dehornoy's geometric monoid is a special kind of inverse $\Omega$-algebra of mine. Appeared in IJAC. ps-file  pdf-file. 
  4. Representations of the Thompson group F. I show how concrete versions of the group F can be obtained from representations of the polycyclic monoid on two generators. Essentially, elements of the group F can be represented as disjoint unions of elements of the polycyclic monoid. The construction owes a lot to the work of Peter Hines. In his thesis, and reported in my book `Inverse semigroups', he developed in a mathematical setting some ideas of Girard in GOIII. Although these ideas were interesting, we did not originally know quite what to make of them. The connection with F came about through reading a paper by Dehornoy.  pdf-file.
  5. Finite automata. This is a tutorial on the theory of finite transducers written at the graduate or advanced undergraduate level. It will appear as an article in the `Handbook of networked and embedded control systems' edited by D. Hristu-Varsakelis and W. S. Levine. Appeared. pdf-file.
  6. In McAlister's footsteps: a random ramble around the P-theorem. (with Stuart Margolis) Accepted for publication. ps-file pdf-file. 
  7. A class of subgroups of Thompson's group V. This is an application of the theory developed in preprint (3). Accepted by SF. ps-file pdf-file.  
  8. Orthogonal completions of the polycyclic monoid. This defines orthogonal completions of arbitrary inverse semigroups with zero, and then computes the orthogonal completion of the polycyclic monoid on n generators. A connection with a paper by Birget is made and so with the Thompson groups V_{n,1}. Accepted by CAps-file  pdf-file.
  9. The polycyclic monoids P_{n} and the Thompson groups V_{n,1}. We construct a quotient of the monoid constructed in (8) whose group of units is the Thompson group V_{n}. This paper uses some ideas to be found in a paper by Birget but goes in a different direction with them. Accepted by CA. ps-file pdf-file.  
  10. A monoid associated with a self-similar group action.  We prove that there is a correspondence between self-similar group actions and the class of left cancellative right hereditary monoids satisfying the dedekind height property.  ps-file pdf-file.
  11. Self-similar group actions and a paper of David Rees.  We show that Rees' 1948 paper on the structure of left cancellative monoids stands at the threshold of the theory of self-similar group actions; this paper can be seen as an introduction to my longer paper (10) above.  ps-file pdf-file.
  12. Zappa-Szep products of free monoids and groups.  We characterise abstractly the Zappa-Szep products of free monoids and groups, and so complete an argument linking faithful self-similar group actions and a class of fundamental left cancellative monoids.  ps-file. pdf-file.
  13. A correspondence between a class of monoids and self-similar group actions I. Replaces (10), (11) and (12) above. ps-file. pdf-file.
  14. Semigroups related to subshifts of graphs. A generalisation of (12) using some semigroups suggested by work of Krieger.  ps-file pdf-file.
  15. Primitive representations of the polycyclic monoids and branching function systems. The papers (8), (9) and (15) form a sequence. psfile  pdf-file.
  16. Morita equivalence of semigroups with local units.  ps-file pdf-file.
  17. Characterizations of Morita equivalent inverse semigroups. With J. Funk and B. Steinberg. psfile  pdf-file.
  18. Inverse semigroup enlargements of inverse monoids. UCNW Preprint 92.27. pdf-file.
  19. A non-commutative generalization of stone duality. psfile pdf-file.