Preprints
- 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with Johannes Kellendonk on tiling semigroups and by some unpublished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.
- 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.
- 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.
- 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.
- 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.
- In
McAlister's footsteps: a random ramble around the P-theorem. (with Stuart Margolis)
Accepted for publication. ps-file
pdf-file.
- 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.
- 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 CA. ps-file pdf-file.
- 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.
- 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.
- 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.
- 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.
- A
correspondence between a class of monoids and self-similar group
actions I. Replaces (10), (11) and (12) above. ps-file. pdf-file.
- Semigroups
related
to subshifts of graphs. A generalisation of (12) using
some semigroups suggested by work of Krieger. ps-file pdf-file.
- Primitive representations of the
polycyclic monoids and branching function systems. The papers (8), (9) and (15) form a
sequence. psfile pdf-file.
- Morita
equivalence
of semigroups with local units. ps-file pdf-file.
- Characterizations of Morita
equivalent inverse semigroups. With J. Funk and B. Steinberg. psfile pdf-file.
- Inverse semigroup
enlargements of inverse monoids. UCNW Preprint
92.27. pdf-file.
- A non-commutative
generalization of stone duality. psfile pdf-file.