09.30-10.20

S. Pride, Homological, finiteness conditions for groups, monoids and algebras

Abstract: Recently Alonso and Hermiller introduced a homological finiteness condition bi-FPn (here called weak bi-FPn) for monoid rings, and Kobayashi and Otto introduced a different property, also called bi-FPn (we adhere to their terminology). From these and other papers we know that: bi-FPn => left and right FPn => weak bi-FPn; the first implication is not reversible in general; the second implication is reversible for group rings. We show that the second implication is reversible in general, even for arbitrary associative algebras, and we show that the first implication is reversible for group rings. We also show that the all four properties are equivalent for connected graded algebras. A result on retractions is proved, and some questions are raised.

10.20-11.00

Coffee

11.00-11.50

D. Munn, Contracted algebras of McAlister monoids

Abstract:  Let X be a set with at least two elements. The notion of a McAlister monoid on X was introduced in the book by Mark Lawson (`Inverse semigroups' (1998), Section 9.4). It can be defined as a certain Rees quotient M of the free inverse monoid on X and, as such, it inherits many of the properties of the latter: thus it is a combinatorial completely semisimple inverse monoid whose nonzero principal factors are finite semigroups of matrix units. A further characterisation of M, based on the Scheiblich description of the free inverse monoid on X, will be supplied.
The contracted (semigroup) algebra of M over a field F has been studied by Mike Crabb and the author. For an arbitrary choice of F, this algebra is prime if and only if X is infinite; and, for the case in which F is the complex field, the contracted algebra is *-primitive (primitive, with an additional property) if and only if X is countably infinite. These results will be discussed in some detail.

12.00-12.50

F. Almeida, A presentation for one-dimensional tiling semigroups

Abstract: Tiling semigroups were first considered by J. Kellendonk and M. Lawson following the work of Kellendonk regarding the modelling of solids using gap labelling.
In this talk, we will give a presentation for the inverse semigroup associated with a factorial language, which is a generalization of the tiling semigroup of a one-dimensional tiling, and discuss some conditions on the tiling in order for the presentation to be finite.
This is a joint work with D. B. McAlister.

12.50-15.00

Lunch

15.00-15.50

P. Silva, Profinite methods for infinite algebras

Abstract: Automata groups constitute nowadays the most promising class of self-similar groups to be studied. They act faithfully on an infinite uniform tree and can be seen as projective limits of finite groups acting on the various uniform subtrees of finite depth. This idea can be adequately expressed with the help of wreath products, and the advantage of this approach to study these infinite groups consists on allowing the use of recursion to solve several problems.
We shall describe how these ideas can be adapted to profinite limits of bimachines to give a new insight into Turing machine computation and Complexity Theory.

16.00-16.50

J. Fountain, Reflection monoids

Abstract: Let V be a finite dimensional Euclidean space. A reflection is a linear isomorphism of V determined by a non-zero vector v Î V which maps v to −v and fixes all vectors in the hyperplane v┴ of all vectors orthogonal to v. A reflection group is a subgroup of GL(V ) which generated by reflections.

The inverse monoid of all partial linear isomorphisms of V is denoted by ML(V ). A partial reflection is defined to be the restriction of reflection to a subspace of V , and a reflection monoid is a factorisable inverse submonoid of ML(V ) generated by partial reflections. A reflection monoid can be characterised by two pieces of data: a reflection group W and a collection of subspaces of V that forms a W-invariant semilattice. The talk will be a report of work in progress (joint with Brent Everitt) outlining the basic properties of reflection monoids, providing examples, giving connections with Renner monoids and hyperplane arrangements, and determining the orders of, and presentations for, some of the monoids.

16.50-17.20

Coffee

17.20- 18.10

D. McAlister, Generating Finite Transformation Semigroups

Abstract: In this talk we will discuss the process of generating finite transformation semigroups and analysing their structure. The approach taken is that used in the stand alone program SgpWin which is written in C++. We will discuss some of the algorithms and data structures used in the program. If time permits, we will also speak, briefly, about the problem of generating the congruences on such semigroups.