Representation Theory of Semigroups

Given any mathematical structure on a set, the collection of structure-preserving maps of the set to itself is an example of an abstract algebraic `object' called a semigroup. Thus, semigroups pervade mathematics. On the other hand, given an abstractly defined semigroup, when can it be represented as a semigroup of maps of a mathematical structure? If so, we say that it is represented by actions.Three main strands of research will be pursued in our three institutions: automaticity of actions (St. Andrews), the use of actions and partial actions in the structure and classification of semigroups (York), and actions of inverse semigroups (Heriot-Watt). However, as explained in our Case for Support, there are many interactions between these strands. We aim to draw together existing material, place it in a common framework, and use our combined expertise to solve a number of outstanding problems. This will be done in a collaborative way, together with leading researchers in the area from across the globe. Studying algebras such as semigroups using automata builds a bridge between algebra and theoretical computer science, allowing us to define infinite algebras using finite state automata. Automatic groups and semigroups are now widely studied, but although the notion of action is heavily relied upon, the study of automatic actions (introduced by Dombi) is in its infancy. Geometric results for automatic groups, such as the equivalence to the fellow traveller property, do not carry over for semigroups. We aim to use automatic actions to develop new notions of automatic semigroup, which will go some way to bridging these gaps. We will consider subsequent properties, establishing new undecidability results and algorithms to calculate semigroups. Inverse semigroups are the algebraic versions of the pseudogroups of transformations that form the foundation for describing local structures in geometry. With each inverse semigroup one can asssociate an etale topological groupoid and from such groupoids one can construct C*-algebras. Thus inverse semigroups, etale topological groupoids, and C*-algebras are closely related, forming an important ingredient in non-commutative geometry. The guiding idea is that the representation theory of inverse semigroups provides a unifying framework for studying partial symmetries. This can be seen as a far-reaching generalization of the way in which the representation theory of groups provides a unifying framework for studying symmetries. For example, inverse semigroups can be associated with aperiodic tilings, and the groupoids that result form part of a non-commutative generalization of Stone duality. Furthermore, the representations of the tiling semigroups are known to control the structure of the groupoids, and hence the associated C*-algebras.The question of when a partial map of a set (roughly speaking, a map not everywhere defined) can be extended (in a suitable way) to a global map, is central to aspects of algebra and model theory. Partial actions of semigroups on sets and ordered structures are used implicitly in many structure theorems, but yet have not been exploited. We will investigate when the partial action of a semigroup on a set with structure can be `globalised', and, in the finite case, whether this question is decidable. We believe this is the key to solving outstanding questions, such as, does every finite inverse semigroup has a finite F-inverse cover? We will also use our combined expertise to try to crack long unsolved questions from the classical theory of actions.The project will involve 5 permanent researchers: the three proposers, a Research Assistant and a PhD student. It will also involve a string of research visits and collaborations with leading experts in the field. We will organise an early Workshop to begin the collaborative process and to ensure we take an inclusive approach to our research.

As outlined in `Academic Beneficiaries', the research will benefit workers belonging to a variety of mathematical disciplines: naturally, semigroup theorists, but also workers in theoretical computer science, model theory, universal algebra, the theory of C*-algebras and further mathematical areas. In addition to finding solutions to the problems laid out in our list of Objectives, we will develop a range of tools and a library of examples that will be applicable throughout semigroup theory, computational algebra and beyond. For example, the intended work on representations of inverse semigroups will provide an algebraic underpinning to the theory of self-similar structures in mathematics and physics. In our immediate environments, the impact will be a boost to the culture of our research groups. Already active, they will benefit from the three-way exchange of ideas and visitors envisaged by the project. Any spin-off from our work that relates to linear representations of semigroups will feed into the strong team in York, headed by Steve Donkin, working in that area. The project will dovetail with the EPSRC funded `Automata, Languages and Decidability in Algebra', currently running at St. Andrews; the algebraic expertise of the Principal Investigators will complement its combinatorial slant. The project will unify a disparate but active field, providing a coherent focus and making the UK a lead nation for work in this area. The prestige resulting from high-quality, well presented research benefits the centres involved, their institutions, their funding body as well as the greater academic community. Such prestige has a wider impact on the culture of our educational institutions and, eventually, society at large. This process will not stop with the ending of the project, as we intend that it will provide a platform for further collaboration, both national and international. Mathematicians and administrators in the developing world continue to value the training and rigour that research into pure mathematics brings, and look to nations such as the UK to provide it - let us continue to do so. York, Heriot-Watt and St. Andrews are all leading teaching institutions as well as leading research institutions, with a strong international presence. The enthusiasm generated by an active research programme filters down to the undergraduate community, finding its way through our teaching and interactions with students. A body of engaged undergraduates, who wish to take a vibrant mathematical ethos out into the wider world, will help combat the poor public face that mathematics sometimes appears to have in the UK. Added to this, there is a woeful shortage of female role models in mathematics, which may discourage young women from the field. A team of workers with a female mathematician at its centre can only be good for the wider mathematical and social culture.