Representation Theory of Semigroups

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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.