The concept of an inverse semigroup was first defined in the 50s, independently by Preston and Wagner, in an attempt to describe the algebraic structure of bijections between subsets of a fixed set. A set of such partial bijections closed under composition and taking inverses forms an inverse semigroup, and conversely, every inverse semigroup can be represented this way. As such, the relationship between inverse semigroups and partial symmetries is a generalization of the relationship between groups and symmetries. Furthermore, similarly as to how groups act by bijections, inverse semiroups act by partial bijections.
During the fellowship, we investigated various facets of actions of inverse semigroups by partial maps. One of these is such actions can be used to construct associative and C*-algebras. We gave a full characterization of the simplicity of the former (which also has implications for the latter), and applied these results to obtain an algorithmic characterization when the action is by certain partial maps on infinite regular trees, these are called Nekrashevych algebras. These results settle a 40-year-old open problem posed by Douglas Munn, conclude a decade research on the simplicity of étale groupoid algebras, and make very significant progress in understanding the simplicity of Nekrashevych algebras and their C*-algebra counterparts. These results have been summarized in two papers:
Simplicity of inverse semigroup and étale groupoid algebras (with B. Steinberg), Adv. Math. 386 (2021) 107611, freely accessible on arXiv:2006.13787
On the simplicity of Nekrashevych algebras of contracting self-similar groups (with B. Steinberg), submitted to Journal für die reine und angewandte Mathematik, freely accessible on arXiv:2008.04220
We also investigated several question about inverse semigroups defined by geometric properties. Namely, motivated by the success of geometric group theory, we studied the algorithmic properties of inverse semigroups that have a hyperbolic, and respectively, tree-like geometry. We have proved several results, in particular, managed to extend one direction of the celebrated Müller-Schupp theorem for groups. The work is summarized in the following paper:
Algorithmic properties of inverse monoids with hyperbolic and tree-like Schützenberger graphs (with R. Gray and P. V. Silva), freely accessible on arXiv:1912.00950
In addition to the papers, we have disseminated the results at several seminars and conferences:
Simplicity Nekrashevych algebras of contracting self-similar groups: seminar talks at the Mathematical Institute, Göttingen, (online, Jan 14, 2021), Western Sydney University, (online, Oct 22, 2020), Alfréd Rényi Institute of Mathematics, (online, Sept 28, 2020).
Simplicity of contracted inverse semigroup algebras: seminar talks at University of Szeged, (Hungary, Sept 2, 2020), University of York, (online, May 20, 2020), conference talk at International Conference on Semigroups and Applications, (Cochin, India, Dec 9, 2019).
Algorithmic properties of tree-like and hyperbolic inverse monoids: seminar talks at University of York, (UK, Oct 17, 2019), Heriot-Watt University, (Edinburgh, UK, May 22, 2019), City University of New York, (USA, May 10, 2019), University of St Andrews, (UK, Apr 4, 2019), conference talk at Semigroups and Groups, Automata, Logics, (Cremona, Italy, Jun 10, 2019).
