North British Semigroups and Applications Network

NBSAN is a network of researchers in Scotland and Northern England with interests in semigroup theory and its applications. It is funded chiefly by a grant from the London Mathematical Society, with additional contributions from host departments and occasionally from other sources. It holds two meetings per year in different locations around the region, plus occasional online activities. The main participant universities are East Anglia, Manchester, St Andrews and York, but our activities are open to all interested researchers. Attendance at all meetings is free. We especially welcome graduate students.

For enquiries about individual meetings, please contact the relevant local organiser as described below. To enquire about the network please contact the central coordinator Nóra Szakács. To join the mailing list please contact Mark Kambites.

Next Meeting

The first meeting of the 2025-26 academic year will be held in Durham on Monday 15th December 2025. The provisional timetable is:

• 12:30 - Early arrivers congregate for lunch (in the foyer café of the Mathematics and Computer Science building)
• 13:30 - Luna Elliott - Thompson Monoids ( abstract hide abstract )

  Abstract. Thompson's group \(V\) was the first known example of a finitely generated infinite simple group. It is also a natural infinite analogue of the finite symmetric groups, the topological full group of the one-sided shift, the automorphism group of the Jónsson-Tarski algebra with 2 unary operations and is in some sense the symmetry group of associativity and commutativity. As a result, this group has received a lot of attention over the past 60 years of research. In this talk I discuss the monoids which play the same four roles as \(V\) except from the perspectives of partial transformations, transformations, and partial bijections (where \(V\) itself corresponds to bijections). In particular, I discuss their generation, presentation, and simplicity. Joint work with Reinis Cirpons, Alex Levine, and James Mitchell.

• 14:30 - Yayi Zhu - Relational depth of semigroups with chain-like \(\mathcal{J}\)-classes ( abstract hide abstract )

  Abstract. Stemming from research on presentations for classical transformation semigroups \(T_n\), \(I_n\), and \(PT_n\), there have been interesting results on presentations for their subsemigroups. Examples include the smallest presentations for the subsemigroups that contain \(S_n\), and presentations for proper ideals such as \(I_n \setminus S_n\), and \(T_n \setminus S_n\). The \(\mathcal{J}\)-classes of these semigroups form a chain, and the relations in these presentations lie in some specific \(\mathcal{J}\)-classes. We are interested in the minimal \(\mathcal{J}\)-class, J, whose elements are involved in the presentations, and we use 'depth' to describe the position of J within the chain. In this talk, I will present results on the relational depth of some semigroups with chain-like \(\mathcal{J}\)-classes.

• 15:15 - Coffee
• 15:45 - Daniel Heath - Pretzel Monoids ( abstract hide abstract )

  Abstract. Left adequate monoids generalise the well-studied class of inverse monoids, with a graphical description of free structures given by Kambites. It is natural then to consider graphical interpretations of presentations, akin to work by Munn and Stephen for inverse monoids. In general, we believe this to be very hard; left adequate monoids do not seem to play well with presentations (they do not form a variety, for example). Here, we take tentative steps towards understanding specific presentations, namely ones resembling the famed Margolis-Meakin expansions. We introduce and study an interesting class of monoids, defined by natural combination operations on birooted, edge-labelled directed graphs, and show that they are examples of left adequate monoids with such presentations. We term the resulting monoids pretzel monoids, due to a resemblance between the diagrams we use to represent their elements and certain baked goods... We construct each monoid from a right cancellative monoid \(C\) and, with this viewpoint, one may understand pretzel monoids as a left adequate analogue of the \(E\)-unitary Margolis-Meakin expansions. This is joint work with Mark Kambites and Nóra Szakács.

• 16:30 - Sarah Rees - Studying monoids that model concurrency ( abstract hide abstract )

  Abstract. I'll discuss joint work of mine with with Ascencio-Martin, Britnell, Duncan, Francoeur and Koutny to set up and study algebraic models of concurrent computation.
  
  Trace monoids were introduced by Mazurkiewicz as algebraic models of Petri nets, where some pairs of actions can be applied in either of two orders and have the same effect. Abstractly, a trace monoid is simply a right-angled Artin monoid. More recently Koutny et al. introduced the concept of a step trace monoid, which allows the additional possibility that a pair of actions may have the same effect performed simultaneously as sequentially.
  
  I shall introduce these monoids, discuss some problems we'd like to be able to solve, and the methods with which we are trying to solve them. In particular I'll discuss normal forms for traces, comtraces and step traces, and generalisations of Stallings folding techniques for finitely presented groups and monoids.

• 17:30 - Close
• 18:30 - Dinner at the Cosy Club, for those able to stay (see registration form below for details)

All talks will take place in room MCS0001, on the ground floor of the Mathematics and Computer Science building, adjacent to the foyer.

If you would like to attend please complete the registration form. To attend the dinner, registration is required by Friday 28th November.

Please contact Max Gadouleau with any queries.

Future Meetings

To receive announcements about future meetings, please email Mark Kambites and request to be added to the NBSAN mailing list.

Past Meetings

For the archive of information about past meetings look here.

Other (non-NBSAN) Events and Semigroup News

• There will be lots of semigroups at AAA 109 and SAL 2026 in Ljubljana from 5th-8th June 2026.