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 next NBSAN meeting will be held in Manchester on Thursday 20th and Friday 21st June 2024, as a satellite to the 75th British Mathematical Colloquium held between 17th and 20th June. The BMC itself will feature lots of semigroups content, and we expect the conjunction of the two events to attract a significant international semigroups audience for the week. The semigroup programme is organised by Marianne Johnson, Mark Kambites, Dmitry Kudryavtsev, Alex Levine and Nóra Szakács.

The BMC runs from 13:00 on Monday to lunchtime on Thursday, with a casual conference dinner (included in the registration fee) on Wednesday evening (note that this is a different day to previously advertised). NBSAN runs from lunchtime on Thursday until 16:30 on Friday, with a dinner (paid separately) on Thursday evening. On Saturday, for those able to stay, there will be an unofficial social excursion (paid separately, provisionally to Quarry Bank Mill, an industrial heritage site with a working 18th century cotton mill and picturesque gardens).

Semigroups in the BMC Programme. The semigroups content of the BMC will be spread over the duration of the meeting, and will feature:

• a morning talk by Robert D. Gray on The geometry of the word problem for groups and inverse monoids;
• invited workshop talks by Peter Cameron, Victoria Gould, Marianne Johnson, Stuart Margolis, Yann Peresse and Nik Ruškuc; and
• contributed speed talks by Ádám Budai, Reinis Cirpons, Matthias Fresacher, Martin Hampenberg Christensen, Daniel Heath, Jung Won Cho, Ajda Lemut, Carl-Fredrik Nyberg Brodda, Duarte Ribeiro and Jonathan Warne.

See the BMC website for the full BMC schedule and semigroups workshop schedule.

NBSAN on Thursday 20th June

• 14:00 - António Malheiro - Quasi-crystals and algebraic structures: linking cyrstal bases to semigroups and beyond ( abstract hide abstract )

  Abstract. This talk delves into the intricate world of quasi-crystals, an extension of the well-established theory of crystal bases in representation theory. Originating from the groundbreaking work of Drinfeld, Jimbo, and Kashiwara in the 1980s, quantum groups and crystal bases have played a pivotal role in theoretical physics and various mathematical domains. Kashiwara's development of crystal bases and crystal graphs provided a framework for studying representations of quantum groups, revealing intriguing connections with Young tableaux and the plactic monoid.
  
  The plactic monoid, central to the theory of symmetric polynomials and the Littlewood-Richardson rule, bridges the gap between crystal structures and Schur polynomials. In parallel, the hypoplactic monoid enters the scene in the realm of quasi-symmetric functions, offering an analogue to the classical plactic monoid with applications in quasi-ribbon tableaux.
  
  Building on the quasi-crystal structure introduced with Alan Cain, this talk explores the extension of crystal concepts to quasi-crystals. We present a set of local axioms for quasi-crystal graphs of simply-laced root systems, drawing parallels with Stembridge's work on crystals. The characterization of quasi-crystal graphs arising from the quasi-crystal of type An answers an open question, providing insight into the local structural properties that define these structures.

• 15:00 - Thomas Aird - The meet-stalactic and meet-taiga monoid ( abstract hide abstract )

  Abstract. Each "plactic-like" monoids is defined by some algorithm which takes a word and outputs a combinatorial object. The stalactic monoid and taiga monoid are two such plactic-like monoids, corresponding the stalactic tableaus and binary search trees with multiplicity respectively. In this talk, I introduce two new plactic-like monoids, the meet-stalactic monoid and the meet-taiga monoid. These monoids are defined by running their respective algorithm in both directions on a word producing two combinatorial objects. After introducing these monoids, I will present a number of results about these monoids, including a Robinson-Schensted-like Theorem. This is joint work with Duarte Ribeiro.

• 15:30 - Magdalena Wiertel - Hecke-Kiselman monoids and algebras ( abstract hide abstract )

  Abstract. To every finite oriented graph \(\Theta\) with \(n\) vertices one can associate a finitely presented monoid \(HK_{\Theta}\), called the Hecke--Kiselman monoid. It is a monoid generated by \(n\) idempotents with relations of the form \(xy=yx\) or \(xyx=yxy=xy\), depending on the edges between vertices \(x\) and \(y\) in \(\Theta\). By the Hecke-Kiselman algebra we mean the monoid algebra \(K[HK_{\Theta}]\) of the monoid \(HK_{\Theta}\) over a field \(K\). We investigate the combinatorics and structure of the Hecke-Kiselman monoids and algebras. It turns out that the case of the monoid \(C_n\) associated to an oriented cycle of length \(n\geqslant 3\) plays a crucial role. The unexpected chain of ideals inside \(C_n\) is constructed. This result is then applied to show that the monoid \(C_n\) satisfies a semigroup identity and the associated algebra is a semiprime Noetherian algebra. As a consequence, we characterise all Hecke-Kiselman monoids that satisfy a semigroup identity. Moreover, the Jacobson radical of all Hecke-Kiselman algebras satisfying a polynomial identity is described.

• 16:00 - Coffee break
• 16:30 - Igor Dolinka - Prefix monoids of groups and right units of special inverse monoids ( abstract hide abstract )

  Abstract. Motivated by the arguably most elusive open problem in combinatorial algebra - the question of decidability of the word problem for one-relator monoids, standing unresolved for almost a century - the past few decades have seen a development of a rich theory showcasing intricate and deep connections between decision problems for finitely presented monoids, inverse monoids, and groups. Among those problems we may single out the submonoid membership problem, more generally, the rational subset membership problem, or, more specifically, membership problems for some particular submonoids such as the prefix monoid (of a finitely presented group) or the monoid of right units (of a finitely presented special inverse monoid). This talk will present recent results in this vein obtained in collaboration with Robert Gray (University of East Anglia), and is, in a broad sense, related to Gray's Morning Talk at the BMC. We obtained a full characterisation of prefix monoids of finitely presented groups, as well as some further information on these monoids, such as the groups of units and Schützenberger groups. Much less is known about the monoids right units of finitely presented special inverse monoids, although some partial information is available. In particular, the universality result for the Schützenberger groups does still hold true. I will present selected results of ongoing research in this direction, as well a number of open questions.

• 17:30 - Towards Dinner....

NBSAN on Friday 21st June

• 09:00 - Ganna Kudryavtseva - Duality theory for Boolean right restriction semigroups ( abstract hide abstract )

  Abstract. We generalize the duality between Boolean right restriction monoids and ample source-etale categories by Cockett and Garner to the non-unital and locally compact setting. Our approach upgrades the widely known construction of the tight groupoid of an inverse semigroup as the groupoid of germs. Elements of a supported Boolean right restriction semigroup are represented by right compact slices of their attached right ample categories. In these categories, the domain map is a local homeomorphism, but the range map is not even open in general, so that ranges of elements of the category are not 'seen' in the attached right restriction semigroup. In the special case where the range map of the right ample category is open, the right restriction semigroup has the additional structure of a left Ehresmann semigroup. Specializing further to the case where the range map of the category is a local homeomorphism, the category has the additional property that every compact right slice is a finite join of compact two-sided slices. On the algebraic side, this brings Boolean right restriction Ehresmann semigroups with the extra property that every element is a finite join of deterministic elements. These semigroups are a natural generalization of Garner's groupoidal right restriction monoids.

• 10:00 - Itamar Stein - The algebra of the monoid of order-preserving functions and other reduced E-Fountain semigroups ( abstract hide abstract )

  Abstract. With the monoid of all order-preserving functions on an \(n\)-set, we associate a category with partial composition - a category in which the composition of two morphisms might be undefined even if the range of the first equals the domain of the second. We will show that the algebra of the monoid is isomorphic to the algebra of the category over any commutative unital ring. Similar results have been useful in the study of algebras of many other finite semigroups: inverse semigroups, the monoid of partial functions, the Catalan monoid etc.
  
  More generally, with every reduced E-Fountain semigroup which satisfies the generalized right ample condition we will show how to associate a category with partial composition. Under some assumptions, we will prove an isomorphism of algebras between the semigroup algebra and the category algebra.
  
  This is a simultaneous generalization of a former result on reduced E-Fountain semigroups which satisfy the congruence condition, a result of Junying Guo and Xiaojiang Guo on strict right ample semigroups and a result of Benjamin Steinberg on idempotent semigroups with central idempotents.
  
  If time allows, we will discuss additional examples.

• 10:30 - Coffee break
• 11:00 - Luna Elliott - E-disjunctive inverse semigroups ( abstract hide abstract )

  Abstract. A congruence on an inverse semigroup is called idempotent-pure if it never relates an idempotent to a non-idempotent. These congruences preserve much more of the structure of the semigroup than most congruences do. The well-known class of E-unitary inverse semigroups is rich in structure because they are precisely the inverse semigroups with a group idempotent-pure quotient.
  
  Dually to this, it is thus natural when studying more general inverse semigroups to focus on the class of inverse semigroups which have "already been quotiented by all of their idempotent-pure congruences". This is precisely the class of E-disjunctive inverse semigroups. I will introduce E-disjunctive inverse semigroups and speak about the above structure preservation in more detail. This exploration is based on some recent work done with my collaborators Alex Levine and James Mitchell.

• 11:30 - Herman Goulet-Ouellet - Profinite bridges between semigroup theory and symbolic dynamics ( abstract hide abstract )

  Abstract. My talk will explore the relationship between free profinite semigroups and symbolic dynamics, a line of research which goes back to the work of Almeida in the early 2000s. I will start by presenting Almeida's fundamental theorem, which gives a bijection between minimal shift spaces (a central object of symbolic dynamics) and maximal regular J-classes of free profinite semigroups. I will then survey a number of interesting features of this bijection from the point of view of semigroup theory, and discuss interesting applications from the point of view of symbolic dynamics.

• 12:00 - Lunch
• 13:30 - Jan Philipp Wächter - Decision problems for automaton semigroups and groups ( abstract hide abstract )

  Abstract. We will discuss the decidability and complexity of the most important decision problems for semigroups and groups generated by finite-state, letter-to-letter transducers (which are usually simply referred to as "automata" in this context). In such an automaton, every state induces a function mapping input to output words. The closure of these functions under composition is the generated semigroup. If the automaton has additional properties (such as being complete and/or invertible), we obtain groups and inverse semigroups. The study of these objects is motivated by some groups with exotic properties that can be found in this class (where Grigorchuk's group is usually considered to be the most prominent one). In the talk, we will recall the basic definitions and mainly give an overview of the most important results with regard to decision problems and also discuss some recent developments.

• 14:00 - Tara Macalister Brough - Preserving self-similarity in free products of semigroups ( abstract hide abstract )

  Abstract. I will give an update on an ongoing project I first talked about at the 12th NBSAN meeting in 2012. I have been trying to determine the extent to which the class of automaton semigroups is closed under various standard semigroup constructions, with a special focus on the free product. In recent work with Jan Philipp Wächter and Janette Welker, we not only substantially improved on my previous free product results with Alan Cain, but also observed that our constructions all work for self-similar semigroups. I will give a brief introduction to self-similarity, describe the evolution of my understanding of free products of automaton and self-similar semigroups, with a sketch of the proof of our latest result, and discuss where to go from here.

• 14:30 - Matthew Brookes - Congruences on direct products of simple semigroups ( abstract hide abstract )

  Abstract. Direct products and congruences are basic building blocks for semigroup theory, so it is natural to ask what are the congruences for a direct product. In fact, congruences themselves are subsemigroups of direct products. One construction for subsemigroups of direct products is fibre products, so one family of congruences on a direct product may be obtained as fibre products of congruences on the factors. For simple monoids this approach describes all the congruences. As a consequence it is possible classify when every congruence on a direct product decomposes as a product of congruences on the factors. One natural question asks what is the maximum length of a chain of congruences? Generalising the fibre congruence construction to describe congruences on simple actions allows us to answer this question for direct products, in terms of the maximum lengths of chains of congruences on the factors.

• 15:00 - Coffee break
• 15:30 - Tim Stokes - Can constellations shed light on semigroups? ( abstract hide abstract )

  Abstract. A constellation is a partial algebra that is a one-sided generalised category, with each element \(s\) having a domain \(D(s)\) but no range in general. They model composition of functions (or morphisms) \(f\circ g\) where the image of \(f\) is contained (as a substructure) in the domain of \(g\), and codomains play no role.
  
  Constellations were developed by Gould and Hollings as a tool in semigroup theory for finding a category-like counterpart of left restriction semigroups. Those constellations corresponding to left restriction semigroups are called inductive, by analogy with the ordered groupoid/inverse semigroup case.
  
  General constellations have proved of interest for their own sake. A process of right canonical extension of a constellation (which puts back in the missing codomains!) gives a category, leading to novel ways to think about the familiar concrete categories of mathematics.
  
  We shall explain how one can use a left-sided version of the canonical extension idea to obtain constellations from monoids. If the constellation is inductive, one obtains a left restriction monoid. This leads to new ways to build some familiar left restriction monoids from their subsemigroups of elements of domain \(1\). The left restriction monoid of partial functions on a set \(X\) can be obtained in this way from the transformation monoid on \(X\). Other examples arise from the theories of binary relations and partitions.

• 16:30 - Close

Contact/queries. For questions about the NBSAN meeting and the semigroups content of the BMC please email Marianne Johnson. (For general queries about the BMC please see the BMC website or email bmc2024@manchester.ac.uk.)

Sponsors. The NBSAN meeting is generously supported by the Engineering and Physical Sciences Research Council, the Heilbronn Institute for Mathematical Research, the London Mathematical Society and the Department of Mathematics at the University of Manchester. Sponsors for the BMC meeting are detailed on the BMC website.

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

• The 75th British Mathematical Colloquium, to be held in Manchester in the week commencing 17th June 2024, will feature lots of relevant activity, including a special session dedicated to semigroups - see above for details of the main meeting and the NBSAN satellite meeting.
• There is a conference on Semigroups, Groupoids and C*-algebras in Seoul, South Korea from 1st-5th July 2024.
• A conference on Theoretical and Computational Algebra, including lots of semigroups, will take place in Aveiro, Portugal from 30th June - 5th July 2024.