North British Semigroups and Applications Network - Past Meetings

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) and holds 2-3 meetings per year in different locations around the region. 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.

This page contains an archive of information about past activities. For information about current activities look here.

The thirty-fifth NBSAN meeting was held in York on Friday 17th November 2023, organised by Craig Miller. The speakers were:

• Peter Faul - Fibrational approaches to Clifford Semigroups ( abstract hide abstract )

  Abstract. Since it is possible to represent Clifford semigroups as functors \(\mathrm{L} \to \mathrm{Grp}\), it is natural to ask if semigroup homomorphisms between them can be conceptualised in terms of maps between their associated functors. Phrased more precisely, is there a category whose objects are the functors so described and which is equivalent to the category of Clifford semigroups? While this problem has been solved, we argue in this talk that there is a more principled categorical approach that leads directly to the solution, involving the notion of a fibration. The advantage of this approach is that it brings along for the ride a number of interesting structures. For instance, this fibration may be exploited to gain a greater understanding of the morphisms between Clifford semigroups. In particular, each orthogonal factorisation system on the codomain lifts to an orthogonal factorisation system on the domain category. We are able to conclude that every homomorphism between Clifford semigroups can be factored as a `surjective cartesian map' followed by an `idempotent-fixing injection'. This restricts easily to the subcategory of commutative inverse semigroups.
  
  

• Robert Gray - Symmetries of Schützenberger graphs and subgroups of special inverse monoids ( abstract hide abstract )

  Abstract. In the 1960's Higman was able to characterize the finitely generated subgroups of finitely presented groups. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present some results of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the maximal subgroups of inverse monoids defined by presentations where all the defining relators are of the form \(w=1\). I will explain the motivation for studying this class of inverse monoids, and also indicate how we established these results by developing new geometric approaches to maximal subgroups, exploiting the fact that they are isomorphic to automorphism groups of Schutzenberger graphs. I will also say something about some related recent joint work with Igor Dolinka (Novi Sad) on, so-called, prefix monoids and their connection with the class of recursively presented group-embeddable monoids.
  
  

• Jung Won Cho - Subsemigroups of the free monogenic inverse semigroup ( abstract hide abstract )

  Abstract. Abstract: In 1975, Boris Schein proved that free inverse semigroups are not finitely presentable as semigroups. In particular, he showed that the free monogenic inverse semigroup is not finitely presentable using two particular partial transformations. Motivated by this result, I will give a proof that a subsemigroup (not necessarily inverse) of the free monogenic inverse semigroup is finitely presented if and only if it contains only finitely many idempotents. This is joint work with Nik Ruškuc.
  
  

• Robert Kropholler - Multiple context-free groups are closed under free products ( abstract hide abstract )

  Abstract. A group is multiple context-free, if its word problem is a multiple context-free language (MCFL). MCFLs sit between the class of context-free languages and context sensitive languages. By a theorem of Muller and Schupp a group is context-free if and only if it is virtually free. In 2015 Salvati showed that \(\mathbb{Z}^2\) is a multiple context-free group. From there, Ho has shown that all virtually abelian groups are in this class. It would be of interest to have a classification similar to that of Muller and Schupp. I will discuss one step of this, namely showing that this class is closed under free products. I will mostly focus on proving this result for context-free languages using pushdown automata and discuss the modifications and automata needed for the multiple context-free case. This is joint work with Davide Spriano.
  
  

• Catarina Monteiro - Formations and i-Fitting classes of inverse semigroups, congruences and languages ( abstract hide abstract )

  Abstract. A class of finite groups \(\mathcal{F}\) closed under quotients and finitary subdirect products is said to be a formation of finite groups. A class of finite groups \(\mathcal{C}\) closed under subnormal subgroups and for groups generated by two normal subgroups which are in \(\mathcal{C}\) is said to be a Fitting class of finite groups. A natural question that arises is whether there are corresponding concepts for congruences and languages. This question has been addressed for formations of groups, of monoids and of many-sorted algebras, and bijections between formations of semigroups, of congruences on the associated free object and of languages have been obtained. In this talk, based on recent join work with G. Gomes, I will show that there exist bijections between formations of inverse semigroups, formations of congruences on inverse semigroups, and formations of congruences on free inverse semigroups. Further, I will prove the existence of bijections between i-formations of inverse semigroups, of (idempotent separating) congruences and languages (contained in the centraliser of the semigroup). This study appears in "Formations and i-Fitting classes of inverse semigroups, congruences and languages", and it was motivated by those developed in "Formations of inverse semigroups" and "Formations of Orthodox Semigroups", where formations associated to idempotent-separating congruences play an essential role. Based on these results, and with Fitting classes being the dual concept of formations, it is natural to look for similar results to those obtained for formations. I will prove the existence of bijections between i-Fitting classes of Clifford semigroups, of (idempotent separating) congruences, and of languages (contained in the centralizer).
  
  

• Nasir Sohail - Amalgamating inverse semigroups over ample semigroups ( abstract hide abstract )

  Abstract. It was shown by Howie that a semigroup amalgam \( (S;T_1,T_2) \) does not embed if \( T_1 \) and \( T_2\) are both groups but \( S \) is not a group. Generalizing this result, Rahkema and Sohail showed that \((S;T_1,T_2)\) is non-embeddable if \(T_1\) and \(T_2\) are completely regular (respectively, Clifford) semigroups but \(S\) is not completely regular (respectively, Clifford). In this talk we shall consider the amalgams \( (S;T_1,T_2) \) such that \(T_1\) and \( T_2 \) are inverse semigroups while S is non-inverse but ample.
  
  

The thirty-fourth NBSAN meeting was held in Norwich on Thursday 15th and Friday 16th June 2023, organised by Robert D. Gray and Islam Foniqi, and with a focus on decision problems in group and monoid theory. Full details are on the dedicated meeting webpage. The speakers were:

• María Cumplido Cabello - Solving the word problem in Artin groups without braid relations
• Laure Daviaud - Some applications of semigroup theory in formal verification
• Lorna Gregory - Representation type and decidability of theories of modules of finite-dimensional algebras
• Martin Hampenberg Christenen - Submonoids of the symmetric inverse monoid
• Alan Logan - Post's correspondence problem in group theory
• Carl-Fredrik Nyberg Brodda - One relator, many problems
• Igor Potapov - Matrix semigroups, equations and linear maps
• Daniel Turaev - Deciding the first order theory of the plactic monoid

The thirty-third NBSAN meeting was held in Manchester on Friday 14th April 2023, organised by Mark Kambites and Nóra Szakács. The speakers were:

• Ruy Exel - The tight \(C^*\)-algebra of an inverse semigroup ( abstract hide abstract )

  Abstract. I'll describe the motivation for and the construction of the tight \( C^* \)-algebra of an inverse semigroup as well as a few examples where the construction leads to \( C^* \)-algebras of interest.
  
  

• Islam Foniqi - Decision problems for one-relator monoids and groups ( abstract hide abstract | slides )

  Abstract. The word problem for one-relator monoids is still open. This motivates the study of other decision problems for monoids, inverse monoids and groups. We will provide some recent results on submonoid, and rational subset membership problems in algebraic structures, and a discussion of their implications and connections.
  
  

• Victoria Gould - Pseudo-finite semigroups and diameter ( abstract hide abstract | slides hide abstract )

  Abstract. A semigroup \( S \) is said to be (right) pseudo-finite if the universal right congruence \( \omega_S:= S\times S \) can be generated by a finite set \( U\subseteq S\times S \), and there is a bound on the length of derivations for an arbitrary pair \( (s,t)\in S\times S \) as a consequence of those in \( U \). The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets. The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Żelazko for Banach algebras. The property also arises from a number of other sources.
  
  Pseudo-finiteness is a very unpredictable condition. In the presence of cancellativity, e.g. for a group, it forces finiteness and, consequently, diameter 1. Any monoid with zero is easily seen to be pseudo-finite with diameter at most 2. Rather trivially, both kinds of examples have minimal ideals. Further, any full transformation monoid \( \mathcal{T}_X \) (for any cardinality of \( X \) ) is pseudo-finite with diameter 1, and has a minimal ideal \( I \) of constant maps on which \( \mathcal{T}_X \) acts with a high degree of transitivity.
  
  The talk introduces the curious world of pseudo-finite semigroups. It has two focusses: exploring the existence and nature of a minimal ideal in a pseudo-finite semigroup, and the size of the diameter. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools will all play a part. This research sits in the wider framework of a study of finitary conditions for semigroups.
  
  This is joint work with James East, Craig Miller, Nik Ruškuc and Tom Quinn-Gregson.
  
  

• Mark Kambites - Subgroups of special inverse monoids ( abstract hide abstract )

  Abstract. I will talk about recent joint work with Robert Gray on the subgroup structure of finitely presented special inverse monoids.
  
  

• Xin Li - \(C^*\)-algebras and dynamical systems arising from semigroups ( abstract hide abstract )

  Abstract. The first part of the talk will be an overview, with a focus on examples, of semigroup \(C^*\)-algebras, which are \(C^*\)-algebras generated by left regular representations of semigroups. The second part of the talk will discuss underlying dynamical systems which are interesting in their own right.
  
  

The thirty-second NBSAN meeting was held in York on Friday 24th June 2022, organised by Victoria Gould and in conjunction with a workshop on Topics in Geometric Semigroup Theory the same week. The speakers were:

• Fabienne Chouraqui - About Garside monoids and M-braces ( abstract hide abstract )

  Abstract. We define an algebraic structure similar to that of a semiring, but without some of the requirements. As it is somehow also similar to the structure of left brace, we call it an \( \mathscr{M} \)-brace. We present a connection between Garside monoids and more generally lcm-monoids with this algebraic structure. An lcm-monoid \( M \) is a left-cancellative monoid such that \( 1 \) is the unique invertible element in \( M \), and every pair of elements in \( M \) admit an lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids.
  
  

• Tom Quinn-Gregson - Minimum degrees of variants of full transformation semigroups ( abstract hide abstract )

  Abstract. Every finite semigroup \( S \) embeds into some transformation semigroup \( \mathcal{T}_n \), and we call the least such \( n \) the (minimal transformation) degree of \( S \), denoted \( \mathrm{deg}(S) \). In this talk we examine how taking a variant affects the degree of \( S \), where for a fixed element \( a\in S \), the variant of \( S \) is the semigroup \( S^a=(S,\star) \) with sandwich product \( x\star y = xay \). By finding the degrees of null semigroups and right null semigroups (right zero semigroups of null semigroups), we are able to answer two questions of J. East from "Transformation Representations of Sandwich Semigroups". Namely, we show that it is possible to have \( \mu(S^a)< \mu(S) \) and deg \( (\mathcal{T}_n^\alpha) < 2n - \mathrm{rank}(\alpha) \).
  
  This is joint work with P. Cameron, J. East, D. FitzGerald, J. Mitchell, and L. Pebody.
  
  

• Stuart Margolis - Translational hulls, combinatorics and discrete geometry ( abstract hide abstract )

  Abstract. In combinatorics and algebra morphisms are usually defined by preserving direct image: "A morphism between graphs (simplicial complexes) is a function that maps edges to edges (faces to faces), for example.
  
  But in topology, morphisms are defined by preservation of inverse image of designated subsets, the open or closed sets.
  
  More generally, an incidence structure is a pair \(\mathcal{I}= (V,B) \) where \( V \) is a finite set and \( B \) is a collection of subsets of \( B \). The incidence matrix \( M(\mathcal{I}) \) is the \( V \times B \) matrix over \( \{0,1\} \) that has a 1 in position \( (v,b) \) if \( v \in b \) and \( 0 \) otherwise.
  
  A partial function on \( V \) is called continuous if for each \( b \in B, bf^{-1} \in B \) whenever \( bf^{-1} \) is non-empty. Thus for the incidence structure consisting of a topolgical space \( V \) and its collection Bof open sets, the monoid of total continuous functions are the continuous functions on \( V \) in the usual sense.
  
  The monoid \( C(\mathcal{I}) \) of continuous partial functions provides a link between combinatorics, discrete geometry and semigroup theory. It is easy to show that \( C(\mathcal{I}) \) is the translational hull of the 0-simple semigroup whose structure matrix is \( M(\mathcal{I}) \). This gives geometric and combinatorial tools to study translational hulls and their subsemigroups.
  
  We illustrate this straightforward idea in the cases of graphs, simplicial complexes, Steiner triple systems and pairwise balanced designs.
  
  

• Xavier Mary - Semigroups with chains of associate idempotents ( abstract hide abstract )

  Abstract. It was discovered very recently that many classes of rings can interestingly be defined in terms of chains of associate idempotents. The purpose of this talk is to explore such connections in the semigroup case. Precisely, we will study non-regular semigroups with the property that any two \( \mathcal{D} \)-related idempotents are related by a sequence of size \( n \) of alternatively \( \mathcal{L} \)- or \( \mathcal{R} \)- related idempotents. We will mainly focus on the cases \( n=1, 2 \) and \( 3 \). In the particular case of \( n=2 \), we obtain a semilattice decomposition in terms of extensions of completely simple semigroups by poor semigroups.
  
  

• Carl-Fredrik Nyberg Brodda - Free products and word-hyperbolic monoids ( abstract hide abstract )

  Abstract. Language-theoretic methods in group theory have been used successfully to characterise many different algebraic properties, including virtual freeness and hyperbolicity. In the past 20 years, there has been an increased effort to explore such methods in semigroup theory. A staple method for constructing new semigroups and monoids from old is by taking free products. Free products are well known to interact well with context-free languages in groups - for example, Anisimov proved that the free product of two groups with context-free word problem is again context-free, and it is easy to show that free products of hyperbolic groups are hyperbolic. In this talk, I will give an overview of how free products of semigroups and monoids can interact with formal language theory in the context of the word problem and word-hyperbolicity. Many existing techniques for studying this require some fixed implementation of the context-free languages (e.g. push-down automata). The methods used here instead use closure properties of the classes of languages involved, yielding much more general results. For example, we recover and generalise a recent result of Brough, Cain and Pfeiffer that the class of context-free monoids are closed under free products. Furthermore, new purely group-theoretic results can be obtained, e.g.: the free product of two groups with ET0L word problem again has ET0L word problem.
  
  

• (Itamar Stein was also invited to speak but was sadly unable to attend; he has kindly made his slides available.)

The thirty-first NBSAN meeting was a "Young NBSAN" organised by and with a special focus on early career researchers. It was held in Manchester on 25th March 2022, organised by Thomas Aird, Daniel Heath, Dmitry Kudryavtsev, Carl-Fredrik Nyberg Brodda and Georgia Schneider. The speakers were:

• Thomas Aird - Semigroup identities in tropical matrices
• Ashley Clayton - Counting subdirect powers of finite commutative semigroups (slides)
• Ambroise Grau - Translational hulls and isomorphisms with endomorphism monoids
• Craig Miller - The \( \mathcal{R} \)-height of semigroups and their bi-ideals (slides)
• Nik Ruškuc - Direct and subdirect products in combinatorial semigroup theory (slides)
• Georgia Schneider - Semigroups of inverse quotients (slides)
• Maria Tsalakou - Computing congruences of finitely presented semigroups
• Bill de Witt - The number of subdirect powers of unary algebras

In Summer 2021 we hosted some another short series of "eNBSAN" seminars:

• 14th July - James East - Congruences of twisted partition monoids ( abstract hide abstract )

  Abstract. This is a report on a project with Nik Ruškuc, in which we classify the congruences of the twisted partition monoids, and describe the lattice formed by them.
  
  

• 21st July - Student Talks Session:
  • Alex Levine - Describing solutions to equations using EDT0L languages ( abstract hide abstract )

    Abstract. Ever since Makanin proved that the satisfiability of systems equations in free monoids and groups was decidable, there has been significant interest in the study of equations in groups and semigroups. More recently, Ciobanu, Diekert and Elder proved that solutions to systems of equations in free monoids and free groups can be expressed as EDT0L languages. Following this there have been several attempts widen the class of groups and semigroups for which solutions can be expressed as an EDT0L language. We give a brief introduction to the work that has been done in this area.
    
    

  • Georgia Schneider - Semigroups of left I-quotients ( abstract hide abstract )

    Abstract. A subsemigroup \( S \) of an inverse semigroup \( Q \) is a left I-order in \( Q \) if every element in \( Q \) can be written as \( a^{-1} b \) where \( a, b \in S \) and \( a^{-1} \) is the inverse of \( a \) in the sense of inverse semigroup theory. If, in addition, we can insist that \( a \) and \( b \) are \(\mathcal{R}\)-related in \( Q \), then we say that \( S \) is a straight left I-order in \( Q \). We give necessary and sufficient conditions for a semigroup to be a straight left I-order.
    
    

  • Maria Tsalakou - Computational methods for small overlap monoids ( abstract hide abstract )

    Abstract. Small overlap monoids are monoids with finite presentations that satisfy simple combinatorial conditions. Finitely presented monoids that satisfy sufficiently strong small overlap conditions have decidable word problem and lexicographic normal forms that can be computed in linear time. We will give a brief description of the properties and the main results that led to the development of practical computational methods for small overlap monoids.
    
    

• 28th July - António Malheiro - Identities in plactic-like monoids: a crystal approach ( abstract hide abstract )

  Abstract. The Plactic monoid, whose elements can be identified with semi-standard Young tableaux, has long been considered an important monoid. It admits several descriptions, one of them in terms of, so-called, Kashiwara crystals. Kashiwara crystals are certain recursively-constructed edge-colored directed graphs, built from a given crystal basis. The way in which crystals give rise to monoids can be described in purely combinatorial terms. In previous work, we modified this construction allowing us to characterize other monoids, referred to as plactic-like, using this approach.
  
  The question of identities satisfied by plactic-like monoid is a subject of an active study. In this talk, we will present a brief report of the results in this area and show how crystals can be used to study them. Abstract text here
  
  

In September 2020 we hosted a series of online expository lectures by Ashley Clayton (St Andrews) on "Direct and Subdirect Products in Groups, Semigroups and Algebras". The series, which was aimed chiefly at PhD students, was generously supported by an extra grant from the London Mathematical Society. You can access:

• Videos of the Lectures (on the LMS Youtube channel).
• Ashley's Lecture Notes

• Thomas Aird - Permutability of tropical matrices
• Carl-Fredrik Nyberg Brodda - The word problem for special monoids
• Scott Carson - Right ideal Howson semigroups
• Bill De Witt - Residual finiteness of monounary algebras and their direct products
• Luke Elliot - Automorphisms of the Brin-Thompson groups nV
• Ambroise Grau - The translational hull of an "almost completely zero-simple" ideal in an independence algebra
• Veronica Kelsey - Base sizes of permutation groups
• Finn Smith - Computing finite semigroups

To conclude the series we held a special eNBSAN seminar on 7th October, following on from the topic of Ashley's lectures:

• Wolfram Bentz - Direct and Subdirect Products in Groups, Semigroups and Algebras - Congruences of Direct Products

In Summer 2020, in place of physical meetings cancelled due to the COVID-19 pandemic, we held a series of online eNBSAN Seminars:

• 24th June - Igor Dolinka - The prefix membership problem for one-relator groups, and its semigroup-theoretical cousins ( abstract hide abstract | slides )

  Abstract. In early 1930s, Wilhelm Magnus proved his famous and celebrated result that the word problem is decidable for all one-relator groups: given two group words \( u,v \) over the alphabet \( X \), there is an algorithm deciding whether \( u \) and \( v \) represent the same element of the group given by a presentation of the form \( \langle X \mid w=1 \rangle \). This result is based on another important theorem proved earlier by Magnus, the Freiheitssatz, which, roughly speaking, locates many free subgroups in one-relator groups. Later on, this inspired investigations of the word problem for other algebraic structures defined by a single relation. For example, in the 1960s Shirshov proved that the word problem is decidable for all one-relator Lie algebras. Surprisingly, the problem whether the word problem is decidable for all one-relator monoids is still open (although several important cases have been resolved by Adjan in 1966, and Adyan and Oganessyan in 1987).
  
  An important intermediate class of algebraic structures lying between groups and monoids are that of inverse monoids. In 2001 Ivanov, Margolis and Meakin highlighted the importance of investigating one-relator inverse monoids by showing that the (conjectured) decidability of the word problem for one-relator special inverse monoids (the ones defined by a relation of the form \( w=1 \)) would imply a positive solution of the word problem for all one-relator monoids. They also showed that, under a condition that is very familiar within the inverse semigroup theory realm, in the \( E \)-unitary case, solving the word problem for the inverse monoid \( M \) given by the presentation \( \langle X \mid w=1 \rangle \) is equivalent to solving the prefix membership problem for the group \( G \) given by the same presentation. This problem asks whether there exists an algorithm deciding whether a given word \( u \) represents an element of the submonoid \( P_w \) of \( G \) generated by all prefixes of the relator word \( w \).
  
  In this talk I will present several recent results, obtained in collaboration with Robert D. Gray (UEA Norwich), pertaining to the prefix membership problem for a class of one-relator groups -- and thus having implications for the word problem for special one-relator inverse monoids. These results will be formulated in terms of the classical constructions in combinatorial group theory: amalgamated free products and HNN-extensions.
  
  

• 1st July - Stuart Margolis - Degree 2 Transformation Semigroups as Continuous Maps on Graphs: Foundations and Structure ( abstract hide abstract | slides )

  Abstract. We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of each point has at most two elements. We show that the graph of fibers of such an action gives a deep connection between semigroup theory and graph theory. It is known that the Krohn-Rhodes complexity of a degree 2 action is at most 2. We show that the monoid of continuous maps on a graph is the translational hull of an appropriate 0-simple semigroup. We show how group mapping semigroups can be considered as regular covers of their right letter mapping image and relate this to their graph of fibers. This is joint work with John Rhodes.
  
  

• 8th July - Yingying Feng - Min network of congruences on an inverse semigroup ( abstract hide abstract | slides )

  Abstract. A congruence on an inverse semigroup S is determined uniquely by its kernel and trace. Denoting by \( \rho_k \) and \( \rho_t \) the least congruence on S having the same kernel and the same trace as \( \rho \), respectively, and denoting by \( \omega \) the universal congruence on S, we consider the sequence \( \omega, \omega_k, \omega_t, (\omega_k)_t, (\omega_t)_k, \cdots \)
  
  We call these congruences, together with the inclusion relation for congruences, the min network of congruences on S. The quotients \( \{S/\omega_k\}, \{S/\omega_t\}, \{S/(\omega_k)_t\}, \{S/(\omega_t)_k\}, \cdots \), as S runs over all inverse semigroups, form quasivarieties.
  
  In this talk, I will talk about the repeated patterns in the resulting quotient semigroups. These patterns help us not only determine the quasivarieties to which the quotient semigroups belong, but also obtain relationships among these quasivarieties. This is joint work with Li-Min Wang, Lu Zhang, Hai-Yuan Huang and Zhi-Yong Zhou.
  
  

• 15th July - Robert Gray - Solving equations in one-relator monoids ( abstract hide abstract | slides )

  Abstract. By the Diophantine problem we mean the algorithmic problem of determining if any given system of equations has a solution. Two classical results due to Makanin (1977, 1983) show that the Diophantine problem is decidable in any free monoid, and in any free group. It is natural to ask to what extent these results can be extended to classes of monoids and groups which are "close to being free", in some sense. In this talk I will present some results from joint work with Albert Garreta (Bilbao) in which we investigated this question for one-relator monoids.
  
  

• 22nd July - Benjamin Steinberg - Homological finiteness of one-relator monoids and related monoids ( abstract hide abstract | slides )

  Abstract. The word problem for one-relator monoids is a longstanding open question. Kobayashi asked in 2000 whether every one-relator monoid admits a finite complete rewriting system. A necessary condition to have a finite complete rewriting system is to satisfy the homological finiteness condition \( FP_{\infty} \). Kobayashi also asked in 2000 whether every one-relator monoid is of type \( FP_{\infty} \) and proved that each one-relator monoid is of type \( FP_3 \).
  
  In this talk, we discuss our proof that every one-relator monoid is indeed of type \( FP_{\infty} \). Our techniques combine methods from algebraic topology (monoids acting on CW complexes), homological algebra and the theory of monoid van Kampen diagrams. This is joint work with Bob Gray.
  
  

• 29th July - Marianne Johnson - Tropical representations and identities of plactic monoids ( abstract hide abstract | slides )

  Abstract. I will report on some joint work with Mark Kambites (https://arxiv.org/abs/1906.03991). We show that the plactic monoid of every finite rank has a faithful representation by upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okniński that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank \( n \) is satisfied by the monoid of \( n \times n \) upper triangular tropical matrices. In particular this implies that the variety generated by the \( 3 \times 3 \) upper triangular tropical matrices coincides with that generated by the plactic monoid of rank \( 3 \).
  
  

The thirtieth meeting was held in Manchester on 12th July 2019, organised by Marianne Johnson and Mark Kambites. The speakers were:

• James East - Semigroups generated by idempotents and one-sided units (slides)
• Bakh Khoussainov - Open questions on automatic structures (joint talk with Manchester Pure Mathematics Colloquium and LMS Aitken Lecture) (slides)
• Finn Smith - Translational hulls of ideals (slides)
• Nóra Szakács - Inverse monoids and immersions of cell complexes (slides)

The twenty-ninth meeting was held in York on 10th and 11th January 2019, organised by Victoria Gould. The speakers were:

• Wolfram Bentz - The existential transversal property and its impact on the regularity of finite transformation semigroups
• Matthew Brookes - The lattice of left congruences on an inverse semigroup
• Scott Carson - The Howson property for one-sided ideals of a semigroup (slides)
• Laura Ciobanu - Solving equations in semigroups and groups
• Fernando Flores Brito - Congruences on End F_n(G) (slides)
• Robert Gray - A Lyndon's identity theorem for one-relator monoids
• Peter Hines - Card shuffles and Cantor space: an inverse semigroup perspective (slides)
• Marianne Johnson - Tropical matrix fountains
• Dandan Yang - Semigroups with finitely generated universal left congruence (slides)

The twenty-eighth NBSAN meeting was held in St Andrews on 14th and 15th June 2018, as a satellite meeting to the 2018 British Mathematical Colloquium. The organisers were Markus Pfeiffer and Nik Ruškuc. The speakers were:

• Tara Brough - Two applications of monoid actions to cross-sections
• Tom Coleman - Generation of intermediate and partial map monoids of first-order structures (slides)
• James East - Sandwich semigroups in locally small categories
• Justine Falque - The orbit algebra of oligomorphic permutation groups with polynomial profile - proof of a conjecture of Cameron (slides)
• Florent Hivert - The 0-rook monoid and its friends (slides)
• Marianne Johnson - Unitriangular matrix varieties
• Itamar Stein - The global dimension of the algebra of the monoid of all partial functions on an n-set (slides)

The twenty-seventh meeting was held in York on 7th and 8th January 2018, organised by Victoria Gould and Rida-E Zenab, and in honour of the 70th birthday of László Márki. Above is a group photo. The speakers were:

• Ashley Clayton - Subdirect products of free semigroups (slides)
• Igor Dolinka - Universal locally finite maximally homogeneous semigroups (slides)
• Victoria Gould - Coherency and purity for monoids (slides)
• Tatiana Jajcayová - Inverse semigroups of partial automorphisms of combinatorial structures: report on recent progress
• Valdis Laan - Morita equivalence for factorizable semigroups
• László Márki - Morita equivalence of semigroups revisited: firm semigroups (slides)
• John Meakin - Adian groups, semigroups and inverse semigroups
• Craig Miller - Right noetherian semigroups
• P. A. Azeef Muhammed - The cross-connection of the inductive groupoid of a regular semigroup (slides)
• Pham Ngoc Ánh - A generalization of Clifford's theorem (slides)
• Mária B. Szendrei - Extensions and I-semidirect products of inverse semigroups (slides)
• Abdullahi Umar - Some remarks on semigroups of contraction mappings of a finite chain (slides)
• Rida-E Zenab - Pseudo-finite monoids and semigroups (slides)
• Xia Zhang - On injective constructions of S-semigroups

The twenty-sixth NBSAN meeting was held in St Andrews on 13th July 2017, organised by Julius Jonusas. The speakers were:

• Nick Gilbert - Extensions and cohomology of inverse semigroups (slides)
• Zur Izhakian - Tropical plactic algebra and semigroup representations
• Mark Kambites - A random walk through random walks (slides)
• Matt McDevitt - Insertion relations on words
• Munazza Naz - Extensions of Green's relations on semigroups of tropical matrices (slides)

The twenty-fifth NBSAN meeting took place in York on Wednesday 18th January 2017, organised by Victoria Gould and incorporating a meeting of the York Semigroup. The speakers were:

• Wolfram Bentz (Hull) - Congruences on the product of two full transformation monoids (abstract | slides)
• Tara Brough (Lisbon) - Word problems of free inverse monoids (abstract)
• Julius Jonusas (St Andrews) - Universal words and sequences (abstract | slides)
• Emma McDougall (Heriot-Watt) - Relation modules for inverse monoids
• Stuart Margolis (Bar Ilan) - Projective indecomposable modules and quivers for a class of Fountain monoids with applications to descent algebras (abstract)
• Peter Neumann (Oxford) - Galois and his groups (York Semigroup Talk, abstract)

The twenty-fourth NBSAN meeting was held in York on Wednesday 11th May 2016, organised by Victoria Gould and with a focus on semigroup algebras and connections with functional analysis. The speakers were:

• Garth Dales (Lancaster) - Ideals in βS (slides)
• Robert Gray (East Anglia) - Cellular and standardly based semigroup algebras (slides)
• Chris Hollings (Oxford) - A brief history of semigroup representations (slides)
• Mark Kambites (Manchester) - Some conditions related to amenability
• Catarina Santa-Clara (Lisbon) - The semigroup ring of a restriction semigroup with an inverse skeleton (slides)
• Dona Strauss (Leeds) - Idempotents in βS (slides)

The twenty-third NBSAN meeting was held in St Andrews on Thursday 17th and Friday 18th March 2016, organised by Markus Pfeiffer. The speakers were:

• Ying-Ying Feng (Foshan and York) - Two kinds of congruence networks on regular semigroups (slides)
• Marianne Johnson (Manchester) - Face monoid actions and tropical hyperplane arrangements
• Mark-Jan Nederhof (St Andrews) - A short proof that O_2 is an MCFL (slides)
• Michael Torpey (St Andrews) - Computing with semigroup congruences (slides)
• Rebecca Waldecker (Martin Luther Halle-Wittenberg) - Applying permutation group theory
• Wilf Wilson (St Andrews) - Computing maximal subsemigroups of finite semigroups (slides)

• Peter Fenner (Manchester) - The gossip monoid
• Maximilien Gadouleau (Durham) - Universal simulation of automata networks (slides)
• Stuart Margolis (Bar Ilan) - Poset cohomology, CW decompositions and the global dimension of left regular band algebras (slides)
• James Mitchell (St Andrews) - Semigroups in GAP: theory and practice (joint session with the CoDiMa training school)
• Colva Roney-Dougal (St Andrews) - Relations relating to generation of groups and semigroups

The twenty-first NBSAN meeting was held at the University of Essex, in Colchester, on 10th July 2015, organised by Peter Higgins, Alexei Vernitski and Gerald Williams. The speakers were:

• Tara Brough - Automaton semigroups: some new examples and non-examples
• Jack Button (Cambridge) - A summary of recent results on 1-relator groups (abstract)
• Andrew Duncan (Newcastle) - Equations over hyperbolic groups and semigroups (abstract)
• Nick Loughlin (Newcastle) - Understanding idempotents in diagram semigroups and algebras (abstract)
• Esamaldeen Mohamed (Essex) - Isomorphisms amongst cyclically presented groups (abstract)
• Jim Renshaw (Southampton) - Free products and amalgams of pomonoids (abstract)

The twentieth NBSAN meeting was held in St Andrews on 22nd and 23rd April 2015, organised by Markus Pfeiffer. Peter Cameron wrote a very nice blog post on the meeting. The speakers were:

• Paul Bell (Loughborough) - Complexity of reachability, mortality and freeness problems for matrix semigroups (slides)
• Tom Bourne (St Andrews) - Tackling the generalised star-height problem (slides)
• Alan Cain (Universidade Nova de Lisboa) - Computations and conjugacy in hypoplactic and sylvester monoids and other homogenous monoids (slides)
• Peter Cameron (St Andrews) - Finding where you are: automata, graph endomorphisms and foldings (slides)
• Rachael Carey (St Andrews) - Graph automatic semigroups (slides)
• Robert Gray (East Anglia) - Crystal monoids (slides)
• Ruth Hoffmann (St Andrews) - Geometric grid permutation classes and regular languages
• Matthew Taylor (Manchester) - A classification of 2-generated T-modules

The nineteenth NBSAN meeting was held in York on 14th January 2015, organised by Victoria Gould. Speakers were:

• Lovkush Agarwal (Leeds) - The reducts of the generic digraph
• Bernard Bainson (Heriot-Watt) - Khovanov's presheaf on some ordered groupoids (slides)
• Wolfram Bentz (Lisbon) - New results on synchronization
• Igor Dolinka (Novi Sad) - Variants of semigroups - the case study of finite full transformation monoids (slides)
• Rowena Paget (Kent) - A representation theory approach to the rook monoid (slides)
• Goncalo Pinto (College of the Bahamas) - Residuated completely simple semigroups (slides)
• Tom Quinn-Gregson (York) - ℵ₀-categorical semigroups (slides)
• Itamar Stein (Bar-Ilan) - The ordinary quiver of the algebra of the monoid of all partial functions on a set (slides)

The eighteeth NBSAN meeting was held in Edinburgh on Monday 21st July 2014, organised by Nick Gilbert. The speakers were:

• Karl Auinger (Vienna) - On the membership problem for pseudovarieties (slides)
• James East (Western Sydney) - Idempotent generators in diagram semigroups (slides)
• Ganna Kudryavtseva (IMFM Ljubljana) - On non-commutative frame theory (slides)
• Laurent Poinsot (Paris Nord XIII) - From combinatorial monoids to algebras, bialgebras and Hopf algebras, functorially (slides)

The seventeenth NBSAN meeting was held in Norwich on Monday 14th and Tuesday 15th April 2014, organised by Robert Gray. The speakers were:

• Catarina Carvalho (Hertfordshire) - Constraint satisfaction problems and dualities (slides)
• Cong Chen (Leeds) - The word norm
• Igor Dolinka (Novi Sad) - Free idempotent generated semigroups: maximal subgroups and the word problem (slides)
• Jan Foniok (Warwick) - Constraint satisfaction problems with tree duality (slides)
• Peter Higgins (Essex) - Languages that require full scanning of words to determine membership (slides)
• Rick Thomas (Leicester) - Word problems of groups, formal languages and decidability (slides)
• Alexei Vernitski (Essex) - From knot groups to knot semigroups (slides)
• Dandan Yang (York) - Free idempotent generated semigroups and endomorphisms (slides)

The sixteenth NBSAN meeting was held in York on Wednesday 20th November 2013, organised by Victoria Gould. The speakers were:

• Jenni Awang - Semigroups, Cayley graphs and finite presentation (abstract)
• Alan Cain - Endomorphisms of semigroups: growth and interactions with subsemigroups (abstract | slides)
• Robert Gray - Ideals and finiteness conditions for subsemigroups (abstract)
• Valdis Laan - Congruence extension property for ordered algebras (abstract | slides)
• John Truss - Some monoids and notions of homomorphism-homogeneity (abstract)
• Yanhui Wang - Left restriction semigroups from incomplete automata (abstract | slides)

The fifteenth NBSAN meeting was held in Manchester on Wednesday 24th July 2013, organised by Marianne Johnson and Mark Kambites. The speakers were:

• Alex Bailey - Subsemigroup growth of finitely generated free semigroups (slides)
• John Meldrum - Some aspects of endormorphism semigroups
• Markus Pfeiffer - Deciding word problems with finite state devices
• Uday Reddy - Transformation monoids in programming language semantics (slides)
• Alistair Wallis - K-theory of inverse semigroups (slides)
• Rida-E Zenab - Semigroups with skeletons and Zappa-Szép products (slides)

The fourteenth NBSAN meeting was held in St Andrews on Tuesday 9th April 2013, organised by Yann Peresse. The speakers were:

• Pham Ngoc Ánh - Divisibility, ideal and valuation theory of classical commutative rings
• Victoria Gould - Coherent monoids (slides)
• James Hyde - Sierpinski rank and universal sequences (slides)
• Julius Jonusas - Maximal subsemigroups of the transformation monoid
• Max Neunhöffer - Generalisations of small cancellation (slides)
• David Wilding - Green's D relation for matrices over semirings

The thirteenth NBSAN meeting was held in York on Wednesday 21st November 2012, organised by Victoria Gould. The speakers were:

• Igor Dolinka (Novi Sad) - Finite groups are big as semigroups (slides)
• David Janin (Bordeaux) - Walking in free inverse monoids (slides)
• Peter Jones (Marquette) - The semigroups B_2 and B_0 are inherently nonfinitely based (slides)
• Alex McLeman (St Andrews) - Cayley automaton semigroups (slides)
• Dandan Yang (York) - Free idempotent-generated semigroups (slides)

The twelfth NBSAN meeting (which was the first Young NBSAN meeting) was held at ICMS in Edinburgh on Monday 23rd July 2012, organised by Alex Bailey, Alistair Wallis and Yanhui Wang. The speakers were....

• Suhear Alwan (Essex) - Languages that require full scanning of words to determine membership
• Tara Brough (St Andrews) - Automaton semigroup constructions
• Zsofia Juhasz (Essex) - Congruences induced by Green's relations and the corresponding egg-box pictures
• Mark Kambites (Manchester) - An introduction to the Cerny conjecture (expository talk)
• Bana Al Subaiei (Southampton) - Amalgamation in pomonoids and dominions
• Alina Vdovina (Newcastle) - Short introduction to expander graphs

The eleventh NBSAN meeting was held in Manchester on Wednesday 11th April 2012. The speakers were:

• Des FitzGerald (Tasmania) - Categorification for binary relations (abstract | notes)
• Miklos Hartmann (York) - E-unitary almost factorizable orthodox semigroups (slides)
• Peter Higgins (Essex) - Products in transformation semigroups: work with the late John M. Howie (slides)
• Sarah Rees (Newcastle) - Artin groups and monoids; normal forms and the group word problem (slides)

The tenth NBSAN meeting was held in York on Wednesday 23rd November 2011. The speakers were:

• Peter Cameron - Permutation groups and transformation semigroups (abstract | slides)
• Jean-Eric Pin - Expansions and covers (abstract | slides)
• Rick Thomas - The word problem for semigroups (abstract | slides)
• Yanhui Wang - Weakly U-abundant semigroups (abstract | slides)

The ninth NBSAN meeting was held in Manchester on Tuesday 30th August 2011. The speakers were:

• Alex Bailey (Southampton) - Flat covers of acts over monoids
• Victoria Gould (York) - Restriction semigroups (an expository talk) (slides)
• Robert Gray (Lisbon) - Structures with lots of symmetry (handout slides | overlay slides)
• Christopher Hollings (Oxford) - The perils of taking shortcuts: embedding semigroups in the 1930s (slides)
• Lubna Shaheen (York) - Model companions of S-posets

The eighth NBSAN meeting was held at the University of St Andrews on Thursday 19th and Friday 20th May 2011. The speakers were:

• Alan Cain (Porto) - Hyperbolic and word-hyperbolic semigroups (slides)
• Si Craik (St Andrews) - Ends of semigroups (slides)
• Andrew Duncan (Newcastle) - Rewriting systems and the geodesic problem for words and cycles
• John Fountain (York) - An inverse monoid approach to Thompson's group V and generalisations (slides)
• Marianne Johnson (Manchester) - Green's J-order and the rank of tropical matrices (slides)
• Dave Jones (Heriot Watt) - Graph Inverse Semigroups (slides)
• Mark Kambites (Manchester) - An Introduction to Tropical Matrix Semigroups
• Alistair Wallis (Heriot Watt) - Rees monoids, self-similar groups and fractals (slides)

The seventh NBSAN meeting was held at the University of York on Wednesday 24th November 2010. The speakers were:

• Bassima Afara (Heriot-Watt) - The Rees theorem and its generalizations (slides)
• Peter Jones (Marquette) - Varieties of P-restriction semigroups (slides)
• Volodymyr Mazorchuk (Uppsala) - Kiselman type semigroups
• Maria Szendrei (Szeged) - Some results on almost factorisable semigroups (slides)
• Alexei Vernitski (Essex) - Graphs to semigroups (slides)

The sixth NBSAN meeting was held at the University of Manchester on Monday 26th July 2010. The speakers were:

• Zur Izhakian (Bar Ilan) - Supertropical algebra
• Erzsi Dombi (Glasgow) - Automatic semigroup acts
• Rick Thomas (Leicester) - FA-presentable semigroups

The fifth NBSAN meeting was held at the University of St Andrews on Friday 7th and Saturday 8th May 2010, organised by James Mitchell. Speakers were:

• Andreas Distler (St Andrews) - How to store 10 semigroups in a bit
• James East (Sydney) - Generators and relations for monoids of block bijections and partitions
• Attila Egri-Nagy (Hertfordshire) - Coordinatizing finite permutation groups and transformations semigroups - computational considerations
• Victoria Gould (York) - Bisimple inverse semigroups as semigroups of quotients
• Zsofia Juhasz (Essex) - The smallest operation-compatible quasiorders containing Green's quasiorders
• Zak Mesyan (Ben Gurion) - Conjugation of injections by permutations
• Markus Pfeiffer (St Andrews) - Semigroups with easily solveable word problem
• Steve Pride (Glasgow) - On diagram groups
• Abdullahi Umar (Sultan Qaboos) - Some combinatorial properties of semigroups of partial transformations

The fourth NBSAN meeting was held at the University of York on Wednesday 25th November 2009, organised by Victoria Gould. Speakers were:

• Claire Cornock (York) - Structure theorems for proper restriction semigroups
• Nick Gilbert (Heriot-Watt) - Approaches to tiling semigroups
• Michael Hoffmann (Leicester) - Hyperbolic monoids
• Marianne Johnson (Manchester) - Tropical matrix algebra
• Jim Renshaw (Southampton) - Adequate transversals of abundant semigroups (abstract | slides)
• Lubna Shaheen (York) - Perfection for pomonoids

The third NBSAN meeting was held on 23rd July 2009 at Heriot-Watt University in Edinburgh, organised by Mark Lawson. The speakers were:

• Chris Eilbeck (Heriot-Watt) - Weierstrass arithmetic semigroups and algebraic curves
• Jennie Hansen (Heriot-Watt) - Using probability theory to study the full transformation monoid
• Peter Higgins (Essex) - Set products in transformation semigroups
• Stuart Margolis (Bar-Ilan) - On some monoids associated to Coxeter groups
• James Mitchell (St Andrews) - Generating uncountable transformation semigroups

The second NBSAN meeting was held in St Andrews on 16th and 17th April 2009. The speakers were:

• João Araújo (Lisbon) - Guess Who
• Alan Cain (Lisbon) - Automaton Semigroups
• Robert Gray (St Andrews) - Homotopical and Homological Finiteness Conditions of Monoids and their Subgroups
• Mark Kambites (Manchester) - Free Adequate Semigroups
• Vesna Kilibarda (Indiana Northwest) - Ends for Semigroups and Monoids
• Valdis Laan (Tartu) - Tensor Products and Preservation of Weighted Limits for S-posets
• Mark Lawson (Heriot-Watt, expository talk) - The Morita Theory of Regular Semigroups
• László Márki (Rényi Institute) - A Matrix Construction for certain Inverse Monoids with an Application in Universal Algebra

The inaugural NBSAN meeting was held in York on 28th January 2009, and was dedicated to the memory of Professor Douglas Munn (1929-2008). The speakers were:

• John Fountain (York, expository talk) - The Mathematical Work of Douglas Munn and its Influence
• Nik Ruškuc (St Andrews, open problem talk) - One Relation Semigroups
• Rebecca Noonan Heale (Heriot-Watt) - Green's Relations for Semigroup Graph Expansions
• Yann H. Peresse (St Andrews) - Generating Transformation Semigroups using Endomorphisms of Preorders, Graphs and Similarity Relations
• Pedro V. Silva (Porto) - The Generalized Conjugacy Problem for Virtually Free Groups
• Jorge Almeida (Porto) - The Mysteries of Free Profinite Semigroups: a Sneak Preview