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The conference program

Each of the invited speakers will give a 50 minutes lecture. In addition, there will be contributed talks of 20 minutes by the participants. Full timetable will be available closer to the beginning of the conference.

A printable pdf file of the abstracts is also available.

Invited lectures

Jorge Almeida (University of Porto)
Title: Schützenberger groups of primitive substitutions are decidable
Abstract: Joint work with Alfredo Costa. It is well known that the words that appear as factors in the iteration on the letters of a primitive endomorphism (substitution) $\varphi$ of the free semigroup on a finite alphabet $A$ is the language of blocks of a minimal symbolic dynamical system (subshift) ${\cal X}_\varphi\subseteq A^\mathbb{Z}$. I showed that, associating to a minimal subshift $\cal X$ over the alphabet $A$ the set $J(\cal X)$ of all infinite pseudowords in the closure of its language of blocks in the profinite semigroup freely generated by $A$, one obtains a bijection between minimal subshifts and $\cal J$-maximal regular $\cal J$-classes. Alfredo Costa showed that, viewed as an abstract group $G(\cal X)$, the maximal subgroups of $J(\cal X)$ constitute a conjugacy invariant. I also showed that, if $\varphi$ induces an automorphism of the free group on $A$, then $G(\cal X_\varphi)$ is a free profinite group, while there are examples for which $G(\cal X_\varphi)$, which is always finitely generated, is not a free profinite group.

Rhodes and Steinberg proved that the closed subgroups of a free profinite semigroup are projective as profinite groups. Hence, as observed by Lubotzky, if finitely generated, they admit finite presentations, as profinite groups, in which the relations simply state that each generator is a fixed point of a retract of the free profinite group. I conjectured that, under special conditions on the primitive substitution $\varphi$, the group $G(\cal X_\varphi)$ admits such a presentation in which the retract is obtained as a (profinite) idempotent iterate of a positive finite continuous endomorphism $\varphi'$ of the free profinite group, where $\varphi'$ can be effectively computed from $\varphi$. The interest in such presentations stems from the fact that the relations can be effectively checked in a given finite group, so that the group with such a retract presentation has decidable finite quotients. It turns out that the conjecture holds for every primitive substitution $\varphi$. It is therefore decidable whether a finite group is a quotient of $G(\cal X_\varphi)$. The proof of the conjecture in such a wide setting depends on a synchronization result of Mossé for (biinfinite) fixed points of primitive substitutions.

Carlos André (University of Lisbon)
Title: Schur algebras and representations of the rook monoid
Abstract: The rook monoid $R_{n}$ is the monoid consisting of $n \times n$ matrices having entries from $\{0,1\}$ with at most one nonzero entry in each row and column. It contains an isomorphic copy of the symmetric group $S_{n}$ as the rank $n$ (permutation) matrices. The representation theory of $R_{n}$ was originally determined by W.D. Munn and furthered by L. Solomon. A ``Young's natural representation'' for $R_{n}$, using rook-monoid analogues of Young symmetrizers, was computed by C. Grood. In particular, Solomon discovered a Schur-Weyl duality between $R_{n}$ and the general linear group $GL_{m}(\mathbb C)$ on tensor space $(V \oplus\mathbb C)^{\otimes n}$ where $n \leq m$ and $V = \mathbb C^{m}$ is the natural module for $GL_{m}(\mathbb C)$; this duality is a generalisation of the original Schur-Weyl duality between $S_{n}$ and $GL_{m}(\mathbb C)$ on tensor space $V^{\otimes n}$. Later, R. Paget proved that the rook monoid algebra $K[R_{n}]$ (over an arbitrary field $K$) is a cellular algebra, and showed it is Morita equivalent to the direct sum $K[S_{n}] \oplus K[S_{n-1}] \oplus \cdots \oplus K[S_{1}] \oplus K$. This means that the representation theory of the $K[R_{n}]$ is the same as that of the above direct sum of symmetric group algebras; in the case where $K$ has characteristic zero, this result was already present in Munn's work and plays an essential role in the representation (and character) theory of $R_{n}$.

In virtue of the Schur-Weyl duality, it is natural to expect that the representation theory of $R_{n}$ can be derived from the (polynomial) representation theory of $GL_{m}(\mathbb C)$. We work over an arbitrary infinite field $K$. The coefficient space $\mathcal{A}$ of the natural representation of $GL_{m}(K)$ on $(V \oplus K)^{\otimes n}$ has the structure of a coalgebra, and hence the dual space $\mathcal{S} = \mathcal{A}^{\ast}$ has the dual structure of a (finite-dimensional) associative algebra; in fact, it is a direct sum of Schur algebras as defined by J.A. Green. The polynomial representations of $GL_{m}(K)$ of degree $\leq m$ are in one-to-one correspondence with representations of $\mathcal{S}$. In this talk, we follow the Schur algebra approach and use Schur functors to show how the representation theory of $R_{n}$ is naturally obtained from the representation theory of $\mathcal{S}$. In particular, in the characteristic zero situation, we obtain Grood's analogue of the Spetch modules for $R_{n}$ (and recover the Munn-Paget reduction to the representation theory of the direct sum $K[S_{n}] \oplus K[S_{n-1}] \oplus \cdots \oplus K[S_{1}] \oplus K$.)

In a final part of the talk, we outline a Lie theoretic approach to the representation theory of $R_{n}$ which extends the methods introduced by Okounkov and Vershik for symmetric groups. To be specific, we use natural embeddings $R_{0} \subseteq R_{1} \subseteq \cdots \subseteq R_{n-1} \subseteq R_{n}$ to define the \textit{branching graph}, which turns out to be isomorphic to the Young graph for the rook monoid.

Joint work with Inês Legatheaux Martins.

Peter Cameron (Queen Mary, University of London)
Title: Dixon's theorem and the probability of synchronization
Abstract: The probability that a single permutation of $\{1,\ldots,n\}$ generates a transitive subgroup of the symmetric group $S_n$ is trivially seen to be $1/n$. According to the easy part of Dixon's theorem, the probability that two permutations generate a transitive subgroup is $1-1/n+O(1/n^2)$. (Incidentally, the number of such pairs is equal to $(n-1)!$ times the number of connected permutations on $\{1,\ldots,n+1\}$, but a bijective proof is not known.)

A transformation monoid on $\{1,\ldots,n\}$ is synchronizing if it contains an element whose image has cardinality $1$. The probability that a random element of the full transformation monoid $T_n$ generates a synchronizing submonoid is $1/n$: the simple proof uses Joyal's proof of Cayley's formula for trees. I conjecture, but cannot yet prove, that the probability that a random pair of transformations generates a synchronizing monoid is close to $1$ for large $n$, perhaps $1-1/n+O(1/n^2)$ as in Dixon's theorem.

A proof of this would involve an analysis of maximal non-synchronizing submonoids.

Bettina Eick (Technical University of Braunschweig)
Title: Computing associative algebras satisfying a polynomial identity
Abstract: Let $A(d,n,F)$ denote the maximal associative $F$-algebra on $d$ generators satisfying the identity $x^n = 0$. Then $A(d,n,F)$ can be considered as an analogue to the Burnside group $B(d,n)$. We show how $A(d,n,F)$ can be computed for all fields $F$ of a given characteristic and we discuss some applications.

John Fountain (University of York)
Title: Inverse monoids, Thompson's group $V$ and generalisations
Abstract: In the 1960s Richard Thompson constructed the infinite groups $F < T < V$ in connection with his work in logic. Since then it has been discovered that these groups arise in many contexts and have many different descriptions. They have been widely studied because of their interesting algebraic and combinatorial properties.

One way of defining $V$ is as the maximum group homomorphic image of an inverse monoid obtained from the free monoid on two generators. This approach is due to Lawson and (independently) Birget and evolved from descriptions of $V$ by Higman, then Scott and then Birget. Higman's original construction works for $n$ generators and so do the subsequent descriptions. The inverse monoid involved consists of all (right ideal) isomorphisms between finitely generated essential right ideals, and the multiplication is composition of partial maps.

The construction can be generalised by starting with an arbitrary right cancellative monoid, and several of the generalisations of $V$ can be obtained in this way.

Robert Gray (University of Lisbon)
Title: Free Idempotent Generated Semigroups and their Maximal Subgroups (joint work with Nik Ruskuc)
Abstract: Let $S$ be a semigroup with set $E(S)$ of idempotents, and let $\langle E(S) \rangle$ denote the subsemigroup of $S$ generated by $E(S)$. We say that $S$ is an idempotent generated semigroup if $S = \langle E(S) \rangle$. Idempotent generated semigroups have received considerable attention in the literature, and there are many natural examples of semigroups with this property, such as the semigroup of all non-invertible mappings from a finite set to itself, and also the semigroup of all non-invertible $n \times n$ matrices over an arbitrary field.

The set of idempotents of an arbitrary semigroup has the structure of a so called biordered set (or regular biordered set in the case of von Neumann regular semigroups). These structures were studied in detail in work of Nambooripad (1979) and Easdown (1985). There is a free object $IG(E)$ in the category of idempotent generated semigroups with biordered set $E$ for an arbitrary biordered set $E$. Clearly, it is important to understand $IG(E)$ if one is interested in understanding an arbitrary idempotent generated semigroup with biordered set of idempotents $E$.

For semigroup theoretic reasons, much of the structure of $IG(E)$ comes down to understanding the structure of its maximal subgroups. It was conjectured by McElwee in 2002 that the maximal subgroups of such a semigroup are all free (in fact, this had been conjectured since the early 1980s). The first counterexample to this conjecture was given by Brittenham, Margolis and Meakin (2009), where it was shown that the free abelian group of rank 2 is a maximal subgroup of the free idempotent generated semigroup arising from the set of idempotents of a certain 72-element semigroup.

In this talk I will present some further recent developments in our understanding of free idempotent generated semigroups and their maximal subgroups.

Zur Izhakian (Bar-Ilan University)
Title: Supertropical algebra and representations
Abstract: Traditionally, matroids and semigroups have been represented by using matrices defined over fields, we extend this notion by introducing new representations by matrices over semirings, more precisely over supertropical semirings. In order to bypass the lack of additive inverses in semirings, we consider a supertropical semiring - a "cover" semiring structure that has a distinguished "ghost ideal" taking the place of the zero element in many of the theorems. This supertropical structure is rich enough and permits a systematic development both of polynomial algebra and matrix algebra, yielding direct analogs to many results and notions from classical commutative algebra. These provide a suitable algebraic framework allowing natural representations of matroids, hereditary collections, and semigroups.

Olga Kharlampovich (McGill University)
Title: Cayley automatic groups
Abstract: This is a joint work with B. Khoussainov and A. Miasnikov. We introduce the concept of Cayley automatic groups. This class satisfies many group theoretic properties. Cayley automatic groups are invariant under the change of generators, closed under finite extensions, the direct and free products, and certain types of amalgamated products. Cayley automatic groups include the class of automatic groups in the sense of the book by Epstein and others "Word Processing in Groups", and shares many algorithmic properties. For instance, the word problem in Cayley automatic groups is decidable in quadratic time just like for Thurston-automatic groups. As nice examples, we prove that all nilpotent groups nilpotency class of at most two and Baumslag-Solitar groups $B(1,n)$ are Cayley automatic.

Michael Kinyon (University of Denver)
Title: Inverse Moufang semiloops and their transformation semigroups
Abstract: A Moufang semiloop is a magma (set with a binary operation) satisfying the Moufang identities. Examples include semigroups, Moufang loops and the multiplicative magmas of alternative rings. The study of Moufang semiloops is an area in which ideas from both semigroup/monoid theory and quasigroup/loop theory blend. It is also an area which, so far, has remained virtually unexplored. In this talk, I will discuss inverse Moufang semiloops, which are those in which each element has a unique inverse, or equivalently, those which are regular (in the sense most appropriate for Moufang semiloops) and have commuting idempotents. Green's relations and the natural partial order generalize well to inverse Moufang semiloops. If $M$ is an inverse Moufang semiloop, then there is no right regular representation for $M$ per se, so instead we consider the right multiplication semigroup $S$ generated by all right translation maps $\rho_x : M\to M; y \mapsto yx$. It turns out that $S$ is an inverse semigroup. In addition, $M$ has a Vagner-Preston representation as a Moufang semiloop of partial mappings on $M$, which generate an inverse semigroup isomorphic to $S$, which is a subsemigroup of the symmetric inverse monoid on $M$. (This isomorphism is a nontrivial fact compared to the associative case). Each $\mathcal{H}$-class of an inverse Moufang semiloop is a Moufang loop. Every E-unitary inverse Moufang semiloop embeds into a semidirect product of a Moufang loop and a semilattice. In general, Green's theory, Rees' theory, McAlister's theory and Meakin's theory transfer very well to inverse Moufang semiloops. In the unlikely event there is time at the end of the talk, I will suggest some research directions for semigroupists who wish to think nonassociatively.

Charles Leedham-Green (Queen Mary, University of London)
Title:: Towards classification of $p$-groups by coclass up to isomorphism
Abstract: The coclass project is joint work with very many people. The recent developments I wish to discuss are joint work with Bettina Eick, Mike Newman, and Eamonn O'Brien.

The coclass project is a sister of the project of Evgeny Khukhro and his collaborators, in that both projects give structure theorems about all finite $p$-groups. A $p$-group of order $p^n$ and nilpotency class $c$ has coclass $r=n-c$. The coclass project studies $p$-groups by regarding the coclass as the primary invariant. Given a $p$-group $P$, typical coclass theorems give strong structure theorems concerning $P/N$, where $N$ is some `small' subgroup of $P$; that is to say, the order of $N$ will be bounded by some given function of $p$ and $r$. Unfortunately, this bound will often exceed the order of a given $p$-group of coclass $r$; so although our theorems apply to all $p$-groups, they only apply usefully to some $p$-groups. Similar remarks apply to Khukhro's project, where one studies $p$-groups with small automorphism groups and small (in various senses) fixed point sets. Again his theorems apply to all $p$-groups, but only apply usefully to some.

Our present aim is to prove a different kind of theorem, that will give a recipe for describing all finite $p$-groups of coclass $r$ up to isomorphism. There are infinitely many such groups for any given prime $p$ and coclass $r$; but we hope (optimistically) to be able to show how all these groups may be defined by a finite number of parametrised presentations.

The catch is that we will not, in almost all cases, have any hope of producing these presentations; so the recipe, if it exists, will consist of instructions that, while algorithmic, cannot be carried out as the output would, in general, be ridiculously large.

The cases $p=2$ (for any $r>0$) and $(p,r)=(3,1)$ are the easiest. Indeed the recipe for constructing the parametrised presentations for $p=2$ is known, and for $(p,r)=(3,1)$ they are easily produced. It seems that the only other cases in which it will be practical to produce these presentations, with $p$ odd, are the cases $(p,r)=(3,2)$ and $(p,r)=(5,1)$. I shall discuss the case $(p,r)=(3,2)$ as a prototype for the general case.

Alex Lubotzky (Hebrew University)
Title: Sieve methds in group theory
Abstract: Sieve methods have played an important role in number theory, usually as a method to find primes within various sets of integers. It has been realized recently that these classical methods can be extended to 'non-commutative' setting, giving a methods 'to measure' various subsets of finitelly generated groups. Here property 'tau' and expanders are playing an important role. We will describe some of these developments and their applications.

Cheryl Praeger (University of Western Australia)
Title: Finding Involutions in finite classical groups and symmetric groups
Abstract: Estimating the proportions of important classes of elements in infinite families of finite simple groups is a crucial ingredient for justifying Monte Carlo algorithms for computing with these groups. I will discuss how Lie-theoretic methods, originally developed to study the representations of Lie type simple groups, have been developed further into a useful estimation theory for Lie type groups. I will use this approach to answer three questions related to finding involutions in simple groups. Solving one of these problems led to the discovery of new results on element distributions in finite symmetric groups.

Ákos Seress (Ohio State University and University of Western Australia
Title: Constructive recognition of linear groups in all characteristic
Abstract: We describe an algorithm for the constructive recognition of the groups $SL(n,q)$, in their natural representation. The algorithm is not based on centralizer computations, and works in all characteristics. The running time is $O^~(n^3)$, that is, $n^3$ multiplied by some $\log n, \log q$ factors, plus one instance of the constructive recognition of $SL(2,q)$ in its natural representation.

Pedro Silva (University of Porto)
Title: Monoids acting on trees
Abstract: Given an arbitrary monoid $M$, we can consider an aperiodic expansion $M'$ of $M$ and build a (possible infinite) rooted tree $T$ so that $M'$ acts on $T$ in a very nice way. The tree $T$ and the action are built by means of a Lyndon-Chiswell length function on $M'$ which can be canonically recovered from the action itself. It turns out that this action of $M'$ on $T$ yields a particularly favourable (by means of the Zeiger property) embedding of $M'$ into a (possibly infinite) wreath product of partial transformation monoids consisting of either permutations or constant mappings. If $M$ is finitely generated, these partial transformation monoids are all finite, and $T$ is locally finite. Moreover, $T$ is finite if and only if $M$ is finite. These results lead to alternative proofs of the following generalizations of the Krohn-Rhodes decomposition theorem, where a right zero monoid is supposed to be a right zero semigroup with an identity adjoined:

Theorem 1. Every monoid divides a (possibly infinite) wreath product of groups and right zero monoids.

Theorem 2. Every finitely generated monoid divides a (possibly infinite) wreath product of finite groups and finite right zero monoids.

This talk presents joint work with John Rhodes.

Mikhail Volkov (Ural Federal University)
Title: Relatively inherently non-finitely-based varieties and quasivarieties
Abstract: A locally finite variety $\mathbf{V}$ is \emph{\infb} if no locally finite finitely-based variety can contain $\mathbf{V}$. A finite algebra $A$ is said to be inherently non-finitely-based if so is the variety $\mathsf{Var}\, A$ generated by $A$. The term "inherently non-finitely-based" was suggested by Perkins in 1984, while the very first example of an inherently non-finitely-based finite algebra (in fact, a 3-element groupoid) was exhibited by Murskii in 1979. One of the highlights of the theory of semigroup varieties is a classification of inherently non-finitely-based semigroups obtained by Sapir in 1987.

If a semigroup $S$ is inherently non-finitely-based, then each interval between $\mathsf{Var}\, S$ and a locally finite variety containing $S$ consists of non-finitely-based varieties. Studying the finite basis problem for various classes of finite semigroups that cannot be inherently non-finitely-based (for instance, for $\mathcal{J}$-trivial semigroups) reveals a similar picture that, however, cannot be explained within the concept of inherent non-finite basability. Looking for a suitable relaxation of the concept, Jackson and the author have suggested (in the context of quasivarieties) the notion of \emph{relative} inherent non-finite basability. Let $\mathcal{P}$ be a property of varieties. A variety $\mathbf{V}$ with this property is said to be inherently non-finitely-based relative to $\mathcal{P}$ if no finitely-based variety with property $\mathcal{P}$ can contain $\mathbf{V}$. When $\mathcal{P}$ is the property of being locally finite, we obtain the notion of inherently non-finitely-based variety. When $\mathcal{P}$ is the property of being finitely-generated, we obtain another notion that has been studied in the literature, namely, the notion of strongly non-finitely-based variety. We present several new examples of relatively inherently non-finitely-based varieties and quasivarieties and formulate a few open problems.

Efim Zelmanov (University of California)
Title: Complexity of semigroups and algebras
Abstract: We will discuss complexity functions associated with finitely presented semigroups and algebras. In particular, we will focus on complexity of Groebner-Shirshov algorithms.

Contributed talks

Alex Bailey (University of Southhampton)
Title: Flat covers of acts over monoids
Abstract: In 1959 H Bass characterised those rings over which every module has a projective cover, these are known as perfect rings. Similar work was done by J Fountain in 1976 for perfect monoids, that is where all their acts have projective covers. In 2001 Bican, Bashir and Enochs finally solved the long standing conjecture that every module over a ring has a flat cover. There has been some very recent work on flat covers of acts over monoids, but there are still many unanswered questions.

Sergey Bakulin (Stevens Institute of Technology)
Title: Covers of Rees-Sushkevich varieties
Abstract: In this paper all covers of the variety generated by all completely 0-simple semigroups over groups of exponent dividing $n$ are described in the lattice of all semigroup varities. Also we proved that there exists a limit finitely generated combinatorial semigroup variety.

Tara Brough (University of Kiel)
Title: Proving groups not to be poly-context-free
Abstract: A finitely generated group is called poly-context-free if its word problem is an intersection of finitely many context-free languages. The only known poly-context-free groups are virtually finitely generated subgroups of direct products of free groups. Thus the only known soluble poly-context-free groups are the virtually abelian groups.

I will outline some methods for proving a language not to be poly-context-free and explain how these are applied to prove certain groups not to be poly-context-free, resulting in a proof that a torsion-free soluble group is poly-context-free if and only if it is virtually abelian.

Eliana Castro (University of Lisbon)
Title: The construction of groups of fourth-power-free order
Abstract: The problem of constructing all finite groups of a given order up to isomorphism has a long history; it was initiated by Cayley in 1854. It is a problem with a simple formulation: given a positive integer $n$, the objective is to find, up to isomorphism, a list of groups of order $n$. However, the answer to this problem is not evident and it depends on the number $n$ we are considering. More precisely, it depends on the factorization of $n$ into prime numbers.

Gaschütz (1953) suggested an approach to construct finite groups which, in 2005, was applied by Dietrich and Eick to determine groups of cube-free order. Our main goal is to extend the existing algorithms in order to construct groups of fourth-power-free order. The algorithm will be first applied to construct all Frattini-free groups of order dividing a positive fourth-power-free integer $n$. Then, for each one of these groups, it will be used to obtain the Frattini extensions of order $n$.

In this talk, we will present the main steps of the Gaschütz approach applied to our case.

Serena Cicalo (University of Cagliari)
Title: An effective version of the Lazard correspondence
Abstract: The Lazard correspondence defines a bijection between $p$-groups of order $p^n$ and nilpotency class less than $p$, and Lie rings of the same order and nilpotency class. We show how we can effectively set up this correspondence for a given $p$-group. This involves computing the inverse BCH-formulas up to a given weight. We discuss some applications: computing Deep-Thought polynomials, and computing faithful modules. This is joint work with Willem de Graaf and Michael Vaughan-Lee.

Brian Corr (University of Western Australia)
Title: Constructive Recognition of the Alternating Square Module of Classical Groups
Abstract: The Matrix Group Recognition Project aims to produce algorithms to recognise the group generated by an arbitrary set of matrices acting on a vector space. The overarching strategy is to decompose the group into parts based on geometric invariants (for example, invariant subspaces of $V$) following Aschbacher's Theorem. The `building blocks', then, are the finite simple groups, and once we have reduced all we can we are left with a simple group acting irreducibly. There exist efficient algorithms dealing with Classical Groups acting in their natural dimension, and in large dimensions. For representations of `moderate' degree there are only a few modules to consider. We describe a one-sided Monte Carlo algorithm for constructive recognition of the Special Linear Group in one particular case, the Alternating Square Module - an algorithm which can be adapted to each Classical Group (or Simple Group of Lie Type).

Joao Pita Costa (University of Ljubljana)
Title: On Skew Ideals
Abstract: Due to their nice properties, ideals have an important role both in Lattice Theory and Semigroup Theory which are the main motivations for the study of skew lattices (double regular bands of semigroups satisfying certain absorption laws). In this talk we present the ideals and filters of skew lattices and the intersections of the theory with the coset structure of a skew lattice. Our motivation is due to their important role in the study of Skew Boolean Algebras where ideals are closely related with congruences.

M. R. Darafsheh (University of Tehran)
Title: Recognition of finite groups by spectrum
Abstract: Let $G$ be a finite group. The set of orders of elements of $G$ is denoted by $\omega(G)$ and is called the spectrum of $G$. This set is partially ordered by divisibility and therefore it is completely determined by the subset $\mu(G)$ of the maximal elements of $\omega(G)$. For a natural number $n$ we let $\pi(n)$ to be the set of all the prime divisors of $n$, and for a finite group $G$ we set $\pi(G) = \pi(|G|)$.

The Gruenberg-Kegel graph of $G$ is a simple graph with vertex set $\pi(G)$ where two distinct vertices $p$ and $q$ are joined by an edge if $pq \in\omega(G)$. This graph is denoted by $GK(G)$ and sometimes is called the prime graph of $G$. The connected components of $GK(G)$ are denoted by $\pi_1,\pi_2,\ldots,\pi_{s(G)}$, where $s(G)$ denotes the number of connected components of $G$. If $G$ is a group of even order, then we choose $\pi_1$ to be the component containing the prime 2 and in this case the components $\pi_i$, $2\leq i\leq s(G)$ are called the odd components of $GK(G)$.

By a well-known theorem of Gruenberg and Kegel if $GK(G)$ is disconnected then the structure of $G$ is known. A finite group $G$ is said to be recognisable by the spectrum if for any group $H$ with $\omega(H) = \omega(G)$ we have $G\cong H$. In general a finite group $G$ is called $k$-recognisable by the spectrum if there are $k$ non-isomorphic groups $H$ with $\omega(H) = \omega(G)$. The history of research on recognition of finite groups by spectrum goes back to roughly 25 years ago. It is proved by Shi the simple group $\mbox{PSL}_2(7)$ is recognizable by spectrum. But we proved that the group $\mbox{PSL}_5(5)$ is 2-recognizable. The recognizability of all the sporadic groups and a few alternating groups and also families of simple groups of Lie type have been proved by several authors. In this talk we will explore all the research works

Manuel Delgado (University of Porto)
Title: Mal'cev products with pseudovarieties of $2$-groups
Abstract: Pseudovarieties of groups are classes of finite groups closed under taking subgroups, homomorphic images and finite direct products. Pseudovarieties of monoids are defined analogously. Computing Mal'cev products of pseudovarieties of monoids with pseudovarieties of groups is tightly related with computing type II semigroups (also named kernels) of finite monoids relative to the pseudovarieties of groups in consideration.

The pseudovarieties of monoids that will be considered in this talk are pseudovarieties generated by monoids of injective partial transformations on a finite set. Specifically, we shall consider monoids of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations on a finite chain. The respective associated pseudovarieties of monoids are denoted by $\mathsf{POI}$, $\mathsf{PODI}$, $\mathsf{POPI}$ and $\mathsf{PORI}$. The pseudovarieties of groups with interest for this talk are the decidable pseudovarieties of abelian groups, an example being the class of $2$-groups, denoted here by $\mathsf{H}_2$,.

We shall address the problems of comparing the pseudovarieties of monoids $\mathsf{PODI}$ with $\mathsf{POI}ⓜ \mathsf{H}_2$ as well as $\mathsf{PORI}$ with $\mathsf{POPI}ⓜ \mathsf{H}_2$.

Computing the kernels of monoids generating, respectively, $\mathsf{POI}$ and $\mathsf{POPI}$ relative to $\mathsf{H}_2$, one can deduce the left to right inclusions.

The groups in the pseudovarieties $\mathsf{POI}ⓜ \mathsf{H}_2$ and $\mathsf{PODI}$ are the same, but whether the pseudovarieties are equal is an open problem.

The analogous question for the other pair of pseudovarieties seems to be even more challenging. Whether the groups in both pseudovarieties are the same has been reduced to a description of the pseudovariety of groups generated by the dihedral groups. Joint work with Edite Cordeiro and Vítor H. Fernandes.

Andreas Distler (University of Lisbon)
Title: On presentations of nilpotent semigroups
Abstract: A semigroup $S$ is nilpotent if the set $S^r$ of all products of length $r$ has size $1$ for some natural number $r$. The smallest such number is called the nilpotency class of $S$, and $|S|-r$ is called the coclass.

I give a complete classification of nilpotent semigroups of coclass 1 and 2 including finite presentations for all types of such semigroups. A generalisation of these results leads to parametrised families of presentations for certain nilpotent semigroups of higher coclass.

David Easdown (University of Sydney)
Title: Presentations of maximal subgroups of semigroups generated by idempotents subject to relations that come from the underlying biordered set
Abstract: We discuss a geometric construction, using fundamental groups of graphs, to produce presentations of maximal subgroups of semigroups generated by elements of a biordered set, subject to relations associated with basic products. We relate this to recent results of Gray and Ruskuc.

Luda Markus Epstein (Technion, Israel)
Title: Word Problem for Inverse Monoids Presented by a Single Relator
Abstract: Since Magnus it has been well known that one-relator groups have a decidable word problem. However, solvability of the word problem in one-relator monoids is far from being completely studied. Only few examples of inverse monoids with solvable word problem are known. Recently, the solvability of the word problem in inverse monoids with a single sparse relator has been announced by Hermiller, Lindblad and Meakin.

We consider certain one-relator inverse monoids. In our attempt to solve the word problem, we rely on the result of Ivanov, Margolis and Meakin which states that the word problem for the inverse one-relator monoid is decidable if the membership problem for the corresponding prefix monoid is decidable. Thus, we first solve the membership problem for the prefix monoid and then apply the theorem to solve the word problem. Our methods involve van Kampen diagrams and word combinatorics.

Ramon Esteban-Romero (Universitat Politècnica de València)
Title: Permutability properties with GAP
Abstract: In this talk we describe some algorithms implemented in GAP which allow us to identify groups in some classes defined in terms of normality, permutability, and Sylow permutability.

Stephen Glasby (Central Washington University)
Title: $p$-groups with few characteristic subgroups and `interesting' automorphism groups
Abstract: The most interesting automorphisms of a $p$-group are commonly those of order coprime to $p$. A results of Helleloid and Martin in 2007 says that almost all $p$-groups (in a technical sense) have no interesting automorphisms. This talk considers $p$-groups $G$ which are atypical in the sense that $|\mbox{Aut}(G)|$ has a very large $p'$-part. We focus on $p$-groups $G$ with three characteristic subgroups only, namely $G$, $\Phi(G)$, and $1$. In small cases $\mbox{Aut}(G)$ induces on $G/\Phi(G)$ a linear group which is interesting such as $\mbox{GL}_4(p)$, $\mbox{GSp}_4(p)$, $\Gamma\mbox{L}_2(p^2)$, $\mbox{GL}_2(p)\otimes \mbox{GL}_2(p)$, etc. These linear groups are commonly (not always) maximal linear subgroups. This is joint work with P.P. Pálfy and Csaba Schneider.

Eddy Godelle (Universite de Caen)
Title: Generic Hecke Algebra for Renner monoids
Abstract: We provide a proof on a conjecture of L. Solomon on the presentation of Iwahori-Hecke algebras of finite reducible monoids.

Peter Higgins (University of Essex)
Title: Burrows-Wheeler transform of a permutation
Abstract: The Burrows-Wheeler transform of a primitive word is a very effective tool in data compression. In this talk we introduce the idea of a BW-transform of a permutation by representing the permutation as a union of one-to-one order-preserving partial mappings. By specialising to the case of a transitive cycle we show that the reason why the BW transform of a primitive word is easy to invert is due to the fact that a partial one-to-one order-preserving mapping on a finite set is entirely determined by its domain and range.

Christopher Hollings (University of Oxford)
Title: A. H. Clifford and unique factorisation: some forgotten early steps into semigroup theory
Abstract: In this historical talk, I will discuss the notion of prime factorisation in semigroups, with a special emphasis on the work of A. H. Clifford in the 1930s. Clifford's work was inspired by that of Emmy Noether in the ring case, but his approach also owes much to the work of his doctoral supervisors, E. T. Bell and Morgan Ward, and their attempts to provide a sound, postulational basis for arithmetic. I will describe the origins of Clifford's work, which was not only Clifford's first work in semigroup theory, but was also amongst the first semigroup theory of any kind.

Mohamad Hadi Hooshmand (Islamic Azad University - Shiraz Branch)
Title: Upper Periodic Subsets of Semigroups and Groups: Generalization of Ideals
Abstract: Suppose $A$ and $B$ are subsets of a semigroup or group. As a new definition, we call $A$ left [right] upper $B$-periodic if $BA\subseteq A$ [$AB\subseteq A$]. If $BA=A$ [$AB=A$]), then we call it left [right] $B$-periodic. Also, we call $A$ (two sided) upper $B$-periodic if is both left and right upper $B$-periodic (equivalently $BA\cup AB\subseteq A$). In this talk a new topic about a vast class of subsets of semigroups, groups and even binary systems which contains all ideals, periodic subsets and sub-semigroups, is introduced and studied. In fact the "upper periodic subsets" can be considered as a generalization of the conception "ideals". We prove a fundamental theorem which states if $A$ is a upper $B$-periodic subset of semigroup $S$, then under some conditions, it has the unique direct representation $A=\mathfrak{B}\cdot D\dot{\cup} B^1\cdot E$, where $B^1=B\cup \{ 1\}$, $B\subseteq \mathfrak{B}\leq S$. Especially we prove a unique direct representation for upper $T$-periodic subsets and applying it we classify all sub-semigroups of $S$ containing the fixed element $T$ to three classes. This classification gives us more interesting properties for the real semigroups. At last we characterize upper $T$-periodic subsets of semigroups and groups.

Rowland Jiang (University of Sydney)
Title: Exceptional $p$-groups of order $p^5$
Abstract: The minimal degree of a finite group $G$, $\mu(G)$, is defined to be the smallest natural number $n$ such that $G$ embeds inside $\mbox{Sym}(n)$. The group $G$ is said to be exceptional if there exists a normal subgroup $N$ such that $\mu(G/N)>\mu(G)$. We will investigate the smallest exceptional $p$-groups, when $p$ is an odd prime. In 1999 Lemiuex showed that there are no exceptional $p$-groups of order strictly less than $p^5$ and imposed severe restrictions on the existence of exceptional groups of order $p^5$. In fact he showed that if any were to exist, they must come from central extensions of four isomorphism classes of groups of order $p^4$. Then in 2007 he exhibited an example of an exceptional group of order $p^5$. I will be demonstrating the existence of two more exceptional groups arising in such a fashion and also ruling out the possibility of the remaining case.

Marianne Johnson (University of Manchaster)
Title: Green's $\mathcal J$-order and the rank of tropical matrices
Abstract: In this talk I will present some recent joint work with Mark Kambites. We give a characterisation of Green's $\mathcal J$-order on the multiplicative semigroup of all $n\times n$ matrices with entries in the tropical semiring $T=(R\cup\{−\infty\},\oplus,\otimes)$. This is the semiring whose ``addition" denoted $\oplus$ is given by maximisation, $a\oplus b:=\max(a,b)$, and whose ``multiplcation" denoted $\otimes$ is given by the usual addition of real numbers $a\otimes b:=a+b$, with the conventions that $−\infty\oplus a=a\oplus−\infty=a$ and $−\infty\otimes a=a\otimes−\infty=−\infty$ for all $a\in T$.

In the case of the multiplicative semigroup of matrices with entries in a field, the $\mathcal J$-relation corresponds exactly to the idea of rank, that is, two matrices are $\mathcal J$-related if and only if they have the same rank. Furthermore it is also known that the $\mathcal J$-relation is equal to the $\mathcal D$-relation in this case. Recall that the rank of a matrix with entries in a field has many equivalent definitions. For tropical matrices however, the analogous definitions cease to be equivalent and give rise to several different notions of rank. Many of these notions of rank turn out to be $\mathcal J$-class invariants, perhaps suggesting that the tropical analogue of the rank of a matrix should not be a single natural number, but rather a $\mathcal J$-class. We note that Hollings and Kambites have given a characterisation of the $\mathcal D$-relation, whilst Izhakian and Margolis have shown that $\mathcal D\neq \mathcal J$ in $M_n(T)$ for $n\geq 3$. In contrast to this, we also show that $\mathcal D=\mathcal J$ in the subsemigroup of matrices without $−\infty$ entries.

Ruanglak Jongchotinon (Chulalongkorn University)
Title: Isomorphism Theorems for Variants of Semigroups of Linear Transformations
Abstract: If $S$ is a semigroup and $a\in S$, the semigroup $(S,\circ )$ defined by $x\circ y = xay$ for all $x,\ y \in S$ is called a variant of $S$ and $(S,\circ)$ is denoted by $(S,a)$. In 2003-2004, Tsyaputa characterized when two variants of the following transformation semigroups are isomorphic: the symmetric inverse semigroup, the full transformation semigroup and the partial transformation semigroup on a finite nonempty set. In this paper, we consider the semigroups under composition of all linear transformations of a finite-dimensional vector space over a finite field. We determine when its variants are isomorphic. We also obtain as a consequence in the same matter for the full $n\times n$ matrix semigroup over a finite field. The was supported by the Development and Promotion of Science and Technology Talents Project (DPST), Thailand.

Mark Kambites (University of Manchaster)
Title: Tropical Matrix Duality and Green's $D$ Relation
Abstract: I will discuss some recent joint work with Christopher Hollings, giving a complete description of Green's $D$-relation for the semigroup of all $n \times n$ matrices over the tropical semiring. The main tool is a new variant on the phenomemon of tropical matrix duality, which gives a necessary and sufficient condition for two tropical convex sets to be the row and column space of a matrix.

Ondřej Klíma (Masaryk University Brno)
Title: Identity Checking Problem for Monoids of Transformations
Abstract: We study the computational complexity of checking identities in a fixed finite monoid. We prove that this problem is coNP-complete for the monoid of all transformations of a four-element set. This result completes the description of the complexity of checking identities in monoids of transformations.

Miroslav Korbelář (Masaryk University Brno)
Title: Additively divisible commutative semirings
Abstract: A (commutative) semiring is an algebraic structure with two commutative and associative binary operations (an addition and a multi- plication) such that the multiplication distributes over the addition. It is well known that a commutative field is finite provided that it is a finitely generated ring. Consequently, no finitely generated commutative ring (whether unitary or not) contains a copy of the field $\mathbb Q$ of rational numbers. On the other hand, it seems to be an open problem whether a finitely generated (commutative) semiring $S$ can contain a copy of the semiring (parasemifield) $\mathbb Q^+$ of positive rationals. Anyway, if $S$ were such a (unitary) semiring with $1_S = 1_{Q^+}$ , then the additive semigroup $S(+)$ should be divisible. So far, all known examples of finitely generated additively divisible commutative semirings are additively idempotent. Hence a natural question arises, whether a finitely generated (commutative) semiring with divisible additive part has to be additively idempotent. We show validity of this conjecture for bounded semirings and several types of one-generated semirings.

Olga Macedonska (Silesian University of Technology)
Title: A semigroup Length of Groups
Abstract: Let $G$ be a group generated by $k$ (and no less) elements. A subsemigroup $S \subseteq G$ generated by any $k$ generators in $G$ is called a base semigroup of $G$: Every element in $G$ is a product of elements from $S \cup S^{-1}$. A possible equality $G = S^{-1}S\cdots S^{-1}S$ allows to define an $S$-length $l(G,S)$ of the group $G$. If there is no such an equality, we say that $l(G, S) = \infty$. If $\mathcal X(G)$ is the set of all base semigroups in $G$ then the $S$-length of $G$ is defined as $l(G) = \sup\{ l(S,G) : S\in\mathcal X(G)\}$. For example, if $G$ is a finite group, $l(G) = 1$. If $G$ is an abelian group, $l(G) = 2$. If $G$ has no free noncyclic subsemigroup, $l(G)\leq 2$: We show that for every $n\in\mathbb N$ there is a polycyclic group with $S$-length equal to $n$. However, a free group and a free soluble groups have an finite $S$-length. A relatively free group has the $S$-length $n\leq 2$ if and only if it satisfies a positive law. The problem on values of $S$-length of other relatively free groups is open. We show that it cannot be $3$.

Inês Legatheaux Martins (University of Lisbon)
Title: Schur-Weyl duality for the rook monoid
Abstract: Following J. A. Green's approach to the polynomial representations of the general linear group $\mbox{GL}_m(\mathbb C)$ through Schur algebras, we will explore an important correspondence between these representations and those of the rook monoid, $R_n$. In addition, we will also link this correspondence to the Schur-Weyl duality for $R_n$ and $\mbox{GL}_m(\mathbb C)$ presented in 2002 by L. Solomon.

This approach will allow us to obtain new proofs of known results about the representations of the rook monoid. We also expect to discuss some applications to problems of Multilinear Agebra.

Peter Mayr (University of Lisbon)
Title: Checking membership in direct powers
Abstract: For a fixed finite algebraic structure $A$ (e.g., a group, a ring, $\dots$) we consider the following Subpower Membership Problem:
INPUT: tuples $a_1,\ldots, a_k$, and $b$ in $A^n$
PROBLEM: Is $b$ in the subalgebra of $A^n$ that is generated by $a_1,\ldots, a_k$?
There exist algebraic structures for which this problem is Exptime-complete (Kozik, 2007). However, if $A$ is a group, then an adaptation of a permutation group algorithm works in polynomial time (Furst, Hopcroft, Luks, 1980). More generally, for rings, algebras over fields, Lie rings, $\dots$ all finite groups with additional multilinear operations the Subpower Membership Problem is in P.

I will present a polynomial algorithm for any finite $p$-group with arbitary additional operations and for any nilpotent loop of prime power order. This uses the fact that on a $p$-group every function behaves almost like a homomorphism.

Alexei Miasnikov (McGill University)
Title: Dehn Monsters
Abstract: I will discuss some exotic examples of finitely generated recursively presented infinite groups where the word problem can be decided only on some negligible sets. Even worse, one cannot effectively produce any infinite sequence of pairwise distinct elements of the group. To build such groups we use Golod-Shafarevich construction and Post approach to simple sets.

James Mitchell (University of St Andrews)
Title: Maximal subsemigroups of the semigroup of all functions on an infinite set
Abstract: We consider the problem of finding maximal proper subsemigroups of the full transformation semigroup on an infinite set. Maximal subgroups of infinite symmetric groups have been extensively studied by many authors including Macpherson, Neumann, and Praeger. Examples include the setwise stabilizer of a finite set, the almost stabilizer of a finite partition, the stabilizer of any ultrafilter, and the stabilizers of some classes of non-maximal ideals. In this talk I will present some recent work with J. East and Y. Peresse, where we classified the maximal subsemigroups of the semigroup of all functions on any infinite set that contain the symmetric group, or any of the maximal subgroup of the symmetric group mentioned above.

Scott Murray (University of Canberra)
Title: Constructive membership testing in classical groups
Abstract: The Matrix Group Recognition Project is a major topic of research in computational group theory. An important component of this project is the recognition of classical groups. These algorithms start by finding standard generators of the group, then use them to build a constructive membership test. The constructive membership test is used repeatedly, so it is vital that it be efficient. In this talk, I discuss our algorithms for constructive membership testing in non-orthogonal classical groups. These algorithms have been analysed and an efficient implementation has been made by Csaba Schneider.

This talk is joint work with Sophie Ambrose, Cheryl E. Praeger and Csaba Schneider.

Alireza Najafizadeh (Payame Noor University)
Title: Square subgroup of an abelian group
Abstract: In this seminar we introduce the notion of square subgroup of an abelian group and discuss this subgroup for different abelian groups. In particular we obtain some results for torsion-free abelian groups or rank two.

Conceição Veloso Nogueira (Instituto Politécnico de Leiria)
Title: Tameness of the pseudovariety join of local semilattices with groups
Abstract: A pseudovariety of semigroups is said to be decidable if its membership problem has a solution, that is, if there is an algorithm to test whether a given finite semigroup lies in that pseudovariety. The join $V\vee W$ of two pseudovarieties $V$ and $W$ is the least pseudovariety containing both $V$ and $W$. A well-known result of Albert, Baldinger and Rhodes states that the join of two decidable pseudovarieties may not be decidable. Decidability is not also preserved by some other common pseudovariety operators.

An idea which has been explored by several authors is to impose stronger properties on the pseudovarieties upon which the operators are to be applied that will guarantee that the resulting pseudovarieties will be decidable. In this context Almeida introduced a stronger form of decidability, called hyperdecidability, which was later refined in collaboration with Steinberg, leading to the notion of tameness.

The tameness property is parameterized by an implicit signature $\sigma$. The canonical signature $\kappa$, containing the semigroup multiplication and the $(\omega-1)$-power, is the most commonly used signature. There are several examples of $\kappa$-tame pseudovarieties of the form $V\vee G$ in the literature, where $G$ denotes the pseudovariety of all finite groups. The idea, in this presentation, is to treat the problem of $\kappa$-tameness of the join $\mbox{LSl}\vee G$, where $\mbox{LSl}$ denotes the pseudovariety of all finite semigroups which are locally semilattices, and to present some generalizations. Notice that $\mbox{LSl}$ is associated, in Eilenberg’s correspondence, to the well known variety of locally testable languages.

Péter P. Pálfy (Rényi Institute, Budapest)
Title: Groups of finite abelian width
Abstract: A group $G$ has abelian width $k$, if $k$ is the smallest integer such that $G$ can be written as a product of $k$ abelian subgroups. Finite symmetric groups have abelian width of order in the logarithm of the degree. However, all infinite symmetric groups have bounded abelian width. Also, with the exception of the alternating groups, the abelian width of all finite simple groups is uniformly bounded. Joint work with M. Abért, L. Pyber, and B. Szegedy.

Kataryna Pavlyk (University of Tartu)
Title: On semigroups generated by partial transformations of pisitive cone of a linearly ordered group
Abstract: This is a joint work with Oleg Gutik.

We study inverse semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\overline{\mathscr{B}}(G)$ and $\overline{\mathscr{B}}\,^+(G)$ which are generated by partial monotone injective translations of a positive cone of a linearly ordered group $G$. In case $G$ is the additive group of integers $\mathbb{Z}_+$ with usual linear order $\leqslant$ then the semigroup $\mathscr{B}^+(G)$ is isomorphic to the bicyclic semigroup ${\mathscr{C}}(p,q)$ and if $G$ is the additive group of real numbers $\mathbb{R}_+$ with usual linear order $\leqslant$ then the semigroup $\mathscr{B}^+(G)$ is isomorphic to the enlarged versions of the bicyclic semigroup consisting of the first quadrant of the plane $B^{1}_{[0,\infty)}$ and the semigroup $\mathscr{B}(G)$ is isomorphic to the versions of the bicyclic semigroup $B^{2}_{(-\infty,\infty)}$ consisting of all real plane $\mathbb{R}^2$.

We describe Green's relations on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\overline{\mathscr{B}}(G)$ and $\overline{\mathscr{B}}\,^+(G)$, their bands and show that they are simple, and moreover the semigroups $\mathscr{B}(G)$ and $\mathscr{B}^+(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences on the semigroup $\mathscr{B}(G)$ (and $\mathscr{B}^+(G)$) are group congruences if and only if the group $G$ is archimedean.

Also we describe the structure of group congruences on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\overline{\mathscr{B}}(G)$ and $\overline{\mathscr{B}}\,^+(G)$.

This research was carried out with the support of the ESF and co-funded by Marie Curie Actions, grant ERMOS36.

Dmitrii Pasechnik (Nanyang Technological University, Singapore)
Title: Permutation groups and semidefinite programming
Abstract: Semidefinite programming, i.e. linear optimization on the cone of positive semidefinite matrices, provides an efficient way to compute approximate solutions to hard combinatorial problems. A typical problem of this sort is computing upper bounds on (co)cliques in graphs. In presence of permutational symmetry, one can work with cone of positive semidefinite elements in the centralizer ring of the underlying permutation group. Passing to a suitable low-dimensional representation of the latter allows one to solve problems which are intractable by other means. We discuss classical and recent application of this approach to various problems in combinatorics and algebra, such as bounds on code sizes, bounds on crossing numbers of graphs, and classifying synchronizing permutation groups.

Libor Polak (Masaryk University)
Title: Solving equations with constants in varieties of completely regular and inverse semigroups
Abstract: For each variety $V$ of groups one can consider the following classes. $CR(V)$ is the class all completely regular semigroups with subgroups in $V$; $CS(V)$ is the class of all completely simple semigroups with subgroups in $V$; $SI(V)$ is the class of all strict inverse semigroups with subgroups in $V$. Already the case $V =$ all groups is interesting enough. Concerning the identity problems in $XY(V)$, it is well-known how to translate them into identity problems in $V$. Next step are equations in variables $x,y,\ldots$ and constants $a,b,\ldots$. Again, one can translate the solvability problem from $XY(V)$ to $V$.

J. H. Renshaw (University of Southampton)
Title: Adequate transversals of abundant semigroups and generalisations
Abstract: Inverse transversals of regular semigroups were introduced by Blyth & McAlister in the late 70's in connection with their study of split orthodox semigroups. These were subsequently generalised by El-Qallali to adequate subsemigroups of abundant semigroups and it is this generalisation that I will discuss in this talk. After summarising some of the known results for adequate transversals, I will detail some recent joint work (with Jehan Al-Bar) on the structure of transversals where the abundant semigroup is in fact quasi-adequate and where the structure is based on a spined product of left and right adequate semigroups.

Neil Saunders (University of Sydney)
Title: Minimal Faithful Degrees of Finite Groups
Abstract: The minimal degree of a finite group $G$ is the smallest non-negative integer $n$ such that $G$ embeds in $\mbox{Sym}(n)$. This defines an invariant of the group $\mu(G)$.

In this talk, I will present some interesting examples of calculating $\mu(G)$ and examine how this invariant behaves under taking direct products and homomorphic images. In particular, I will focus on the problem of determining the smallest degree for which we obtain a strict inequality $\mu(G \times H) < \mu(G) + \mu(H)$, for two groups $G$ and $H$. Up until 2010, there was only one known example of this strict inequality and in my this talk, I will present an infinite family of examples which will lead us to more interesting open questions.

Lubna Shaheen (University of York)
Title: Axiomatisability problems for $S$-posets
Abstract: Monoids have been widely studied via their representations as mappings of sets; that is, via the concept of $S$-act. Over the past three decades, an extensive theory of properties of $S$-acts has been developed (involving free, projective, and flat acts of various kinds). A fresh and comprehensive survey of this area was published in 2000 by M. Kilp, U. Knauer and A. Mikhalev in the monograph Monoids, Acts and Categories.

More recently, ordered monoids, known as pomonoids, have also been studied via their representations, this time as order-preserving maps of partially ordered sets, that is, by $S$-posets. To date there is no suitable text on $S$-posets, and only few articles attempting to generalise material from $S$-acts to $S$-posets. But there are major differences, since $S$-acts are merely algebras, whereas $S$-posets are relational structures as well as algebras. Therefore one needs to be very careful when dealing with $S$-posets due to the partial order involved.

It is known that associated to a class of algebras $\mathcal A$ there is a first order language $L$. It is natural to ask, is there a set of sentences say $\Sigma$ in $L$ such that a member $A\in\mathcal A$ has a property $P$ if and only if $A\models\Sigma$. If such a set of sentences exists, we say that the subclass $\mathcal B$ of $\mathcal A$ whose members have the property $P$ is axiomatisable.

Axiomatisability questions of $S$-acts was initiated by Gould in 1985, and continued by Stepanova and Bulman-Fleming. The type of questions they asked involve axiomatisability, completness and model completness of some categorically defined classes of left $S$-acts. We focus on the axiomatisability of different classes of $S$-posets. We have succeeded in determining when the classes of, free, projective, strongly flat, flat, weakly flat, principally weakly flat, po-flat, weakly po-flat and principally weakly po-flat $S$-acts are axiomatisable. We also axiomatised some classes of $S$-posets satisfying conditions such as Condition $(\mbox{W})$, $(\mbox{EP})^\leq$, $(\mbox{Pw})$, $(\mbox{PWP})$ and Condition $(\mbox{PWP}_w)$.

Kunitaka Shoji (Shimane University)
Title: Finitely generated semigroups presented by regular congruence classes
Abstract: In this talk, we give characterizations of finitely generated semigroups presented by congruences such that all their congruence classes are regular languages or finite languages. Moreover some properties of such semigroups are shown.

Nissara Sirasuntorn (Chulalongkorn University)
Title: Left Regular and Right Regular Elements of Semigroups of 1-1 Transformations and 1-1 Linear Transformations
Abstract: An element $x$ of a semigroup $S$ is said to be left-regular [right-regular] if $x = yx^2$ [$x = x^2y$] for some $y \in S$, or equivalently, $x L x^2$ [$x R x^2$]. In this paper, we characterize the left-regular and right-regular elements of the semigroups under composition of all 1-1 transformations of a set and of all 1-1 linear transformations of a vector space. The research was supported by the Development and Promotion of Science and Technology Talents Project (DPST), Thailand.

Michal Stolorz (University of Aberdeen)
Title: On simple modules over twisted finite category algebras
Abstract: We shall present how the recent proof, by Ganyushkin, Mazorchuk and Steinberg, of the parametrisation of simple modules over finite semigroup algebras due to Cliford, Munn and Ponizovskii carries over to twisted finite category algebras. We later observe that the parametrisations of simple modules over Brauer algebras, Temperley-Lieb algebras, and Jones algebras due to Graham and Lehrer, can be obtained as special cases of our main result. We further note that the notion of weights in the context of Alperin's weight conjecture extends to twisted finite category algebras.

István S. Szöllősi (Babes-Bolyai University)
Title: Computing the extensions of preinjective and preprojective Kronecker modules
Abstract: Let $K$ be the Kronecker quiver and $\kappa$ an arbitrary field. The path algebra $\kappa K$ over the Kronecker quiver is the Kronecker algebra. We will consider the category of finite dimensional right modules over this algebra, the category of Kronecker modules, denoted by mod-$\kappa K$. The category mod-$\kappa K$ can and will be identified with the category rep-$\kappa K$ of the finite dimensional $\kappa$-representations of the Kronecker quiver. The indecomposables in mod-$\kappa K$ are divided into three families: the preprojectives, the regulars and the preinjectives. .

In our talk we deal with an important problem in the theory of Kronecker modules, the explicit description of the middle terms in $\mbox{Ext}^1(I,I')$, where $I$ and $I'$ are decomposable preinjective modules. It turns out that the extensions of these modules carry some interesting combinatorial properties, which we plan to cover in some detail. We present a simple linear time algorithm to decide if $I$ is a middle term in $\mbox{Ext}^1(I',I'')$ for $I,\ I',\ I''\in\mbox{mod-}\kappa K$ given preinjective Kronecker modules. We propose an algorithm to generate all the middle terms in $\mbox{Ext}^1(I',I'')$, given $I',\ I''\in\mbox{mod-}\kappa K$. All these results apply dually to preprojective Kronecker modules as well.

We also mention some applications. The Kronecker algebra is an important case of tame hereditary algebra and the category mod-$\kappa K$ is derived equivalent with the category of coherent sheaves on the projective line. Moreover, Kronecker modules correspond to matrix pencil s in linear algebra and our results could constitute some first steps in solving an important open problem, the matrix subpencil problem.

Rick Thomas (University of Leicester)
Title: The word problem for semigroups
Abstract: One of the classic problems in group theory is that of the word problem. Given a presentation for a group $G$, we can ask whether or not a word in the generators represents the identity element of $G$. An alternative formulation is to think of the set $W$ of words that represent the identity and then ask whether or not a given word lies in $W$. This leads to some very interesting connections between group theory and formal language theory, asking which groups have their word problem in some specified class of formal languages.

As it stands, this idea does not generalise naturally to monoids and semigroups. Duncan and Gilman have proposed an elegant definition for the word problem of a semigroup $S$. Given a generating set $A$ for $S$, we pick some new symbol $\#$ (i.e. some symbol not in $A$) and then consider the set of all words of the form $u \# \mbox{rev}(v)$ where $u$ and $v$ represent the same element of $S$ (and where $\mbox{rev}(v)$ is the reversal of the word $v$). This is a natural extension of the definition for groups since, given a family $F$ of languages, the word problem of a group in the group sense lies in $F$ if and only if the word problem of the group in the semigroup sense lies in $F$

In this talk we will introduce these ideas and survey some of what is known about word problems of semigroups.

Yanhui Wang (University of York)
Title: Beyond orthogonal semigroups
Abstract: My aim is to describe weakly $B$-abundant semigroups with (C) from the viewpoint of generalised categories. To construct a weakly $B$-abundant semigroup based on a generalised category, I introduce the concept of inductive ordered generalised category over a band $B$. This is an analogue of inductive cancellative categories developed by Armstrong, who was herself influenced by Nambooripad’s work on the connection between biordered sets and regular semigroups. Conversely, given a weakly Babundant semigroup $S$, I build an inductive ordered generalised category over $B$, defining partial orders $r$ and $l$ on the trace of $S$. Such a result may be regarded as a generalisation of the result of Lawson for Ehresmann semigroups. Finally, I investigate the relationship which exists between certain classes of ordered categories and the class of weakly $B$-abundant semigroups with (C).

Thomas Weigel (University of Milan-Bicocca)
Title: The group of almost automorphisms of a locally-finite tree
Abstract: An almost automorphism of a locally-finite tree $T$ is an equivalence class of isomorphisms $\alpha\colon T_\Lambda\to T_\Gamma$, where $T_\Lambda$ and $T_\Gamma$ are two sub-forests of $T$ which are graph theoretic complements obtained by removing finite, connected, saturated sub-trees. The group of almost automorphisms $\mathbb{A}ut(T)$ coincides with the group of sphero-morphisms introduced by Y. Neretin in case that the tree $T$ is regular, but in general it can be quite different. By construction, $\mathbb{A}ut(T)$ carries always the structure of a totally-disconnected, locally-compact group containing its automorphism group as an open subgroup. But it also contains interesting dense subgroups which can be seen as generalizations of the Higman-Thompson groups $G_{n,r}$, and in case that $T$ is spherically-homogeneous also discrete subgroups isomorphic to the Richard Thompson groups $T_{n,r}$ and $F_{n,r}$.

James B. Wilson (Ohio State University)
Title: Isomorphism in expanding families of indistinguishable groups
Abstract: For every prime power $N$, there is a Heisenberg group of order $N^{1.2+o(1)}$ which has $N^{O(\log N)}$ pairwise non-isomorphic quotients of order $N$. Yet, these groups are virtually indistinguishable. They have isomorphic character tables, the same sizes and numbers of conjugacy classes, they are directly and centrally indecomposable, and the recognized portion of their automorphism groups are isomorphic, of small index, and act isomorphically on the the abelianizations of the groups. Despite all these similarities, there is a polynomial-time algorithm that can determine if a group lies in this class, and determine if two groups in this class are isomorphic.

Pavel Zalesski (Universidade de Brasilia)
Title: Genus for groups
Abstract: We shall introduce a notion of a genus $g(C,G)$ for a class of groups $C$ and $G\in C$. It consists of isomorphism classes of groups from $C$ having the same profinite completion as $G$. We shall discuss finiteness results for $g(C,G)$ for several important families of groups including finitely generated virtually free groups and present formulas for the number of elements in $g(C,G)$ in various cases.

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