Michael Torpey
*University of St Andrews
***Finding a semigroup's congruence lattice
**

Congruences are an important part of a semigroup's structure, providing a full description of its homomorphic images, as normal subgroups do for groups. We present a new computational method for finding all the congruences of a given semigroup, as implemented in the Semigroups package for GAP, and we show how it applies to a few examples, including the Motzkin monoid.

Mohammed Aljohani
*University of St Andrews
***Synchronising and separating association schemes
**

Synchronisation is a recently (10 years ago) defined property for permutation groups. It is related to another concept which is separation. They can be naturally extended to association schemes. In this talk, I will discuss these properties for a particular permutation group (resp. association scheme), namely the group induced by the symmetric group \(Sym(n)\) on the set of \(k\)-element subsets of an \(n\)-set (resp. Johnson association scheme).

Nseobong Uto
*University of St Andrews
***Semi-Latin rectangle: A sophisticated and attractive row-column design
**

Semi-Latin rectangles are row-column designs that generalize the Latin squares/semi-Latin squares. They possess nice combinatorial properties, and are useful in several experimental situations. We present an overview of this class of design; with a view to showcasing some of its nice properties, how useful it is, as well as how it generalizes the Latin square/semi-Latin square.

Feyishayo Olukoya
*University of St Andrews
***Constructing simple groups from asynchronous transducers
**

Generalising existing methods for synchronous transducers, we show how to construct simple groups using asynchronous transducers. Joint with Casey Donoven.

Chris Russell
*University of St Andrews
***Creating a database of 0-simple semigroups
**

0-simple semigroups are in some sense the building blocks of semigroups. Enumerating these semigroups up to isomorphism boils down to finding certain matrices up to a special equivalence relation. I will explain how I created a database of these for orders < 49, focusing on the strategies that allowed me to compute to higher orders.

Adán Mordcovich
*University of St Andrews
***On probabilistic generation of finite classical groups
**

It is a fact that a simple group \(G\) can be generated by two elements, a natural question then follows: if we pick two elements uniformly at random from \(G\) (allowing repetition) what is the probability \(P_2(G)\) that such elements generate the whole of \(G\)? It is also true that a pair of elements of \(G\) do not generate the whole group if and only if there is a maximal subgroup of \(G\) containing both of these elements.

From the above one might suspect that there is a strong relation between the maximal subgroups of \(G\) and the probability \(P_2(G)\); indeed, this is the case. We aim to discuss this relationship in general with an eye to when \(G\) is a finite simple classical group.

Matt McDevitt
*University of St Andrews
***Permutation containment
**

We consider permutations as sequences of integers and introduce a containment relation on them. We then turn our attention to permutation anti-chains and explore some recent classification results.

Nayab Khalid
*University of St Andrews
***New Insights into the Presentations of Thompson's Group F
**

I will present our recent work into the development of a new presentation of R. Thompson's group F, which reflects its permutations and can be generalized to a wider setting.

Finlay Smith
*University of St Andrews
***Computing boolean matrix semigroups
**

Boolean matrices provide a useful representation of binary relations over a finite set. The number of boolean matrices grows extremely rapidly with the dimension of the matrix, so enumerative algorithms for semigroups of boolean matrices are infeasible even for relatively small sizes. In this talk I will discuss the theoretical and practical aspects of more efficient methods.

Gerard O'Reilly
*University of St Andrews
***The Profinite Topology on the Free Group
**

It is a result due to Marshall Hall Jr that a finitely generated subgroup of the free group is closed in the profinite topology. We will give a necessary and sufficient condition for a finitely generated subsemigroup of the free semigroup to be closed in the subspace topology.

Wilf Wilson
*University of St Andrews
***Computing direct products of semigroups
**

Direct products of semigroups are completely straightforward to define. However, to perform many algorithms from computational semigroups theory, we require a generating set for a given semigroup - and direct products are not defined in terms of generating sets. I will talk about some aspects of practically constructing reasonably small generating sets for direct products of semigroups.

Fernando Flores Brito
*University of St Andrews
***More on congruences of EndF_n(G)
**

I will discuss the outcomes of the methodology of my first approach on classifying the congruences of this monoid, the results I have about congruences of its minimal ideal, and some other results about congruences of higher rank.

Craig Miller
*University of St Andrews
***Right noetherian semigroups
**

A semigroup \(S\) is right noetherian if every right congruence on \(S\) is finitely generated. In this talk, we will begin by looking at some known examples of right noetherian semigroups. We will then consider whether the property of being right noetherian is preserved by various semigroup constructions.

Ashley Clayton
*University of St Andrews
***Finitary conditions for fiber products of free objects
**

If \(A\) and \(B\) are two algebras of the same type, then a *subdirect
product* of \(A\) and \(B\) is a subalgebra \(C\) of \(A\times B\), such that the
projections from \(C\) onto \(A\) and \(B\) are surjective. An important tool for
considering subdirect products is via *fiber products* of algebras,
which can be constructed via homomorphisms from two algebras onto a common
image. In this talk, we give some results to the typical finitary questions
for fiber products of free semigroups and monoids, and consider some further
related construction questions.

Rosemary Bailey
*University of St Andrews
***A substitute for the non-existent affine plane of order 6
**

A Latin square of order \(n\) can be used to make an incomplete-block design for \(n^2\) treatments in \(3n\) blocks of size \(n\). The cells are the treatments, and each row, column and letter defines a block. Any pair of treatments concur in 0 or 1 blocks, and it is known that the block design is optimal for these parameters.

If there are mutually orthogonal Latin squares, then the process can be continued, eventually giving an affine plane. But there are no mutually orthogonal Latin squares of order 6, so what should we do if we need a block design for 36 treatments in 30 blocks of size 6?

I will describe how a series of mistakes and wrong turnings in a different research project led to an answer.

Peter Cameron
*University of St Andrews
***Reed--Muller codes and Thomas' conjecture
**

A countable first-order structure is countably categorical if its automorphism group has only finitely many orbits on n-tuples of points of the structure for all n. (Homogeneous structures over finite relational languages provide examples.) For countably categorical structures, we can regard a reduct of the structure as a closed overgroup of its automorphism group. Simon Thomas showed that the famous countable random graph has just five reducts, and conjectured that any countable homogeneous structure has only finitely many reducts. Many special cases have been worked out but there is no sign of a general proof yet. In order to test the limits of the conjecture, Bertalan Bodor, Csaba Szabo and I showed that a vector space over GF(2) of countable dimension with a distinguished non-zero vector has infinitely many reducts. The proof can most easily be seen using an infinite generalisation of the binary Reed--Muller codes.