Semigroups
Workshop

All talks in Bragg Theatre - Schuster Building

15:30 Stuart Margolis
Bar Ilan University
Complexity 1 is decidable for finite semigroups
The complexity of a finite semigroup S is the least number of groups needed in any decomposition of S as a divisor of a wreath product of groups and aperiodic (group-free) semigroups. Such a number is guaranteed by the Krohn-Rhodes Theorem.

We prove that it is decidable if a finite semigroup has complexity 1. This  settles a problem first posed in 1965. The talk will give background in complexity theory and discuss the tools needed to prove the theorem.

This is joint work with John Rhodes and Anne Schilling.

16:10 Peter Cameron
University of St Andrews
Complete mappings of semigroups
A complete mapping of a finite group is a bijective mapping from the group to itself which has the property that the mapping obtained by combining this with left multiplication by the argument is also bijective. Marshall Hall Jr and Lowell Paige conjectured that a finite group has a complete mapping if and only if either it has odd order or its Sylow 2-subgroups are non-cyclic. The proof had to wait more than 50 years before it was settled by Wilcox, Evans and Bray, using the Classification of Finite Simple Groups. Subsequently two different proofs avoiding CFSG have been found.

There is no compelling reason for examining complete mappings of semigroups, but it turns out that this is a very interesting problem, depending on detailed analysis of Rees 0-matrix semigroups. With João Araújo, Wolfram Bentz and Michael Kinyon, we have nearly completed the solution for this problem. The missing piece seems to be purely combinatorial, related to Hall's marriage theorem.

16:50 Yann Peresse
University of Hertfordshire
Polish topological semigroups
Topological Algebra (not to be confused with Algebraic Topology) studies algebraic objects that have been endowed with a compatible topology. For example, a topological semigroup is a semigroup S with a topology on S under which the semigroup multiplication of S is continuous. Topological groups and inverse semigroups are defined analogously and are also required to have continuous inversion.

General semigroups - compared with their peers such as groups, rings, or fields - are rich in examples and variety but poor in structure. So, extra structure in the form of topology is especially welcome when studying semigroups. By combining Algebra and Topology we may ask more meaningful questions and answer them by using concepts and tools from both fields.

For a given semigroup S there are typically many topologies that turn S into a topological semigroup. So, for topological questions and results about S to be meaningful and valid, the topology used should be, in some sense, natural or even canonical. For example, Gaughan showed that the symmetric group on a countably infinite set X has a unique Polish (i.e. completely metrizable and separable) group topology.

I will present some recent results showing that many well-studied semigroups such as the full transformation semigroup on X or the semigroup of partial functions on X have unique Polish semigroup topologies. The symmetric inverse monoid on X has infinitely many Polish semigroup topologies but a unique Polish inverse semigroup topology.

11:30 Victoria Gould
University of York
Translational hulls of semigroups of endomorphisms of an algebra
A right translation ρ of a semigroup S is a self-map of S such that (ab)ρ=a(bρ) for all a,b in S; left translations are defined dually. A pair (λ, ρ) is a bi-translation of S if ρ is a right translation, λ is a left translation and the pair is linked, which means aλ(b) =(aρ)b, for all a,b in S. The set of bi-translations forms a subsemigroup of the direct product of (the opposite of) the full transformation monoid on S with itself, and is known as the translational hull Ω(S) of S.

We consider the translational hull Ω(I) of an arbitrary subsemigroup I of an endomorphism monoid End(A) where A is a universal algebra. We give conditions for every bi-translation of I to be realised by transformations, or endomorphisms, of A. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of I and the idealiser of I within End(A), which in the case where I is an ideal is simply End(A).

This is the first part of a two-part talk. The second will be given by Marianne Johnson. The work presented is joint with Ambroise Grau, Marianne Johnson and Mark Kambites.

13:45 Marianne Johnson
University of Manchester
Translational hulls of certain ideals of endomorphisms
For a universal algebra A and a natural number k, let Ik denote the ideal of the endomorphism monoid of End(A) consisting of those endomorphisms whose image is contained in a subalgebra generated by fewer than k elements of A. Building upon the methods outlined in Victoria Gould’s talk (and motivated by several questions posed by Stuart Margolis concerning the nature of the translational hulls of ideals of some specific endomorphism monoids) we consider the nature of the translational hull Ω(Ik)  where  A is: a free algebra; an independence algebra; a finite symmetric group.  We determine when Ω(Ik) is isomorphic to End(A) in these cases. This is joint work with Victoria Gould, Ambroise Grau and Mark Kambites.

14:25 Nik Ruskuc
University of St Andrews
Free projection-generated regular *-semigroups
Regular *-semigroups are semigroups with a unary operation * satisfying the identities x**=x, xx*x=x, (xy)*=y*x*, and include partition-, Brauer- and Temperley-Lieb monoids as distinguished families of examples.  In their recent work, East and Muhammed have developed a groupoid framework for regular *-semigroups, akin to that of Nambooripad for regular semigroups, but using the so-called projection algebras in place of regular biorders of idempotents. Arising from this  framework is a natural notion of the free regular *-semigroup over a projection algebra, which will be the subject of my talk. I will present  basic definitions, including presentations in terms of generators and defining relations, comparisons between these new free objects and the better-known free (regular) idempotent-generated semigroups, and some intriguing initial examples. This is joint work with J. East, R.D. Gray and P.A.A. Muhammed.

Carl-Fredrik Nyberg Brodda
KIAS (Korea Institute for Advanced Studies)

Jung Won Cho
University of St Andrews

Jonathan Warne
University of East Anglia

Martin Hampenberg Christensen
University of Hertfordshire

Daniel Heath
University of Manchester

Reinis Cirpons
University of St Andrews

Ajda Lemut Furlani
Institute of Mathematics, Physics and Mechanics, Ljubljana

Ádám Budai
University of Szeged

Matthias Fresacher
Western Sydney University

Duarte Ribeiro
Universidade NOVA de Lisboa