Abstract: A semigroup topology for a semigroup S is any topology on S under which the semigroup operation is continuous. If S happens to be an inverse semigroup (for example a group), then an inverse semigroup topology on S is a semigroup topology for S under which inversion is also continuous. 

Given a semigroup, it is natural to wonder what semigroup topologies it admits and if there is, in some sense, a preferred or canonical one.The symmetric group and full transformation semigroup and many of its important subgroups and subsemigroups have a natural topology, the so-called pointwise topology. There are many good reasons to consider the pointwise topology canonical for these two semigroups. For example, it is the unique Polish topology which is a semigroup topology for the full transformation semigroup and a group topology for the symmetric group. 

The Symmetric Inverse Monoid is to inverse semigroups what the symmetric group and the full transformation semigroup are to groups and semigroups, respectively. In this talk, we will consider what topology one should give to the Symmetric Inverse Monoid. Among other results, we will show that the Symmetric Inverse Monoid has infinitely many Polish semigroup topologies but a unique Polish inverse semigroup topology.

Abstract: There exists a vast literature around the idea of constructing groups (and monoids) via some correspondence with finite simple graphs. One well-known example is right-angled Artin groups (RAAGs), where edges in our graph corresponding precisely to commuting generators in our group.

In this talk, we will discuss an adaptation of this group, known as twisted RAAGs (T-RAAGs), where we introduce the option of ‘Klein-bottle’ relations, to correspond to directed edges of our graph. We will survey what we know already about T-RAAGs, and the many open questions about these groups (and monoids).

Abstract: A semigroup S is said to be finitely right equated if its right annihilator congruences

r_{S}(a) = {(u,v) ∈ S x S | au=av} (a ∈ S)

are finitely generated. In this talk we will explain how being finitely right equated arises from weak right coherency. We will also examine the closure under finite direct product of the class of finitely right equated semigroups compared to the finitely right equated monoids, highlighting the increased complexity in the semigroup case. We will provide necessary and sufficient conditions for the direct product of semigroups to be finitely right equated.

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Abstract: Schützenberger graphs are becoming a widely used tool to study inverse semigroups. Given an inverse semigroup S, the maximal group morphism from S to S/σ induces maps from the Schützenberger graphs of S to the Cayley graph of S/σ. It is natural to ask when one can lift certain geometric or algorithmic properties of S/σ to S, which leads us to study the geometric distortion of these maps which we call group distortion. We investigate when this distortion is bounded, and when it is bounded by a recursive function, and explore connections with the structure and the algorithmic properties of the inverse semigroup. This is based on joint work with Mark Kambites.

Abstract: Cayley's Theorem states that any finite monoid can be faithfully represented as a semigroup of transformations (self-maps) of a finite set. The minimum size of such a set is the (minimum transformation) degree of the monoid.

We obtain formulae for the degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid P_n has degree 1 + (B(n+2)-B(n+1)+B(n))/2 for n ≥ 2, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

This is joint work with Reinis Cirpons and James Mitchell, both at Univ St Andrews.

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Abstract: One way to obtain normal forms for elements of a finitely presented monoid is to identify words with combinatorial objects. Perhaps the most well-known example of this appears in the Plactic monoid, where words are identified with Young tableaux according to the Robinson–Schensted insertion algorithm.

In this talk, I will define a monoid due to Hivert, Novelli and Thibon that relates to binary search tree insertion - the Sylvester monoid - and discuss the properties of some of its quotients. With this established, I will highlight similarities with other plactic-like monoids.

Abstract: An important result proved by Magnus in 1932 is that all groups with a single defining relation, now known as one-relator groups, have decidable word problem. In stark contrast, the word problem for one-relator monoids remains unsolved - despite over a century of investigation. It was proved by Adian in 1966 that all one-relator monoids of the form Mon<A | w = 1 > have decidable word problem, while it still remains open for one-relator monoids of the form Mon<A | u=v >. However, throughout the last century, many reduction results have been proven. 

Results of Adian and Oganesian prove that the word problem for one-relator monoids can be reduced to the word problem in certain two-generator one-relator monoids. While Ivanov, Margolis and Meakin proved that it can be reduced to the word problem for one-relator special inverse monoids. Furthermore they proved that, in certain cases the word problem for one-relator special inverse monoids can be reduced to the prefix membership problem for the maximal group image which is a one-relator group.

Recent results have shown that there are one-relator inverse monoids with undecidable word problem and one-relator groups with undecidable prefix membership problem. The constructions used to establish those results always result in examples of groups and inverse monoids with at least three generators. Results for groups and monoids prove that each one-relator group/monoid with n generators embeds into a one-relator group/monoid with two generators. The techniques used in the groups proof fail immediately when we consider special inverse monoids, and it is unknown whether or not the method from the monoid proof works considering special inverse monoids. 

I will present new results that show how one can construct two-generator one-relator groups with undecidable prefix membership problem, and  two-generator one-relator inverse monoids with undecidable word problem, building on recent results by Gray.

Abstract: Graph products of monoids provide a common generalisation of direct and free products. Given a graph Γ = (V,E) and a collection of 'vertex' monoids M = {M_𝛼 : 𝛼 ∈ V }, the graph product of M with respect to Γ is the freest monoid generated by the union of the M_𝛼’s such that elements in 'adjacent' vertex monoids commute. In the case where the vertex monoids are free monogenic, graph products are known as graph monoids (or as free partially commutative monoids, or as right angled Artin monoids), and arise in the study of concurrent processes in computer science.

A finitary condition for a monoid is one possessed by all finite monoids. This talk will consider some finitary conditions for monoids in the context of graph products. We focus on the question of whether certain such conditions pass up and down between the graph product and the vertex monoids. Results obtained immediately apply to direct products and free products of monoids.

The work presented will be joint with Jung Won Cho, Nik Ruškuc and Dandan Yang.

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