Pure Maths Colloquium: Victoria Gould

Studying algebras via their finitary properties is a classic approach stretching back to Noether and Artin in the early part of the last century. Here by a finitary property for a class of algebras \(\mathcal{A}\) we mean a property, defined for algebras in \(\mathcal{A}\), that is guaranteed to be satisfied by any finite member of \(\mathcal{A}\). Of course, the idea is that it will also be satisfied by some infinite algebras in \(\mathcal{A}\), and the fact that it is will ensure that such algebras behave in some way like the finite ones.

This talk will focus on finitary properties for monoids. Many of these arise naturally from the representation of a monoid \(S\) via mappings of sets or, equivalently and more concretely, by \(S\)-acts. A right \(S\)-act is a set \(A\) together with a map \(A\times S\rightarrow A\) where \((a,s)\mapsto as\), such that for all \(a\in A\) and \(s,t\in S\) we have \(a1=a\) and \((as)t=a(st)\). For example, a monoid is right noetherian if every right congruence is finitely generated, and this is equivalent to every monogenic right \(S\)-act being finitely presented.

A finitary property of particular interest to me is that of coherency. We say that a monoid \(S\) is right coherent if every finitely generated \(S\)-subact of every finitely presented right \(S\)-act is finitely presented. Left coherency is defined dually and \(S\) is coherent if it is both right and left coherent. These notions are analogous to those for a ring \(R\) (where, of course, \(S\)-acts are replaced by \(R\)-modules).

Coherency arises naturally from several directions, as this talk will explain, and is closely related to the notion of acts being algebraically closed. We examine the connection with other finitary properties, such as that of being right noetherian. Of course one wants to know which monoids are (right) coherent, and how coherency behaves with respect to some standard constructions. These question turns out to be hard to answer, in spite of us having a condition for coherency analogous to that of Chase for rings.

I will present a selection of the results in this area and also a number of open problems. The talk will be aimed at a non-specialist audience.

The results in this talk are taken from many sources, the most recent being joint with Yang Dandan, Miklós Hartmann, Thomas Quinn-Gregson, Nik Ruškuc and Rida-e Zenab.