Nik Ruskuc (University of St Andrews, UK), Semigroup actions and finiteness conditions

Tue, 7 November 2023, 1:25 pm MT

Let S be a semigroup. A (right) S-act X is a representation of S by transformations on X. S-acts can be regarded as (unary) algebras in their own right. Thus we can talk about subacts, homomorphisms, quotients, generating sets and defining relations for S-acts. This in turn opens up a possibility of attempting to study/understand/classify semigroups according to properties of their S-acts. In this talk I will report on some recent work on some such properties:
- Noetherian: S is Noetherian if every finitely generated S-act is finitely presented;
- Coherent: S is coherent if every finitely generated subact of a finitely presented S-act is finitely presented;
- The trivial S-act is finitely presented;
- Pseudofiniteness: the trivial S-act is finitely presented and there is a uniform bound on the length of derivation sequences.
I will also indicate some intriguing open problems that have arisen in each area. This work is currently being pursued under an EPSRC-funded project held jointly with Victoria Gould from York, and involves several collaborators, including Matthew Brookes, Craig Miller, James East and Thomas Quinn-Gregson.

[slides] [video]