Coherency and purity for monoids

Professor Victoria Gould, University of York

  • Thu 16 May 19

    14:00 - 16:00

  • Colchester Campus

    STEM 3.1

  • Event speaker

    Professor Victoria Gould

  • Event type

    Lectures, talks and seminars
    Mathematical Sciences Departmental Seminar

  • Event organiser

    Mathematical Sciences, Department of

  • Contact details

    Dr Andrew Harrison

This talk will be aimed at a non-specialist audience.

A monoid is a set S, together with an associative binary operation and possessing an identity, that is, there is an element 1 є S such that 1a = a = a1 for all a є S. 

This talk will focus on finitary properties for monoids, by which we mean properties guaranteed to be satisfied by all finite monoids. Many of these properties arise naturally from the representation of a monoid S via mappings of sets, that is, from S-acts.

For example, an S-act is finitely presented if it may be obtained from a finite set of information: a finite set of generators and a finite set of relations between them.

A monoid is right noetherian if every right congruence is finitely generated, and this is easily seen to be equivalent to every right S-act with a single generator being finitely presented. In fact, an old result of Normak tells us that for a right noetherian monoid every finitely generated right S-act is finitely presented.

A finitary property of particular interest to Professor Gould 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. Certainly right noetherian monoids are right coherent, but the converse is far from true. Coherency arises naturally from several directions, as this talk will explain. It is closely related to notions of purity, which arise from considerations of equations over S-acts, and hence to injectivity.

Professor Gould will present a selection of the results in this area and also a number of open problems. 


Professor Victoria Gould was an undergraduate and a PhD student at York, then a temporary lecturer at Bristol and Manchester before holding a Royal Society European Research Fellowship at the Technishe Hochshule, Darmstadt. She returned to York with an SERC (now EPSRC) Postdoctoral Fellowship, later joining the permanent academic staff.

Her research interests are in algebra, at the abstract end: semigroup theory, universal algebra and model theory.  She enjoys working with her students and other collaborators from around the world. Recently, her main projects have involved free idempotent generated semigroups, finitary properties of monoids, free partial algebras and some classical structural approaches to certain classes of non-regular semigroups.