Kollokvium Matematiska vetenskaper

Abstrakt finns enbart på engelska: A monoid is a set S, together with an associative binary operation and possessing an identity. Everyone knows something about monoids: the first mathematical structure we ever encounter is the natural numbers N = {1, 2, · · · } under addition - this is a semigroup - once we are sophisticated enough to add 0, we have a monoid N 0. Just as groups model invertible functions, monoids model arbitrary functions. Indeed, the collection of structure preserving maps of any mathematical structure is a monoid.

Let A be a class of algebraic structures. A finitary condition for A is a condition defined for all members of A that is satisfied by all the finite members of A. The idea is that there are infinite members of A that satisfy the condition and that therefore inherit some of the ‘nice’ behaviour of the finite members. The study of algebras via finitary conditions began early in the last century with the work of Artin and Noether and their names became associated with various finitary conditions. There are different ways of defining ‘noetherianity’ for a monoid; N 0 satisfies them all, even though it is infinite. This is largely by virtue of the fact that for any n ∈ N 0 there are only finitely many elements of N 0 smaller than n.

This talk will focus on finitary conditions for monoids. It will be aimed at a non-specialist audience: rather than giving details I will aim to describe the area, its connections and motivations, and present a number of open problems.