University of Sydney Algebra Seminar

James East (University of Western Sydney)

Friday 29th June, 12:05-12:55pm, Carslaw 175

Idempotent generators in partition monoids

Not everyone knows it, but any non-invertible integer matrix is a product of idempotent matrices (Erdos, 1967). This is one of several results that were inspired by John Howie's 1966 paper in which he investigated the semigroup generated by the idempotents of a full transformation semigroup. In the finite case, one recovers the semigroup of all non-invertible mappings. In the infinite case, one obtains all noninvertible mappings of finite shift plus all the infinite shift mappings that satisfy a certain balancing condition with respect to three parameters that (in some sense) measure how far away a mapping is from being the identity mapping.

In this talk, we consider the analogous problem for the partition monoids. These monoids are defined naturally in terms of diagram concatenation, and form the basis of the partition algebras, which contain many important objects such as the Temperley-Lieb and Brauer algebras. Some very interesting combinatorics arises, and there is a lot of interplay between some classic semigroups such as the full transformation semigroup and the symmetric and dual symmetric inverse monoids.