University of Sydney Algebra Seminar

James East (Estern Sydney University)

Friday 25 May, 12-1pm, Place: Carslaw 375

Congruences on diagram monoids

A congruence on a semigroup is an equivalence relation that is compatible with the semigroup operation. Congruences play a role in semigroup theory akin to that of normal subgroups in group theory; they govern the formation of quotient semigroups, are kernels of semigroup homomorphisms, and so on. In a major 1952 paper, A.I. Mal’cev classified the congruences of a full transformation semigroup: i.e., a semigroup consisting of all self-maps of a fixed set. In the finite case, the lattice of all such congruences forms a chain. In the infinite case, the situation is far more complicated, but Mal’cev gives a succinct description nevertheless. This talk will report on some recent work on congruences on diagram monoids; these include the partition, Brauer and Temperley-Lieb monoids, for example. The finite case is joint work with James Mitchell, Nik Ruskuc and Michael Torpey (all at St Andrews), and the infinite is joint with Ruskuc.