Algebra Seminar: East -- Congruences on diagram monoids

James East (Western Sydney University) 

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

Title: Congruences on diagram monoids.  

Abstract: 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.