Algebra Seminar: Ruskuc -- Congruence diameter of semigroups

Nik Ruskuc (University of St Andrews) will be speaking in the algebra seminar.  We will
go out for lunch after the talk---all are welcome to join! 

When: 12-1pm Friday November 13 

Where: SMRI Seminar Room (Macleay Building A12 Room 301) 

Title: Congruence diameter of semigroups 

Abstract: Actions are a fundamental concept in semigroup theory.  They play the role
analogous to modules in rings and permutation representations for groups.  A closely
related concept is that of a right (or left) congruence - mirroring the role ideals and
subgroups.  In this talk I will discuss an intriguing finiteness condition - that the
full relation SxS on a semigroup S is finitely generated as a right congruence - and a
natural notion of diameter accompanying it.  When S is a group the condition is
equivalent to finite generation of S itself, and the diameter being finite is equivalent
to S itself being finite.  However, in general, there are many 'weird' interesting
examples, many of them 'very infinite'.  

So, let S be a semigroup, and suppose that X is a finite generating set for SxS.  Then,
for any two elements s,t in S, there is a finite chain of X-transformations connecting s
and t.  The longest such chain is the X-diameter of S, and the smallest X-diameter when
X ranges over all finite generating sets is the (right congruence) diameter of S,
denoted D(S).  This parameter can be finite or infinite.  It turns out that it is finite
for many classical semigroups of transformations, linear transformations and partitions,
and is then very small, i.e.  at most 4.  Usually some intriguing combinatorics is
involved in these results, but the underlying reasons for this phenomenon remain
mysterious.  The new results presented in this talk are due to various subsets of J.
East, V.  Gould, C.  Miller, T.  Quinn-Gregson and myself.