## All kinds of mathematics, 3

Wednesday morning’s first speaker was Bob Gray, who told us about his quest, with Igor Dolinka, on a semigroup analogue for Philip Hall’s countable universal homogeneous locally finite group. He started with a clear introduction to Hall’s group. For semigroups, it turns out that no such example can exist; said otherwise, a countable universal locally finite semigroup cannot be homogeneous. So how homogeneous can it be? If the automorphism group of such a semigroup acts transitively on the copies of the finite semigroup S it contains, then S must be an amalgamation base; they call a semigroup maximally homogeneous if the converse holds. There is a unique such semigroup which is a limit of full transformation semigroups (as Hall’s group is a limit of symmetric groups, though the analogue of Hall’s construction fails). It has lots of nice properties; for example, its Graham–Houghton graph is the universal homogeneous graph with bipartition.

Bob started by talking about some of my books and papers which have influenced him. A reminder that any piece of writing, once it leaves the writer’s desk, has a life of its own.

Then António Malheiro told us about the plactic monoid (connected to Young tableaux) and some generalisations, the hypoplactic monoid (related to ribbon tableaux) and the Sylvester monoid (related to binary search trees). He was interested in finding analogues of Kashiwara chrystal bases for these objects.

James Mitchell spoke about semigroups generated by digraphs (where we regard a directed arc (ab) as a rank n−1 idempotent which maps a to b and fixes everything else. He surveyed many old and new results about these semigroups, and remarked that unlike many types of transformation semigroup the correlation of properties between semigroup and digraph is very close. There are connections with sliding-block puzzles (indeed, he quoted the work of Richard Wilson on these). Sliding-block puzzles are more usually represented by a groupoid than a semigroup; there must be connections!

Alexander Hulpke has new code for computing interval lattices in a group G (essentially, all subgroups between H and G for some subgroup H and their inclusions), which is much faster than the current GAP code. I am sure I will find a use for this code, which will be included in GAP 4.9 (hopefully before the end of the year).

And finally Peter Mayr talked about counting finite (universal) algebras up to term equivalence. He introduced the class of algebras with few subpowers (the number of subalgebras of An grows only exponentially with n rather than doubly exponentially, and showed us that such algebras are finitely related, have a polynomial time solution for the constraint satisfaction problem, and various other nice properties.

After lunch, we left for the excursion. First we went to the National Gallery, where the highlights included a polyptych of the generation of Henry the Navigator and the following generation of Portuguese royalty whose interpretation is very controversial, and a wonderful Hieronymus Bosch triptych. I also passed an old acquaintance whom my colleagues will certainly recognise:

Then to the Ajuda palace gardens, where we had a very enjoyable dinner in a tent in the garden, interspersed with some embarrassing recollections (which, fortunately, were not recorded), and so home.