AAA108

Last month I was in Vienna, at AAA108. This was the 108th meeting of the Arbeitsgruppe Allgemeine Algebra (or Workshop on General Algebra), which has been going since 1971, with usually two meetings a year. The meetings are held at weekends, to maximise the chance that people can attend without disrupting their lecturing schedules too much (Friday, Saturday and Sunday morning).

It was founded by Rudolf Wille (whose influence still lingers; there was at least one talk about concept lattices), and over time it merged with other algebra meetings. The next three meetings will be in Ljubljana this summer (followed by Semigroups and Automata), and Dresden and Siena next year. It shares interest and audience with other meetings such as BLAST.

Perhaps the most popular topic at the meeting was the material around clones, polymorphisms, and constraint satisfaction problems. In particular, there was a lot of discussion of the recent proofs by Bulatov and Zhuk of the Dichotomy Theorem for constraint satisfaction problems over finite templates (such a problem is either in P or NP-complete), and the related conjecture of Bodirsky and Pinsker about ω-categorical templates. I wasn’t talking about that. Instead, I took the opportunity to present some thoughts about extending results about graphs on groups to more general algebras.

My theme was that some graphs on groups are inexorably bound to the group structure (such as the commuting and nilpotency graphs), but there is a cluster of graphs which can be defined for arbitrary algebras since the definitions only use the notion of subalgebra (and possibly endomorphism); these are the power graph (and its directed, enhanced and intersection versions), generating graph, independence graph and rank graph. What really amazed me is that some of the results for groups, including very recent ones, extend to arbitrary algebras. For example, a recent result by Daniela Bubboloni, Francesco Fumagalli and Cheryl Praeger asserts that, if the enhanced power graph of a group is a cograph, then it is also a chordal graph. (Not every cograph is a chordal graph!) I was surprised to discover that this result is not special to groups, the same proof works in any algebra whatsoever; and with a minor adjustment, it works not only for the enhanced power graph but also for the intersection power graph as well. The material in my talk is on the arXiv, at 2602.00712; it went live a day or two before the conference.

Anyway, my talk was the opening plenary, so after that the pressure was off and I could listen to other talks. One of the talks in the very next session was by Samir Zahirović, someone I really wanted to meet; he proved something which had eluded me, namely that groups having isomorphic enhanced power graphs also have isomorphic directed power graphs. His talk was about the difference between the enhanced power graph and the power graph, and he has shown that for some groups this graph determines the power graph up to isomorphism.

Quite a few talks at the conference, including Ross Willard’s nice plenary, concerned algebras with a Mal’cev or difference term; this is not something I know much about, but perhaps I was too hasty in saying that the commuting graph really only works for groups. There are algebras which have a good commutator theory, and perhaps there is room for some graph theory there.

One talk I really enjoyed was by Péter Pál Pálfy. He revisited a theorem of Galois, which stated (in modern terminology) that, for a prime number p, the group PSL(2,p) has no subgroup of index less than p+1 (and so the problem of dividing the periods of an elliptic function by p has no resolvent equation of degree smaller than p+1) except for the primes 2, 3, 5, 7, 11. P cubed had studied Galois’ work on this group carefully, and was able to show us a proof of this fact which could well have been the proof Galois came up with: it used methods he was familiar with and extended his argument for the case p = 5.

My colleague James Mitchell gave a beautiful plenary talk (supported by a contributed talk by Luna Elliott) on the extent to which the algebraic structure of a group or monoid determines its topology. For example, the symmetric group of countable degree has a unique Polish topology. For semigroups the position is much more complicated, but James took us through what he and his co-authors have done. For example, the symmetric inverse monoid on a counntable set has infinitely many Polish topologies, forming a join-semilattice with fascinating properties. His style was based on his undergraduate lecturing style, stopping for quizzes from time to time to keep us awake.

Finite dimensional vector spaces over a fixed finite field form a Fraïssé class whose Fraïssé limit is the vector space of countable dimension. Peter Mayr told us a result he and Nik Ruskuc have obtainedd, which says that something very similar holds for finite direct powers of any simple abelian Mal’cev algebra. They have also investigated the automorphism group of the Fraïssé limit.

Misha Muzychuk was supposed to tell us about a connection between Jordan schemes (the Jordan algebra version of association schemes) and ring-alternative Moufang loops. I had looked forward to this, but unfortunately he had not been able to come.

Christian Pech talked on a topic that Jarik Nešetřil and I started some time ago. We introduced the notion of homomorphism-homogeneity, or HH (any homomorphism between finite substructures extends to an endomorphism of the whole structure). This was developed by Lockett and Truss and by Coleman, Evans and Gray; Christian and his coauthors have perhaps finished the job off with a general study of XY for all reasonable choice of X and Y.

Gianluca Paolini talked about the position in the Borel hierarchy of various isomorphism problems; most interesting for me was the problem for oligomorphic groups, which is surprisingly easy. I won’t try to say more.

There was a great deal more of interest at the meeting, but that hopefully gives you some of the flavour.

I prefer to write about meetings like this closer to the time, while they are fresh in my mind; but circumstances didn’t permit that.