Abstract

Let X be a finite set and let o be a binary operation on X that is injective in each variable separately and has the property that
x o (y o z) = (x o y) o z for a positive proportion of triples (x,y,z) with x,y,z in X. What can we say about this operation? In particular, must there be some underlying group structure that causes the partial associativity? The answer turns out to be yes ? up to a point. I shall explain what that point is and give some indication of the ideas that go into the proof, which is joint work with Jason Long. I shall also report on a natural strengthening that one might hope for. We identified a likely counterexample, which was recently proved to be a counterexample by Ben Green, so in a certain sense our result cannot be improved. (However, there are still some interesting questions one can ask.)