Mikhail Berlinkov posted a paper on the arXiv this week proving that two random transformations of an n-set generate a synchronizing semigroup with probability 1-o(1/n) for large n.
His approach was quite different from the one I’d been taking, using much more probability theory and less hands-on combinatorics. In particular, we wondered whether this is the first paper in finite semigroup theory in which a double integral appears!
But this is by no means the end of the story. The semigroup generated by two random transformations surely has many other properties; in proving some of these, I suspect that ideas about graph endomorphims will have a role to play.
So the story continues …
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