Random synchronization

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 …

About Peter Cameron

I count all the things that need to be counted.
This entry was posted in mathematics, open problems and tagged , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.