# Matrices vs. idempotent ultrafilters part 2.5

11 Jan 2010Note: there seems to be some problematic interaction between the javascripts I use and blogspot’s javascripts which prevents longer posts from being displayed correctly. As long as I don’t understand how to fix this, I will simply split the posts.

We can also describe size and the algebraic structure.

- with () generates a right (left) zero semigroup (hence of size , except for ).
- with or generates a semigroup with nilpotent (of size , except for , where we have the null semigroup of size ).
- with generate (isomorphic) semigroups of size . These contain two disjoint right ideals, two disjoint left ideals generated by and respectively.

Luckily enough, we get something very similar from our alternative for .

**Proposition** In case the solutions for being of rank one consist of five one – dimensional families namely (for )

As before we can describe size and structure.

- with () generates a right (left) zero semigroup (as before).
- with or generates a semigroup with nilpotent (as before).
- with generates the same element semigroup (as before).

Finally, it might be worthwhile to mention that the seemingly missing copies of the element semigroup are also dealt with; e.g. generates the same semigroup as etc.

It is striking to see that the orders of all finite semigroups generated by rational idempotent two by two matrices are either or .

At first sight it seems strange that we cannot find other semigroups with two generators like this. As another friend commented, there’s just not enough space in the plane. I would love to get some geometric idea of what’s happening since my intuition is very poor. But that’s all for today. pdf