# Matrices vs. idempotent ultrafilters part 2

07 Jan 2010In an earlier post I gave a short introduction to an interesting finite semigroup. This semigroup could be found in the matrices over .

When I met with said friend, one natural question came up: what other semigroups can we find this way?

The first few simple observations we made were

- If either or is the identity matrix or the zero matrix the resulting semigroup will contain two elements with an identity or a zero element respectively.
- In general, we can always add or to the semigroup generated by and and obtain a possibly larger one.
- generate a finite semigroup iff is of finite order (in the sense that the set of its powers is finite).
- has finite order iff its (nonvanishing) eigenvalue is .
- For of rank we may assume (by base change) that is one of the two matrices

So, as a first approach we thought about the following question.

**Question** If we take to be one of the above, what kind of options do we have for , i.e., if is idempotent and to generate a finite semigroup?

Thinking about the problem a little and experimenting with Macaulay 2 we ended up with the following classification

**Proposition** For the solutions for being of rank one consist of four one – dimensional families, namely (for )

Additionally, we have four special solutions

Note: due to technical problems, this post continues here .

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