Peter Krautzberger · on the web

# Matrices vs. idempotent ultrafilters part 2

In 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.

1. with () generates a right (left) zero semigroup (hence of size , except for ).
2. with or generates a semigroup with nilpotent (of size , except for , where we have the null semigroup of size ).
3. 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.

1. with () generates a right (left) zero semigroup (as before).
2. with or generates a semigroup with nilpotent (as before).
3. 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