Matrices vs. idempotent ultrafilters part 207 Jan 2010
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.
- 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