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
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 .
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
Thinking about the problem a little and experimenting with Macaulay 2 we ended up with the following classification
the solutions 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
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
It is striking to see that the orders of all finite semigroups generated by rational idempotent two by two matrices are either
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