Matrices vs. idempotent ultrafilters part 2.511 Jan 2010
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