Peter Krautzberger on the web

Matrices vs. idempotent ultrafilters part 2.5

Note: there seems to be some problematic interaction between the javascripts I use and blogspot’s javascripts which prevents longer posts from being displayed correctly. As long as I don’t understand how to fix this, I will simply split the posts.

We can also describe size and the algebraic structure.

  1. A with F_1 ( F_2 ) generates a right (left) zero semigroup (hence of size 2 , except for x=0 ).
  2. A with F_3 or F_4 generates a semigroup with AB nilpotent (of size 4 , except for x=0 , where we have the null semigroup of size 3 ).
  3. A with G_i generate (isomorphic) semigroups of size 8 . These contain two disjoint right ideals, two disjoint left ideals generated by A and B respectively.

Luckily enough, we get something very similar from our alternative for A .

Proposition In case A = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} the solutions for B being of rank one consist of five one – dimensional families namely (for x\in \mathbb{Q} )
% <![CDATA[ H_1(x) = \begin{pmatrix} 1 & x \\ 0 & 0 \end{pmatrix}, \\ H_2(x) = \begin{pmatrix} x+1 & x \\ ( – x – 1) & – x \end{pmatrix}, \\ H_3(x) = \begin{pmatrix} 0 & x \\ 0 & 1 \end{pmatrix}, \\ H_4(x) = \begin{pmatrix} ( – x+1) & ( – x+1) \\ x & x \end{pmatrix}, \\ H_5(x) = \begin{pmatrix} ( – x+1) & ( – x – 1 – \frac{2}{x – 2}) \\ x – 2 & x \end{pmatrix} , x \neq 2. %]]>

As before we can describe size and structure.

  1. A with H_1 ( H_2 ) generates a right (left) zero semigroup (as before).
  2. A with H_3 or H_4 generates a semigroup with AB nilpotent (as before).
  3. A with H_5 generates the same 8 element semigroup (as before).

Finally, it might be worthwhile to mention that the seemingly missing copies of the 8 element semigroup are also dealt with; e.g. – G_i generates the same semigroup as G_i etc.

It is striking to see that the orders of all finite semigroups generated by rational idempotent two by two matrices are either 2^k,2^k + 1 or 2^k + 2 .

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