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 \(x\in \mathbb{Q}\))
\[ H_1(x) = \begin{pmatrix} 1 & x \\ 0 & 0 \end{pmatrix}, \\ H_2(x) = \begin{pmatrix} x+1 & x \\ ( – x – 1) & – x \end{pmatrix}, \\ H_3(x) = \begin{pmatrix} 0 & x \\ 0 & 1 \end{pmatrix}, \\ H_4(x) = \begin{pmatrix} ( – x+1) & ( – x+1) \\ x & x \end{pmatrix}, \\ H_5(x) = \begin{pmatrix} ( – x+1) & ( – x – 1 – \frac{2}{x – 2}) \\ x – 2 & x \end{pmatrix} , x \neq 2. \]

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