Category Archives: thelazyscience@blogspot

Good news, bad news

Just so that not another week ends without me writing a post. The bad news is that my departure for Michigan gets closer and the technicalities take up more and more time. Therefore I’m not sure I’ll have much time to post in the next couple of weeks. Additionally, I’ll be attending a winter school in Hejnice in the first week of February so I also need to prepare finish preparing my talk.

So what’s the good news? Well, I have been busy on the blog but nothing has come of it yet. On the one hand I have been studying the Google App Engine so as to move the blog there — which should make the work flow much more efficient (and the code better). On the other hand that there are three blog posts I have not finished — so there’s a chance the dry spell will be over sooner than I think. Finally, I hope to write posts during the winter school reflecting on the (possibly daily) experience.

Well, let me at least throw in some nice links worth a read. Gil Kallai turned a mathoverflow question into the kind of blog posts I really like . Over at the n-Category Cafe David Corfield explains muses over the “sacred” and the “profane” in mathematics (or rather for mathematicians) which made me ponder what my own “bottom line” is.

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

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.