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.

• $A$ with $F_1$ ($F_2$) generates a right (left) zero semigroup (hence of size $2$, except for $x=0$).
• $A$ 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 $3$).
• $A$ 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 0.0.4 In case $A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$ the solutions for $B$ being of rank one consist of five one-dimensional families namely (for $x\in \mathbb{Q}$)

$$H_1\left(x\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill x\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),H_2\left(x\right)=\left(\begin{array}{cc}\hfill x+1\hfill & \hfill x\hfill \\ \hfill -x-1\hfill & \hfill -x\hfill \end{array}\right),$$

$$H_3\left(x\right)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill x\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right),H_4\left(x\right)=\left(\begin{array}{cc}\hfill -x+1\hfill & \hfill -x+1\hfill \\ \hfill x\hfill & \hfill x\hfill \end{array}\right),$$

$$H_5\left(x\right)=\left(\begin{array}{cc}\hfill -x+1\hfill & \hfill -x-1-\frac{2}{x-2}\hfill \\ \hfill x-2\hfill & \hfill x\hfill \end{array}\right),x\ne 2.$$

As before we can describe size and structure.

• $A$ with $H_1$ ($H_2$) generates a right (left) zero semigroup (as before).
• $A$ with $H_3$ or $H_4$ generates a semigroup with $AB$ nilpotent (as before).
• $A$ 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.

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.