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 $x\in \mathbb{Q}$)

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.
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