Montag, 11. Januar 2010

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 F1 (F2) generates a right (left) zero semigroup (hence of size 2, except for x = 0).
  • A with F3 or F4 generates a semigroup with AB nilpotent (of size 4, except for x = 0, where we have the null semigroup of size 3).
  • A with Gi 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 = 11 0 0 the solutions for B being of rank one consist of five one-dimensional families namely (for x )

H1(x) = 1x 0 0 ,H2(x) = x + 1 x -x - 1 -x ,

H3(x) = 0x 0 1 ,H4(x) = -x + 1-x + 1 x x ,

H5(x) = -x + 1-x - 1 - 2 x-2 x - 2 x ,x2.

As before we can describe size and structure.

  • A with H1 (H2) generates a right (left) zero semigroup (as before).
  • A with H3 or H4 generates a semigroup with AB nilpotent (as before).
  • A with H5 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.  - Gi generates the same semigroup as Gi 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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