Donnerstag, 7. Januar 2010

Matrices vs idempotent ultrafilters, part 2

In an earlier post I gave a short introduction to an interesting finite semigroup. This semigroup could be found in the 2 × 2 matrices over .

When I met with said friend, one natural question came up: what other semigroups can we find this way?

The first few simple observations we made were


  • If either A or B is the identity matrix I2 or the zero matrix 02 the resulting semigroup will contain two elements with an identity or a zero element respectively.
  • In general, we can always add I2 or 02 to the semigroup generated by A and B and obtain a possibly larger one.
  • A,B generate a finite semigroup iff AB is of finite order (in the sense that the set of its powers is finite).
  • AB has finite order iff its (nonvanishing) eigenvalue is +- 1.
  • For A of rank 1 we may assume (by base change) that A is one of the two matrices
    10 0 0 , 11 0 0 .

So, as a first approach we thought about the following question.

Question If we take A to be one of the above, what kind of options do we have for B, i.e., if B is idempotent and A,B to generate a finite semigroup.

Thinking about the problem a little and experimenting with Macaulay 2 we ended up with the following classification

Proposition For A = 10 0 0 the solutions for B being of rank one consist of four one-dimensional families, namely (for x )

F1(x) = 1x 0 0 ,F2(x) = 10 x 0 ,

F3(x) = 0x 0 1 ,F4(x) = 00 x 1 . Additionally, we have four special solutions G1 = -11 -2 2 ,G2 = -1-1 2 2 ,

G3 = -12 -1 2 ,G4 = -1-2 1 2 .

Note: due to technical problems, this post continues here.

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 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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