## Organic semigroup theory: ferns growing in the Jones/Temperley-Lieb monoid

What is the interesting part of doing science? When a new idea, or some new technology enable us to see some unexpected connection between seemingly unrelated areas. As the new technology, take the bipartition capabilities of the Semigroups package for GAP, and the related visualisation tools. For the seemingly unrelated fields, take semigroup theory and fractal geometry. Here is the story.

The Jones/Temperley-Lieb monoids are interesting combinatorial representations of semigroups. For degree 4 here are the usual generators:

, , , .

Only pairs of dots are connected, and the diagram is planar as a graph. The multiplication is the usual diagram composition, stacking two diagrams, connecting the dots, and doing the simplification if possible. Unlike in diagram algebras, the `bubbles’ in the middle are simply discarded.

Here is a random element of , exhibiting some typical features.

I said `interesting’ before, so I owe you the justification. In mathematics interesting is often equal to difficult. There are a few open questions regarding the combinatorics of the Jones monoid. For instance, one of the basic questions for semigroups is the number of idempotents. By sheer silicon power and coding finesse we can figure out some low degree values (A225798) but we have no formula for that, so beyond our computational horizon it is just darkness.

In semigroups, when you are looking for idempotents, you simply look at the -class picture, and find the -classes with idempotents. Assuming that you have the luxury of having the -class picture in sight. Luckily GAP + Semigroups + VIZ give you exactly that. So Eastie did so. Ok, we still don’t have the closed formula (it probably requires a bit more work than just looking), but he discovered this:

This is a diagram showing the locations of the idempotents in a -class of . Yes, they look like ferns, or more like iterated attempts to get the shape of the fern right ( in the top left corner). And it really looks like those images generated by iterated function systems (repeated affine transformations of a simple geometric object), for example the Barnsley fern.

This image is of course not the first one we saw, by

gap&gt; Splash(DotDClasses(JonesMonoid(9)));

you get something like this:

but when you increase the degree, the generated images eventually choke your PDF-viewer. Also, we wanted to see the images in high-definition, pixel per idempotent precision, so a little bit of extra coding was needed. The situation is reminiscent of Mandelbrot’s discovery of the Mandlebrot set. The first images were very crude, so they had to sharpen the tools to see them more clearly. Except that in his case he was searching for the set, while for the Jones monoid, James was on his routine patrol.

Needless to say that the image depends on the ordering of the – and -classes, but the algorithm happens to do the right, systematic ordering. Otherwise, the discovery would have been delayed, until someone comes shrewd (or desperate) enough to try reordering.

For the time being, we have no explanation of the ferns growing inside the semigroup structure of the Jones monoid. Knowing that this structure is of interest for physicists, we are wondering about the possible biological connections. In any case, so far we’ve been simply interested in the Jones monoid, but now we are interested and EXCITED! 🙂

## There are 132 069 776 transformation semigroups on 4 points…

…and here they are. If it is not clear why this is so exciting, then this little note may be helpful.

So slightly more than 132 million – up to conjugation, of course. The grand total is 3 161 965 550, but under normal circumstances we are only interested in the conjugacy class representatives.

How do we know? We enumerated all subsemigroups of computationally. It is actually not that difficult, standard search algorithms with some semigroup-specific optimizations, making the search parallel by using Rees-quotients (see draft paper for details). Well, once it’s done, it always looks easy, but it actually took a while (2 years) to finish the calculation. It was a hobby project in the beginning, which became and ‘official’ research project, by the sheer amount of invested time.

The above archive is a bit of a surprise package, the file size is 1.2GB and you need 9.2GB of disk space when uncompressed. The semigroups are classified according to their size and the numbers of , , and -classes. For instance, suppose you are looking for degree 4 transformation semigroups of size 42 with 6 -classes ,13 -classes and 3 -classes. Then you need to find the file

`T4_S042_L006_R013_D003.gens`

then you can fire up GAP and the `Semigroups`

package and use the `ReadGenerators`

function to load the semigroups, or rather the generating sets for them. Nice and easy, isn’t it? Well, we are planning to make this a bit more accessible with some web interface magic, hopefully in the near future.

For the above example, the file contains only one semigroup, stored in exactly 42 bytes. 🙂

If the required file doesn’t exist, then there is no degree 4 transformation semigroup with the specified numbers. Or we made a mistake at some point.

Speaking of that, we believe that the archive has no errors and omissions. The enumeration uses simple graph-search algorithms, so the correctness is easy to see. However, with real-world actual physical computers, you can never know. Please let us know if something looks suspicious.

Also lease let us know if you find this database helpful/interesting/exciting/useful.

Here is a talk summarizing the project (in case you can put up with my heavy accent, the audio quality gets better once the mic is switched on).