# Cayley tables for semigroups of order $\le 8$

I need Cayley Tables for semigroups of order $\le 8$. If someone knows where can I find this information, please let me know.

I know that this information is stored in GAP(Groups, Algorithms, Programming), in the GAP package Smallsemi, but anyway I cannot use it.

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I do not know where you can find this information, but I do know that there are a lot of semigroups of order $\leq 8$. There is $1$ of order $1$, $4$ of order $2$, $18$ of order $3$, $126$ of order $4$, $1160$ of order $5$, $15973$ of order $6$, $836021$ of order $7$, and a whopping $1,843,120,128$ semigroups of order $8$. I believe that it is unknown how many semigroups of order $9$ there are...So, emm, good luck with your quest! – user1729 Nov 15 '11 at 14:41
(Also, those numbers are up to anti-isomorphism, if my notes can be believed, not just isomorphism!) – user1729 Nov 15 '11 at 14:54
You know thatthe information is in GAP but you cannot use it? Why? – Mariano Suárez-Alvarez Nov 15 '11 at 15:52
@Trigran: why don't you ask the GAP developers in their mailing list? I am sure they generated those tables so that people can use them... – Mariano Suárez-Alvarez Nov 15 '11 at 16:23
Thanks. My notes must be our of date. Not that I have ever used these numbers before yesterday... – user1729 Nov 17 '11 at 11:53

Try:

gap> LoadPackage("smallsemi");;
gap> SmallSemigroup(8,10200808);;
gap> Display(RecoverMultiplicationTable(8,10200808));
[ [  1,  2,  3,  4,  5,  6,  7,  8 ],
[  2,  1,  4,  3,  6,  5,  8,  7 ],
[  3,  4,  2,  1,  7,  8,  6,  5 ],
[  4,  3,  1,  2,  8,  7,  5,  6 ],
[  5,  6,  8,  7,  2,  1,  3,  4 ],
[  6,  5,  7,  8,  1,  2,  4,  3 ],
[  7,  8,  5,  6,  4,  3,  2,  1 ],
[  8,  7,  6,  5,  3,  4,  1,  2 ] ]


Turning these into text files seems a bit insane. It will be a very large text file (around 350GB). The compression format used by the package is much better. Try to use GAP to study them. You should find it fairly easy to use.

You should unpack the smallsemi archive inside your pkg directory.

Here is a routine to export the Cayley tables to files. Each Cayley table of a semigroup of order n is represented as n lines of n digits each, so n ≤ 9.

gap> for n in [1..7] do for k in [1..NrSmallSemigroups(n)] do
> AppendTo( Concatenation("cay",String(n),".txt"),
>   JoinStringsWithSeparator( List( RecoverMultiplicationTable( n, k ),
>     row -> JoinStringsWithSeparator( row, "" ) ), "\n" ), "\n\n" );
> od; od;


You'll find the files as cay1.txt etc. in the current directory, with cay7.txt being 47MB. A similar file for cay8.txt would be 125GB.

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Thanks for your reply again. I need an example of semigroup which satisfies some identities but not the others. I wrote a program by C++, which checks the usability of the example by its Cayley tabl. That's why I need the Cayley tables for those small semigroups. If it is possible to turn this information into a text file, please say the way. Maybe it will be sufficient to have the Cayley tables for semigroups of order up to 5 or 6. – Tigran Hakobyan Nov 17 '11 at 13:12
@Tigran: can you give an example text file for a semigroup of order 3? — You may also consider asking GAP if the semigroup satisfies the relationship. It may be possible to check the semigroup without creating its table, but rather checking the precomputed table of properties. This would be a dramatic speedup. – Jack Schmidt Nov 17 '11 at 17:22
The identities to which the semigroup should satisfy are of this sort: $x^4=x^2, x^3yx^3=x^2yx^2$, and I want that semigroup not to be a hyperassociative one. – Tigran Hakobyan Nov 17 '11 at 17:51
Thank you very much!!! It helped me :D – Tigran Hakobyan Nov 19 '11 at 6:41
For readers of this discussion who will try to use Smallsemi under Windows: see "Introduction" chapter from the Smallsemi manual (curent version 0.6.5), which says: "As the data in the library is compressed, 30 MB of disk space will be sufficient to install Smallsemi under Unix. To use the library under Windows the data has to be uncompressed and will then occupy approx. 1.6 GB. ... It should be possible to use Smallsemi under Windows after unzipping all data files. (These are located in the directory data and its subdirectories and have the file extension .gz.)". – Alexander Konovalov Apr 22 '13 at 20:28

You may also find Andreas Distler's PhD thesis on Classification and Enumeration of Small Semigroups useful. This contains a lot of the theory behind what is implemented in GAP, I think.

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Nice reference, at least for me. $+^{+}$ – Babak S. Jul 2 '13 at 6:18

All the semigroups of order 3 are the following:

111
121
113

111
122
123

111
122
133

111
121
333

111
123
333

111
222
333

211
122
122

211
122
123

212
121
212

213
123
333

221
222
123

221
222
223

222
222
221

222
222
222

222
222
223

222
222
333

223
223
333

231
312
123

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