Cayley tables for semigroups of order $\le 8$

Question

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.

Answer 1

Try:

gap> LoadPackage("smallsemi");;
gap> SmallSemigroup(8,10200808);;
#I  Smallsemi: loading data for semigroup properties. Please be patient.
#I  Smallsemi: loading data for semigroups of size 8.
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.

Answer 2

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