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.
 A: 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.
A: 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.
A: 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  
