[GAP Forum] (no subject)

[James Mitchell]

Dear Joao,

If the question is: Let S and T be semigroups of transformations of  
degree n where |S|=10 and |T|=11. Then is S a subsemigroup of T?

Then the answer is yes, just use ForAll(S, x-> x in T);

If the question is: Let S be a semigroup of transformations where |S| 
=10. Then does there exist T such that |T|=11 and S embeds in T?

Then the answer is no, there is no method installed in GAP to handle  
this situation.

If the question is: Let S be a semigroup of degree 10 given by its  
Cayley table and let T be a semigroup of degree 11 given by its  
Cayley table. Then is S isomorphic to a subsemigroup of T?

Then the answer is: it is possible to do this in GAP but the method  
is not efficient. Look at the orbits of the 11 10x10 subtables of the  
Cayley table of T inside the symmetric group on 11 points acting on  
the pairs (i,j) i, j in {1,...,11}. If the Cayley table for S lies in  
any of these orbits, then the answer is yes! Otherwise, the answer is  
no. However it might take a long time to tell you an answer.

Let me know if that helps.

Regards,

James





> Message: 6
> Date: Fri, 15 Jun 2007 14:47:32 +0100 (WEST)
> From: Joao Araujo <mjoao at classic.univ-ab.pt>
> Subject: [GAP Forum] test for subsemigroups
> To: forum at gap-system.org
> Message-ID: <Pine.GSO.4.64.0706151431530.23144 at classic>
> Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed
>
>
> Dear Forum,
>
> I would be grateful if someone could tell me if there is in GAP an
> easy way to check if a 10-elements semigroup of transformations S
> can be embedded in an 11-elements semigroup of transformations T.
>
> I thank in advance,
> Joao