[GAP Forum] (no subject)

[James Mitchell]

On 15 March 2012 10:06, James Mitchell <jdm3 at st-and.ac.uk> wrote:

> Dear Shubh,
>
> The answer to your question is yes and no. There is *no* function in GAP
> named IsRightFactorisableSemigroup, so in this sense the answer is no. On
> the other hand, it should be more or less straightforward for a given
> example of a semigroup to determine whether or not it is right factorisable
> by examining its structure in GAP. If you have the Citrus package loaded,
> then you can do the following:
>
> gap> gens:=[ Transformation( [ 2, 6, 7, 2, 6, 1, 1, 5 ] ),
> >   Transformation( [ 3, 8, 1, 4, 5, 6, 7, 1 ] ),
> >   Transformation( [ 4, 3, 2, 7, 7, 6, 6, 5 ] ),
> >   Transformation( [ 7, 1, 7, 4, 2, 5, 6, 3 ] ) ];;
> gap> s:=Monoid(gens);
> <monoid with 4 generators>
> gap> GroupOfUnits(s);
> Group(())
> gap> IsBand(s);
> false
>
> which proves that this semigroup is not right factorisable. Another
> example:
>
> gap> gens:=[Transformation([2,1,4,5,3,7,8,9,10,6]),
> > Transformation([1,2,4,3,5,6,7,8,9,10]),
> > Transformation([1,2,3,4,5,6,10,9,8,7]),
> > Transformation([9,1,4,3,6,9,3,4,3,9])];;
> gap> s:=Monoid(gens);;
> gap> iter:=Iterator(s);
> <iterator of transformation semigroup>
> gap> g:=GroupOfUnits(s);;
> gap> e:=Idempotents(s);;
> gap> NrIdempotents(s);
> 13655
> gap> t:=Set(Flat(List(e, x-> x*Elements(g))));;
> gap> Length(t);
> 186726
> gap> Size(s);
> 491558
>
> Which shows that the monoid <s> is also not right factorisable.
>
> I hope that that helps.
>
> Regards,
>
> James
>
> > From: "shubh at iitg.ernet.in" <shubh at iitg.ernet.in>
> > Subject: [GAP Forum] Code for Right Factorisable Semigroup
> > Date: 13 March 2012 11:13:51 GMT
> > To: "forum at gap-system.org" <forum at gap-system.org>
> > Cc: "forum-request at gap-system.org" <forum-request at gap-system.org>
> >
> > Dear Sir,
> > Is there a code in GAP to check whether a semigroup is Right Factorisable
> > or not.
> >
> > A semigroup S is right factorisable if S = EG for some subgroup G of S
> and
> > E is the set of all idempotents in S.
> >
> >
> >
> > With Regards,
> > \-Shubh
> --
> James Mitchell
> tinyurl.com/jdmitchell
>
> The University of St Andrews is a charity registered in Scotland : No
> SC013532
>
>
>
>
>


-- 
James Mitchell
tinyurl.com/jdmitchell

The University of St Andrews is a charity registered in Scotland : No
SC013532