[GAP Forum] Misunderstanding about principle ideals of a semigroup

[Robert Morse]

Two elements in a semigroup are J-related if they generate the same
two sided ideal.  There are two elements in your example that generate
the two sided ideal of order 8 and the other 6 elements generate the
same two sided ideal of order 6.  The elements in the equivalent
classes in the J-relations are not ideals.

Robert F. Morse

On Sun, Mar 30, 2014 at 1:01 PM, Mohammad Reza Sorouhesh
<msorouhesh at gmail.com> wrote:
> Dear Forum,
>
> We know that what we call a J-Class in a finite semigroup is really a
> principle ideal generated by an element of the semigroup. Here is a
> finitely presented semigroup of order 8:
>
> gap> f:=FreeSemigroup("a","b");;
>         a:=f.1;; b:=f.2;;
>         s:=f/[[a^3,a],[b^2*a,a*b],[(a*b)^2*b,b]];;
>         e:=Elements(s);
>
> Now, lets the following code calls the  two-sided ideal generated by e[1]:
>
>> Elements(SemigroupIdealByGenerators(s,[e[1]]));
> [ a, a^2*b, a^2, a*b, b*a, a*b*a, b*a*b, a*b*a*b ]
>
> But by doing:
>
>> GreensJClasses(s);
>
> none of the Green J-Classes of semigroup "s" is equal to above set. Simply
> asking why two below codes have different outputs? Or, I am mistaking about
> a very simple fact which is clear?
>
>
> Regards
>
> M.R.Sorouhesh
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