[GAP Forum] Misunderstanding about principle ideals of a semigroup

[Mohammad Reza Sorouhesh]

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