Implement the fact that a finite \(H\)-trivial is aperiodic
sage: Semigroups().HTrivial().Finite_extra_super_categories() [Category of aperiodic semigroups] sage: Semigroups().HTrivial().Finite() is Semigroups().Aperiodic().Finite() True
Implement the fact that an \(H\)-trivial inverse semigroup is \(J\)-trivial.
Generalization for inverse semigroups.
Recall that there are two invertibility axioms for a semigroup \(S\):
One stating the existence, for all \(x\), of a local inverse \(y\) satisfying \(x=xyx\) and \(y=yxy\);
One stating the existence, for all \(x\), of a global inverse \(y\) satisfying \(xy=yx=1\), where \(1\) is the unit of \(S\) (which must of course exist).
It is sufficient to have local inverses for \(H\)-triviality to imply \(J\)-triviality. However, at this stage, only the second axiom is implemented in Sage (see
Magmas.Unital.SubcategoryMethods.Inverse()). Therefore this fact is only implemented for semigroups with global inverses, that is groups. However the trivial group is the unique \(H\)-trivial group, so this is rather boring.
sage: Semigroups().HTrivial().Inverse_extra_super_categories() [Category of j trivial semigroups] sage: Monoids().HTrivial().Inverse() Category of h trivial groups