H-trivial semigroups¶
- class sage.categories.h_trivial_semigroups.HTrivialSemigroups(base_category)[source]¶
- Bases: - CategoryWithAxiom- Finite_extra_super_categories()[source]¶
- Implement the fact that a finite \(H\)-trivial is aperiodic. - EXAMPLES: - sage: Semigroups().HTrivial().Finite_extra_super_categories() [Category of aperiodic semigroups] sage: Semigroups().HTrivial().Finite() is Semigroups().Aperiodic().Finite() True - >>> from sage.all import * >>> Semigroups().HTrivial().Finite_extra_super_categories() [Category of aperiodic semigroups] >>> Semigroups().HTrivial().Finite() is Semigroups().Aperiodic().Finite() True 
 - Inverse_extra_super_categories()[source]¶
- Implement the fact that an \(H\)-trivial inverse semigroup is \(J\)-trivial. - Todo - 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.- EXAMPLES: - sage: Semigroups().HTrivial().Inverse_extra_super_categories() [Category of j trivial semigroups] sage: Monoids().HTrivial().Inverse() Category of h trivial groups - >>> from sage.all import * >>> Semigroups().HTrivial().Inverse_extra_super_categories() [Category of j trivial semigroups] >>> Monoids().HTrivial().Inverse() Category of h trivial groups