Power Semigroups and Polynomial Closure

摘要

We show that the pseudo-variety of semigroups which are locally block groups is precisely that generated by power semigroups of semigroups which are locally groups; that is P(LG) = L(PG) (using that PG = BG). We also will show that this pseudovariety corresponds to the Boolean polynomial closure of the LG-languages which is hence polynomial time decidable. More generally, it is shown that if H is a pseudovariety of groups closed under semidirect product with the pseudovariety of p-groups for some prime p, then the pseudovariety of semigroups associated to the Boolean polynomial closure of the LH-languages is P(LH). The polynomial closure of the LH-languages is similarly characterized.