Algorithmic problems for finite groups and finite 0-simple semigroups

It is shown that the embeddability of a finite 4-nilpotent semigroup into a 0-simple finite semigroup with maximal groups from a pseudovariety Mathematical script capital V is decidable if and only if the universal theory of the class Mathematical script capital V is decidable. We show that it is impossible to replace 4 by 3 in this statement. We also show that if the membership in Mathematical script capital V is decidable then the membership in the pseudovariety generated by the class of all finite 0-simple semigroups with subgroups from Mathematical script capital V is decidable while the membership in the quasi-variety generated by this class of 0-simple semigroups may be undecidable. © 1997 Published by Elsevier Science B.V.