  We investigate two operators on classes of regular languages: polynomial
closure (Pol) and Boolean closure (Bool). We apply these operators to classes
of group languages G and to their well-suited extensions G+, which is the least
Boolean algebra containing G and the singleton language containing the empty
word. This yields the classes Bool(Pol(G)) and Bool(Pol(G+)). These classes
form the first level in important classifications of classes of regular
languages, called concatenation hierarchies, which admit natural logical
characterizations. We present generic algebraic characterizations of these
classes. They imply that one may decide whether a regular language belongs to
such a class, provided that a more general problem called separation is
decidable for the input class G. The proofs are constructive and rely
exclusively on notions from language and automata theory.
