Any semigroup S can be embedded into a semigroup, denoted \phi S, having some remarkable properties. For general semigroups there exists a strong relationship between local submonoids of S and \phi S. For a number of usual properties P, S and \phi S simultaneously satisfy P or not, this does not include complete regularity. In that case, a suitable subsemigroup of \phi S does the trick. For regular semigroups, the relation of S and \phi S is even closer. In addition, every element of \phi S is a product of at most four idempotents. Using the subsemigroup of \phi S aluded to above, for completely regular semigroups this induces a new operator on their varieties.