A structure preserving embedding of semigroups into regular semigroups

For any semigroup S a regular semigroup C(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schtzenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.