For any inverse semigroup S we construct an inverse semigroup S(S), which has the following universal property with respect to dual prehomomorphisms from S: there is an injective dual prehomomorphism ιS: S → S(S) such that for each dual prehomomorphism θ from S into an inverse semigroup T there exists a unique homomorphism θ* : S(S) → T with ιSθ* = θ. If we restrict the class of dual prehomomorphisms under consideration to order preserving ones, S(S) may be replaced by a certain homomorphic image Ŝ(S) which can be viewed as a natural generalization of the Birget-Rhodes prefix expansion for groups  to inverse semigroups. Recently, Lawson, Margolis and Steinberg  have given an alternative description of Ŝ(S) which is based on O’Carroll’s theory of idempotent pure congruences . It should be noted that our ideas can be used to simplify some of their arguments.
Billhardt, B., On a wreath product embedding and idempotent pure congruences on inverse semigroups, Semigroup Forum45 (1992), 45–54. MR93c:20103
Billhardt, B., On a wreath product embedding and idempotent pure congruences on inverse semigroups, Semigroup Forum45 (1992), 45–54. MR93c:20103)| false