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Given a multirelational database scheme and a relational mapping f transforming it, an important question is whether the resulting scheme is equivalent to the original one. This question was addressed in the literature with respect to those relational schemes that satisfy the so called universal relation assumption; however, no study was ever concerned with multirelational (data base) schemes that do not necessarily satisfy this assumption.We present two general definitions of lossless transformation of the database scheme based on the so-called closed world and open world assumptions. While both definitions seem to be practically justified, the one based on the open world assumption is more \"tractable\" We are able to test losslessness defined in such a way for a wide class of relational expressions and dependencies. An algorithm for testing losslessness of a mappings (which are arbitrary relational expressions built up from projections, cartesian products and restrictions) is presented in the paper. Moreover, given a lossless transformation, our algorithm enables us to explicitly construct an \"inverted\" mapping that restores the corresponding state of the original database. The application of the algorithm to schemes specified by differrent types of dependencies is described. In particular the application of the algorithm for schemes specified by inclusion dependencies is presented. In this case the algorithm works for families of inclusion dependencies having finite chase property. This class of inclusion dependencies is characterized in the paper.