In this chapter we shall discuss several notions of when a relation is adequately represented by its decomposition onto a database scheme. We also introduce a new type of equivalence for database schemes, data equivalence. We first examine both topics for an arbitrary set of relations P, and then state further results for the case where P = SAT(C). NOTIONS OF ADEQUATE REPRESENTATION We state here some notations that will be used throughout the chapter. We want to represent members of a set P of

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... relations. All relations in P are over the scheme U. Q denotes the set of all relations with scheme U. We shall refer to relations in Q as instances, to avoid confusion with relations that are components of databases. R = (Rt, Rz, . . . , RP} will be a database scheme such that U = RIRz ---R,. Let M be the set of all databases over R. We want to represent instances in P as databases in M, so we shall examine the restrictions on R necessary for an adequate representation. In the chapters on normal forms, we were looking for database schemes that eliminated redundancy and gave lossless decompositions. In this chapter, we shall be concerned with enforcing constraints and unique representations. Lossless decomposition frequently will enter into the discussions of the second condition. We have already seen the project-join mapping defined by R, mR. We shall find it useful to separate the projection and join functions. Definition 9.1 The project mapping for R, xn, maps instances in Q to databases in M. For r E Q, we define ?TR(r) = d 195 1% Representation Theory whered = {rr, r2, . . ., rp } is the database in M such that When it is understood we are projecting instances onto databases, we use T for ?Tu. Definition 9.2 The join mapping for R, W, maps databases in M to instances in Q. For database