Normalization of databases generally entails the decomposition of the original relations. Even though the new relations are physically stored, the original relations remain a correct representation of the database. One method of obtaining the original relations is by taking join over the new relations and extending some tuples with null values to account for missing information. The set of relations obtained by this method is known as the representative instance model. When we formulate queries over the original relations, we need algebraic expressions to compute restricted projections (a restricted projection is a projection in which all tuples with null values are removed).

In this paper, the author has given a necessary condition for the existence of a union of tableaux that expresses the restricted projection. More specifically, it is shown that if the set of dependencies (full implicational dependencies) of the database scheme is equivalent to a single join dependency and some equality-generating dependencies, then the restricted projection can be expressed by the union of tableaux. In the case of tuple-generating dependencies, the restricted projection can be expressed by the union of tableaux if and only if the set of dependencies is equivalent to a single join dependency.