The Number of Keys in Relational and Nested Relational Databases

Thalheim, B., The number of keys in relational and nested relational databases, Discrete Applied Mathematics 40 (1992) 265-282. Combinatorial propositions, concerning the maximal number of minimal keys are considered. It is shown that the results of Demetrovics about the maximal number of minimal keys on unbounded domains do not hold for finite domains. Using this result lower bounds for the size of minimum-sized Armstrong relations are derived. It is also shown that the maximal number of minimal keys in databases on nonuniform domains is also precisely exponential in the number of attributes like in the case of uniform domains. In relational database theory and practice, it is often postulated that none of the attributes of the primary key may ever obtain an undefined, unknown value, since otherwise we would not know what entity a tuple with an undefined value of the primary key represents. This assumption could be weakened. Eventually, a new approach based on distinguished tuples is presented. We consider key sets, a generalization of keys and existence constraints. Key concepts in nested relational schemes can be introduced on distinct equality concepts. It is shown that the results of Demetrovics hold for the nested relational model.