Cylindric structures and dependencies in relational databases

It is shown that recent algebraic investigations of cylindric lattices provide counterexamples to Cosmadaki's claim that cylindrics dependencies are finitely (schema) axiomatizable.

Abstract

In this paper, we explore the precise connection between dependencies in relational databases and variants of cylindric algebras, and apply recent algebraic results to problems of axiomatizing dependencies. First, we will consider project-join dependencies and the corresponding class of (representable) cylindric semilattices. Since representable cylindric semilattices have a non-finitely axiomatizable quasi-equational theory, there is no finite axiomatization for n-dimensional unrestricted project-join dependencies. Then we will look at Cosmadakis (in: A.K. Chandra (Ed.), Proc. 28th Annual Symp. on Foundations of Computer Science, IEEE Computer Society Press, Los Angeles, CA, 1987, pp. 411–420) who introduces cylindric dependencies, and makes several claims regarding the structural properties of these dependencies. Using the above-mentioned precise connection, we show that recent algebraic investigations of cylindric lattices provide counterexamples to Cosmadaki's claim that cylindric dependencies are finitely (schema) axiomatizable.