On the expressive power of the relational algebra with partially ordered domains

Assuming data domains are partially ordered, we apply Paredaens' and Bancilhon's Theorem 13, 5] to examine the expressiveness of the relational algebra. We rst extend the relational algebra to the partially ordered re-lational algebra (which we call the PORA) by allowing the ordering predicate v to be used in formulae of the selection operator (). We then show that the PORA expresses exactly the set of all possible relations which are invariant under order-preserving automorphism of databases. The extension is justiied by its consistency with the two important extreme cases of unordered and linearly ordered domains. Moreover, we investigate three hierarchies of: (1) computable queries, (2) query languages and (3) partially ordered domains, and show that there is a one-to-one correspondence between them.