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The natural join and the inner union combine in different ways tables of arelational database. Tropashko [18] observed that these two operations are themeet and join in a class of lattices-called the relational lattices- andproposed lattice theory as an alternative algebraic approach to databases.Aiming at query optimization, Litak et al. [12] initiated the study of theequational theory of these lattices. We carry on with this project, making useof the duality theory developed in [16]. The contributions of this paper are asfollows. Let A be a set of column's names and D be a set of cell values; wecharacterize the dual space of the relational lattice R(D, A) by means of ageneralized ultrametric space, whose elements are the functions from A to D,with the P (A)-valued distance being the Hamming one but lifted to subsets ofA. We use the dual space to present an equational axiomatization of theselattices that reflects the combinatorial properties of these generalizedultrametric spaces: symmetry and pairwise completeness. Finally, we argue thatthese equations correspond to combinatorial properties of the dual spaces oflattices, in a technical sense analogous of correspondence theory in modallogic. In particular, this leads to an exact characterization of the finitelattices satisfying these equations.

Relational lattices via duality