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The natural join and the inner union operations combine relations of adatabase. Tropashko and Spight [24] realized that these two operations are themeet and join operations in a class of lattices, known by now as the relationallattices. They proposed then lattice theory as an algebraic approach to thetheory of databases, alternative to the relational algebra. Previous works [17,22] proved that the quasiequational theory of these lattices-that is, the setof definite Horn sentences valid in all the relational lattices-is undecidable,even when the signature is restricted to the pure lattice signature. We provehere that the equational theory of relational lattices is decidable. That, iswe provide an algorithm to decide if two lattice theoretic terms t, s are madeequal under all intepretations in some relational lattice. We achieve this goalby showing that if an inclusion t $\le$ s fails in any of these lattices, thenit fails in a relational lattice whose size is bound by a triple exponentialfunction of the sizes of t and s.

The equational theory of the natural join and inner union is decidable