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The natural join and the inner union operations combine relations of adatabase. Tropashko and Spight realized that these two operations are themeetand join operations in a class of lattices, known by now as the relationallattices. They proposed then lattice theory as an algebraic approach tothetheory of databases alternative to the relational algebra. Litak et al.proposed an axiomatization of relational lattices over the signature thatextends thepure lattice signature with a constant and argued that thequasiequational theory of relational lattices over this extended signature isundecidable.We prove in this paper that embeddability is undecidable forrelational lattices. More precisely, it is undecidable whether a finitesubdirectly-irreduciblelattice can be embedded into a relational lattice. Ourproof is a reduction from the coverability problem of a multimodal frame by auniversal product frameand, indirectly, from the representability problem forrelation algebras. As corollaries we obtain the following results: thequasiequational theoryof relational lattices over the pure lattice signature isundecidable and has no finite base; there is a quasiequation over the purelattice signature which holds in all the finite relational lattices but failsin an infinite relational lattice.

The quasiequational theory of relational lattices, in the pure lattice signature (embeddability into relational lattices is undecidable)