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

TL;DR

This paper proves that the problem of embeddability into relational lattices is undecidable, demonstrating the complexity of their algebraic structure and implications for the theory of databases.

Contribution

It establishes the undecidability of embeddability for relational lattices and shows that their quasiequational theory over the pure lattice signature is undecidable.

Findings

Embeddability into relational lattices is undecidable.

The quasiequational theory of relational lattices is undecidable and lacks a finite basis.

A quasiequation holds in all finite relational lattices but fails in some infinite ones.

Abstract

The natural join and the inner union operations combine relations of a database. Tropashko and Spight realized that these two operations are themeet and join operations in a class of lattices, known by now as the relational lattices. They proposed then lattice theory as an algebraic approach to thetheory of databases alternative to the relational algebra. Litak et al. proposed an axiomatization of relational lattices over the signature that extends thepure lattice signature with a constant and argued that the quasiequational theory of relational lattices over this extended signature is undecidable.We prove in this paper that embeddability is undecidable for relational lattices. More precisely, it is undecidable whether a finite subdirectly-irreduciblelattice can be embedded into a relational lattice. Our proof is a reduction from the coverability problem of a multimodal frame by a universal product frameand, indirectly, from the representability problem for relation algebras. As corollaries we obtain the following results: the quasiequational theoryof relational lattices over the pure lattice signature is undecidable and has no finite base; there is a quasiequation over the pure lattice signature which holds in all the finite relational lattices but fails in an infinite relational lattice.