On the consistency problem for modular lattices and related structures

TL;DR

This paper proves that the consistency problem, deciding if equations have non-trivial solutions, is unsolvable for various algebraic structures including modular lattices, rings, and relation algebras, impacting multiple fields.

Contribution

It extends the known unsolvability results to modular lattices and related structures, including subspace lattices and algebraic systems used in databases.

Findings

Unsovlability of the consistency problem for finite modular lattices.

Unsovlability results for rings and relation algebras.

Implications for equations in Grassmann-Cayley algebra and database dependencies.

Abstract

The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and Rabin (1958) it is known unsolvable for (the class of) groups and, recently, by Bridson and Wilton (2015) for finite groups. We derive unsolvability for (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite dimensional vector spaces over a fixed or arbitrary field of characteristic $0$. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann-Cayley algebra and to functional and embedded multivalued dependencies in databases.