On the consistency problem for modular lattices and related structures

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. For the variety of all groups, this is the complement of the triviality problem, shown undecidable by by Adyan [Algorithmic unsolvability of problems of recognition of certain properties of groups. (Russian) Dokl. Akad. Nauk SSSR (N. S.) 103 (1955) 533-535] and Rabin [Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958) 172-194]. For the class of finite groups, it amounts to the triviality problem for profinite completions, shown undecidable by Bridson and Wilton [The triviality problem for profinite completions, Invent. Math. 202 (2015) 839-874]. We derive unsolvability of the consistency problem for the class of (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 and expansions thereof, e.g. the class of subspace ortholattices of finite-dimensional Hilbert spaces. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings with unit 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.