On the complexity of equational decision problems for finite height(ortho)complemented modular lattices

Abstract

We study the computational complexity of satisfiability problems for classes of simple finite height (ortho)complemented modular lattices L. For single finite L, these problems are shown to be N P-complete; for L of height at least 3, equivalent to a feasibility problem for the division ring associated with L. Moreover, it is shown that, for the class of finite dimensional Hilbert spaces, the equational theory of the class of subspace ortholattices, as well as that of the class of endomorphism * -rings with pseudo-inversion, is complete for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This result extends to the additive category of finite dimensional Hilbert spaces, enriched by adjunction and pseudoinversion.