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

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 tobe $\mc{NP}$-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 the equational theory of the class of subspace ortholattices as well as endomorphism *-rings (with pseudo-inversion) of finite dimensional Hilbert spaces is complete for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This results extends to the category of finite dimensional Hilbert spaces, enriched by pseudo-inversion.