Orthomodular lattice in Lorentzian globally hyperbolic space-time

An orthomodular lattice without covering law is considered in globally hyperbolic space-time where orthogonality is generated by the chronological relation. In this lattice, the least upper bound and orthocomplementation cannot be interpreted as the disjunction and negation of classical logic. Two-dimensional pictures are presented, demonstrating nonclassical character of the lattice. M3-N5 theorem is used to consider nonmodularity. A comparison of causal logic and quantum logic is discussed.