The logic of causally closed spacetime subsets

The causal structure of spacetime offers a natural notion of an opposite or orthogonal in the logical sense, where the opposite of a set is formed by all points non-timelike related with it. We show that for a general spacetime the algebra of subsets that arises from this negation operation is a complete orthomodular lattice, and thus has several of the properties characterizing the algebra of physical propositions in quantum mechanics. We think this lattice could be used to investigate the causal structure in an algebraic context. As a first step in this direction, we show that the causal lattice is, in addition, atomic, find its atoms, and give necessary and sufficient conditions for irreducibility.