Minimal supports in quantum logics and Hilbert space

It is shown that if a fully atomic, complete orthomodular lattice satisfies a minimal support condition (m.s.c.), then it satisfies Piron's axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice L(A,..sigma..) for a realistic entity (A, ..sigma..) in the sense of Foulis-Piron-Randall. This content provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.