Abstract:The investigation of orthocomplemented lattices with a symmetric difference initiated the following question: Which orthomodular lattice can be embedded in an orthomodular lattice that allows for a symmetric difference\,? In this paper we present a necessary condition for such an embedding to exist. The condition is expressed in terms of $Z_2$-valued states and enables one, as a consequence, to clarify the situation in the important case of the lattice of projections in a Hilbert space.
Keywords: orthomodular lattice, quantum logic, symmetric difference, Boolean algebra, group-valued state
AMS Subject Classification: 06A15 03G12 28E99 81P10