A Generalisation of Stone Duality to Orthomodular Lattices

Abstract

With each orthomodular lattice L we associate a spectral presheaf generalising the Stone space of a Boolean algebra, and show that (a) the assignment is contravariantly functorial, (b) is a complete invariant of L, and (c) for complete orthomodular lattices there is a generalisation of Stone representation in the sense that L is mapped into the clopen subobjects of the spectral presheaf The clopen subobjects form a complete bi-Heyting algebra, and by taking suitable equivalence classes of clopen subobjects, one can regain a complete orthomodular lattice isomorphic to L. We interpret our results in the light of quantum logic and in the light of the topos approach to quantum theory.