Orthomodular lattices with rich state spaces

Abstract

We introduce a new construction technique for orthomodular lattices. In contrast to the preceding constructions, it admits rich spaces of states (= probability measures), i.e., for each pair of incomparable elements a,c there is a state s such that s(a) = 1 > s(c). This allowed a progress in many questions that were open for a long time; among others we prove that there is a continuum of varieties of orthomodular lattices with rich state spaces and solve a problem formulated by R. Mayet in 1985. As a by-product of this research, the uniqueness problem for bounded observables (posed by S. Gudder in 1966) has been solved. As a tool, we introduce also a new construction –identification of atoms in an orthomodular lattice – which may be of separate interest.