Yosida–Hewitt and Lebesgue Decompositions of States on Orthomodular Posets1

Orthomodular posets are usually used as event structures of quantum mechanical systems. The states of the systems are described by probability measures (also called states) on it. It is well known that the family of all states on an orthomodular poset is a convex set, compact with respect to the product topology. This suggests using geometrical results to study its structure. In this line, we deal with the problem of the decomposition of states on orthomodular posets with respect to a given face of the state space. For particular choices of this face, we obtain, e.g., Lebesguetype and Yosida–Hewitt decompositions as special cases. Considering, in particular, the problem of existence and uniqueness of such decompositions, we generalize to this setting numerous results obtained earlier only for orthomodular lattices and orthocomplete orthomodular posets.