Yosida-Hewitt and Lebesgue decompositions of states on orthomodular posets

Orthomodular posets are usually used as event structures of quantum mechani cal systems. The states of the systems are described by probability measure s (also called states) on it. It is well known that the family of all state s on an orthomodular poset is a convex set, compact with respect to the pro duct topology. This suggests using geometrical results to study its structu re. In this line, we deal with the problem of the decomposition of states o n orthomodular posets with respect to a given face of the state space. For particular choices of this face, we obtain, e.g, Lebesgue-type and Yosida-H ewitt decompositions as special cases. Considering, in particular, the prob lem of existence and uniqueness of such decompositions, we generalize to th is setting numerous results obtained earlier only for orthomodular lattices and orthocomplete orthomodular posets. (C) 2001 Academic Press.