Representation of fuzzy quantum posets of types I,II

Let (,M) be a fuzzy quantum poset of type I, II, or FQP of type I, II for short. For Boolean representations of fuzzy quantum spaces, by a representation of (,M) we mean a quantum logic (i.e., an orthocomplemented-orthocomplete orthomodular poset with a homomorphismh: such that for any states onM and any observable ¯X on there is a state ¯s on and observableX onM such that the following diagram commutes where () is a Borel-algebra of the real line ]: We prove that a representation of FQP of type I always exists and a representation of FQP of type II exists in some cases.