Fuzzy quantum spaces and compatibility

A new approach to representation of observables on fuzzy quantum posets

“…A simple example of an observable is a mapping X a , where a is a fixed fuzzy element of M, defined by (1) X a (E)…”

“…It should be noted that a model II of a fuzzy quantum poset with the condition (5) need not be a model I, see the following example. Let X ao > -^6o be two observables on (ft,M) defined via (1). Then it is clear that (ii)X~F(/ 1) ...,/ JV );…”

“…Two models of fuzzy quantum posets were considered by A. Dvurecenskij, F. Chovanec, F. Kopka and L. B. Long in [1,2,3,4,8], where two fuzzy sets a and b are said to be orthogonal, notation aJ_6 iff a + 6 ^ 1, and fuzzy orthogonal, notation a±.pb iff a D b ^ ~. (See also Mesiar [9].)…”

Generalized versions of MV-algebraic central limit theorems

“…However, for systems with infinitely many degrees of freedom (e. g., occurring in quantum statistical mechanics and quantum field theory) the Hilbert space representation is not unique (see [28]). On the other hand, when quantum mechanical events are described only vaguely, a fuzzy approach should be incorporated to the model (see [8]). Therefore besides quantum logics considered in [9,18,29,31] as orthomodular posets, fuzzy models of quantum mechanics were studied in many papers.…”

“…Riečan in [22] used this fuzzy measure for building the theory of F-quantum spaces. The mentioned theory was further developed by many authors (e. g., Chovanec, Dvurečenskij, and Mesiar in [8,11]). In the fuzzy quantum logic model the Zadeh connectives, used in F-quantum spaces, were replaced by their Lukasiewicz counterparts (see [20,21]).…”