Abstract
It is noted that the general relationship between various quantum structures and fuzzy sets is as close as the general relationship between Boolean algebras, which are specific examples of quantum structures, and traditional crisp sets, which are specific examples of fuzzy sets. The representation of quantum structures by families of fuzzy sets allows placing the quantum probability calculus on an equal footing with the classical probability calculus. The only difference between them is the fact that in the latter probability measures are defined on σ-Boolean algebras of crisp sets while in the former they are defined on specific families of fuzzy sets, which are quantum structures. This chapter describes relations between abstract quantum logics and other more general abstract quantum structures and some special families of fuzzy sets endowed with partially defined Lukasiewicz's union and intersection. In specific, Lukasiewicz's union and intersection of fuzzy sets are within the uncountable family of possible union-like and intersection-like operations on fuzzy sets, the only operations that can be used to construct quantum structures of fuzzy sets is of interest to physicists.
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