PROPERTIES OF QUANTUM LOGIC MAPS AS FUZZY RELATIONS ON A SET OF ALL SYMMETRIC AND IDEMPOTENT BINARY MATRICES

A quantum logic is one of possible mathematical models for non-compatible random events. In this work we solve a problem proposed at the conference FSTA 2006. Namely, it is proved that s-maps are symmetric fuzzy relations on a set of all symmetric and idempotent binary matrices. Consequently s-maps are not antisymmetric fuzzy relations. This paper also explores other properties of s-maps, j-maps and d-maps. Specifically, it is proved that s-maps are neither reflexive, nor irreflexive, and nor transitive, j-maps have the same properties as s-maps and d-maps are reflexive and not transitive.