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 dmaps are reflexive and not transitive.