Definability of approximations in reflexive relations

Considering a reflexive relation R on a fixed nonempty set U, four different constructions of lower and upper approximations are described by using the so-called R-successor or/and R-predecessor sets of each object of the set U. The first two of the four constructions of lower and upper approximations are well known, and one pair is presented in this paper for the first time. The lower and upper approximations in each pair are mutually dual, and all the four upper approximations discussed in this paper are extensive and monotonic. If the reflexive relation R is further assumed to be symmetric, the four constructions of lower and upper approximations are induced to the commonly used lower and upper approximations. The primary goal of this paper is to study definability of approximations in reflexive relations via a special kind of neighborhood systems, called total pure reflexive neighborhood systems. It is shown that such neighborhood systems give a unified framework for definability of the four constructions.