Approximation of sets based on partial covering
The paper presents a generalization of the rough set theory. The starting point is an arbitrary nonempty family \(\mathfrak{B}\) of subsets of an arbitrary nonempty universe of discourse \(U\); its elements are called \(\mathfrak{B}\)-sets. In analogy to the definition of the \(\sigma\)-algebra \(\sigma(U/\varepsilon)\), \(\mathfrak{D}_{\mathfrak{B}}\) represents the extension of \(\mathfrak{B}\) with the empty set and all the unions of some \(\mathfrak{B}\)-sets. The notions of lower and upper approximations are straightforward point-free generalizations of Pawlak's same approximations imitating the \(\varepsilon\)-elementary set based formulations; both of them belong to \(\mathfrak{D}_{\mathfrak{B}}\). The entire discussion is within an overall approximation framework whose scope ranges from weak approximation pairs of maps on \(U\) to the notion of a Galois connection on \(2^U\). The common features of the rough set theory and the approach from this paper are treated uniformly, and most notions of Pawlak's rough set theory constitute compound ones and they are split into two or more parts; this framework helps to understand the state of their compound nature and to specify their constituents in a more general context. To demonstrate the effectiveness of the proposed approach a real life biological example is examined. It is based on the META program which is a grid-based, landscape-ecology-oriented, satellite-image supported, field vegetation mapping method of Hungarian habitats.
recognition and evaluation of natural vegetation heritage
scientific article; zbMATH DE number 5964202
scientific article