The characterizations of upper approximation operators based on special coverings (Q521568)

The paper under review is about some characterizations of upper approximation operators based on special coverings. Let us recall some basic definitions of this subject: \(U\) is the universe of discourse and \(P(U)\) denotes the family of all subsets of \(U\). If \(C\) is a family of subsets of \(U\), none of the sets in \(C\) is empty, and \(\bigcup C=U\), then \(C\) is called a covering of \(U\). Definition: (i) A mapping \(n:U\rightarrow P(U)\) is called a neighborhood operator. (ii) A neighborhood system of an object \(x\in U\), denoted by \(NS(x)\), is a non-empty family of neighborhoods of \(x\). The set \(\{NS(x):x\in U\}\) is called a neighborhood system of \(U\), and it is denoted by \(NS(U)\). \(NS(U)\) is said to be serial, if for any \(x\in U\) and \(n(x)\in NS(x),n(x)\) is a non-empty space. \(NS(U)\) is said to be reflexive, if for any \(x\in U\) and \(n(x)\in NS(x)\), \(x\in n(x)\). \(NS(U)\) is said to be symmetric, if for any \(x,y\in U\) and \(n(x)\in NS(x),\) \( n(y)\in NS(y),x\in n(y)\) then \(y\in n(x).\) \(NS(U)\) is said to be transitive, if for any \(x,y,z\in U\) and \(n(y)\in NS(y) \) and \(n(z)\in NS(z),x\in n(y)\) and \(y\in n(z)\) then \(x\in n(z).\) Definition: (Covering approximation space) If \(U\) is a universe and \(C\) is a covering of \(U\), then we call \(U\) together with the covering \(C\) a covering approximation space, denoted by \((U,C)\). Definition: Let \(NS(U)\) be a neighborhood system of \(U.\) The lower and upper operators of \(X\) are defined as follows: \[ \underline{apr}_{NS}(X):=\left\{ x\in U:\exists n(x)\in NS(x)\text{, } n(x)\subseteq X\right\} ; \] \[ \overline{apr}_{NS}(X):=\left\{ x\in U:\forall n(x)\in NS(x)\text{, } n(x)\cap X\neq \varnothing \right\} . \] This study contains 6 sections and a good deal of information is given in the introduction about the subject. The main ideas of generalized rough sets and covering approximations are recalled in the second section. In Section 3, the properties of \(NS(U)\) are given with some examples. The authors study in Section 4 the characterization of \(NS(U)\) for \(\underline{apr}_{NS}\) being a closure operator, while they consider in Section 5 the properties of \(S\) and \(\overline{apr}_{S}\) and they obtain general topological characterizations of a special covering \(S\) for the covering-based upper approximation operator \(\overline{apr}_{S}\) to be a closure operator. Finally, Section 6 concludes the paper.