Decision systems in rough set theory: A set operatorial perspective

In rough set theory (RST), the notion of decision table plays a fundamental role. In this paper, we develop a purely mathematical investigation of this notion to show that several basic aspects of RST can be of interest also for mathematicians who work with algebraic and discrete methods. In this abstract perspective, we call decision system a sextuple S = < U-S, Omega(S), C-S, D-S, F-S, A(S)>, where U-S, C-S, A(S) are non-empty sets whose elements are called, respectively, objects, condition attributes, values, D-S is a (possibly empty) set whose elements are called decision attributes, Omega(S) := C-S boolean OR D-S and F-S :U-S x (C-S boolean OR D-S) -> A(S) is a map. The basic tool of our analysis is the equivalence relation (A) on U-S, depending on the choice of a condition attribute subset A subset of C-S boolean OR D-S and defined as follows: u (A) u' :<--> F-S(u,a) = F-S(u', a) for all a is an element of A. We denote by [u]A the equivalence class of u is an element of U-S with respect to (A). We interpret the classical RST notions of consistency and inconsistency for a decision table in an abstract algebraic set operatorial perspective and, in such a context, we introduce and investigate a kind of local consistency in any decision system S. More specifically, we fix W subset of U-S, A subset of C-S and try to determine in what cases all objects u is an element of W satisfy the condition [u](A )boolean AND W subset of [u]D-S boolean AND W. Then, we build a formal general framework whose basic tools are two local consistency set operators and a global closure operator, the condition attribute set C-S. This paper provides a detailed study of these set operators, of the induced set systems and of the most relevant links between them.