Decision Systems in Rough Set Theory. A Set Operatorial Perspective.

In rough set theory (RST), the notion of {\em 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 {\em decision system} a sextuple $\mcS=\langle U\ms, \Omega\ms, C\ms, D\ms, F\ms, \Lambda\ms \rangle$, where $U\ms$, $C\ms$, $\Lambda\ms$ are non-empty sets whose elements are called respectively {\em objects}, {\em condition attributes}, {\em values}, $D\ms$ is a (possibly empty) set whose elements are called {\em decision attributes}, $\Omega\ms:=C\ms \cup D\ms$ and $F\ms:U\ms \times (C\ms \cup D\ms) \to \Lambda\ms$ is a map. The basic tool of our analysis is the equivalence relation $\equiv_A$ on $U\ms$, depending on the choice of a condition attribute subset $A \subseteq C\ms \cup D\ms$ and defined as follows: $$ u \equiv_A u' :\iff \, F\ms(u,a)=F\ms(u',a) \, \, \forall a \in A. $$ We denote by $[u]_A$ the equivalence class of $u\in U\ms$ with respect $\equiv_A$. We interpret the classical RST notions of {\em consistency} and {\em inconsistency} for a decision table in an abstract algebraic set operatorial perspective and, in such a context, we introduce and investigate a kind of {\em local consistency} in any decision system $\mcS$. More specifically, we fix $W \subseteq U\ms$, $A\subseteq C\ms$ and try to determine in what cases all objects $u \in W$ satisfy the condition $[u]_A \cap W \subseteq [u]_{D\ms} \cap W$. 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 $C\ms$. This paper provide a detailed study of these set operators, of the induced set systems and of the most relevant links between them.