Summary: 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 \(\mathcal{S} = \langle U_{\mathcal{S}}, \Omega_{\mathcal{S}}, C_{\mathcal{S}},\)\(D_{\mathcal{S}}, F_{\mathcal{S}}, \Lambda_{\mathcal{S}} \rangle\), where \(U_{\mathcal{S}}\), \(C_{\mathcal{S}}\), \(\Lambda_{\mathcal{S}}\) are non-empty sets whose elements are called, respectively, objects, condition attributes, values, \(D_{\mathcal{S}}\) is a (possibly empty) set whose elements are called decision attributes, \(\Omega_{\mathcal{S}} : = C_{\mathcal{S}} \cup D_{\mathcal{S}}\) and \(F_{\mathcal{S}} : U_{\mathcal{S}} \times(C_{\mathcal{S}} \cup D_{\mathcal{S}}) \rightarrow \Lambda_{\mathcal{S}}\) is a map.
The basic tool of our analysis is the equivalence relation \(\equiv_A\) on \(U_{\mathcal{S}}\), depending on the choice of a condition attribute subset \(A \subseteq C_{\mathcal{S}} \cup D_{\mathcal{S}}\) and defined as follows: \[u \equiv_A u^\prime : \Leftrightarrow F_{\mathcal{S}}(u, a) = F_{\mathcal{S}}(u^\prime, a) \quad \forall a \in A .\] We denote by \([u]_A\) the equivalence class of \(u \in U_{\mathcal{S}}\) with respect to \(\equiv_A\). 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 \(\mathcal{S}\). More specifically, we fix \(W \subseteq U_{\mathcal{S}}\), \(A \subseteq C_{\mathcal{S}}\) and try to determine in what cases all objects \(u \in W\) satisfy the condition \([u]_A \cap W \subseteq [u]_{D_{\mathcal{S}}} \cap W\). 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 \(C_{\mathcal{S}}\). This paper provides a detailed study of these set operators, of the induced set systems and of the most relevant links between them.