Decision systems in rough set theory: A set operatorial perspective

Abstract

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 [Formula: see text] [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] are non-empty sets whose elements are called, respectively, objects, condition attributes, values, [Formula: see text] is a (possibly empty) set whose elements are called decision attributes, [Formula: see text] and [Formula: see text] is a map.The basic tool of our analysis is the equivalence relation [Formula: see text] on [Formula: see text], depending on the choice of a condition attribute subset [Formula: see text] and defined as follows: [Formula: see text] We denote by [Formula: see text] the equivalence class of [Formula: see text] with respect to [Formula: see text]. 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 [Formula: see text]. More specifically, we fix [Formula: see text], [Formula: see text] and try to determine in what cases all objects [Formula: see text] satisfy the condition [Formula: see text]. 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 [Formula: see text]. This paper provides a detailed study of these set operators, of the induced set systems and of the most relevant links between them.