Algebraic and order properties of maps and structures related to dependence relations arising in topology, algebra and rough set theory

Abstract

In this paper, given an arbitrary set [Formula: see text], we study the main order and algebraic properties of some maps and set structures that are strictly related to dependence set relations on [Formula: see text], which are binary relations between subsets of [Formula: see text] naturally arising when [Formula: see text] is a topological space or an attribute set in rough set theory and granular computing based on information systems. The previous maps, that we call granular maps, have the families of the set systems, set operators, binary set relations or also of information systems on the ground set [Formula: see text] as their domain and codomain. We make use of various algebraic methodologies on granular maps to determine the main order-theoretic and combinatorial properties of specific sub-collections of set systems, binary set relations and set operators naturally arising in the investigation of dependence set relations and of rough set theory. We introduce, in more detail, the notion of granular sub-bijection to formalize in all these situations the undefined notion of cryptomorphism, and through which we exhibit new equivalences between specific families of set systems, binary set relations and set operators strictly related to dependence set relations. By means of suitable granular maps we determine three granular sub-bijections between the family of all the closure operators, that of all the Moore set systems and that of all dependence set relations on the same ground set [Formula: see text]. Next, through a property of adjunctivity, we see that in order to generate a dependence set relation it suffices to consider pointed relations on [Formula: see text], namely collections of pairs in [Formula: see text] ℘([Formula: see text]). Because of that, we study order-theoretical properties of some relevant subclasses of pointed relations and analyze the granular maps on [Formula: see text] which determine two nontrivial granular sub-bijections between two subclasses of set operators and two corresponding subclasses of pointed relations. Next, we show that any dependence set relation has the form [Formula: see text] that is a dependence set relation induced by an information system [Formula: see text] on [Formula: see text] and generalizes the Pawlak dependence set relation frequently used in rough set theory. With regard to this representation result, we characterize some set systems of minimal subsets with respect to the Pawlak indiscernibility relation on information systems. Finally, given an arbitrary binary set relation [Formula: see text] on [Formula: see text], we consider the smallest dependence set relation [Formula: see text] on [Formula: see text] containing [Formula: see text] and call it dependence closure of [Formula: see text]. Then, when [Formula: see text] is a finite set, we show how to generate [Formula: see text] in four different and recursive ways by starting from [Formula: see text]. Moreover, again in the finite case, given an information system [Formula: see text] on [Formula: see text], we also determine a binary set relation [Formula: see text] on [Formula: see text] for which [Formula: see text] agrees with [Formula: see text] and whose cardinality is minimum with respect to that of all binary set relations whose dependence closure agrees with [Formula: see text].