In the present paper, given an arbitrary set $\Omega$, we study the main order and algebraic properties of some maps and set structures that are strictly related to {\em dependence set relations} on $\Omega$, which are binary relations between subsets of $\Omega$ naturally arising when $\Omega$ 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 {\em granular maps}, have the families of the set systems, set operators, binary set relations or also of information systems on the ground set $\Omega$ 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. More in detail, we introduce the notion of {\em granular sub-bijection} to formalize in all these situations the undefined notion of {\em 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 $\Omega$. Next, through a property of adjunctivity, we see that in order to {\em generate} a dependence set relation it suffices to consider {\em pointed relations} on $\Omega$, namely collections of pairs in $\Omega \times \pset (\Omega)$. Because of that, we study order-theoretical properties of some relevant subclasses of pointed relations and analyze the granular maps on $\Omega$ which determine two non-trivial 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 $\textnormal{Dep}\mfp$, that is a dependence set relation induced by an information system $\mfP$ on $\Omega$ 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 $\mcD$ on $\Omega$, we consider the smallest dependence set relation $\mcD^+$ on $\Omega$ containing $\mcD$ and call it {\em dependence closure} of $\mcD$. Then, when $\Omega$ is a finite set, we show how to generate $\mcD^+$ in four different and recursive ways by starting from $\mcD$. Moreover, again in the finite case, given an information system $\mfP$ on $\Omega$, we also determine a binary set relation $\mcL\mfp$ on $\Omega$ for which $\mcL\mfp^+$ agrees with $\textnormal{Dep}\mfp$ and whose cardinality is minimum with respect to that of all binary set relations whose dependence closure agrees with $\textnormal{Dep}\mfp$.
Algebraic and Order Properties of Maps and Structures related to Dependence Relations Arising in Topology, Algebra and Rough Set Theory
Chiaselotti, G.
;Infusino, F. G.
In corso di stampa
Abstract
In the present paper, given an arbitrary set $\Omega$, we study the main order and algebraic properties of some maps and set structures that are strictly related to {\em dependence set relations} on $\Omega$, which are binary relations between subsets of $\Omega$ naturally arising when $\Omega$ 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 {\em granular maps}, have the families of the set systems, set operators, binary set relations or also of information systems on the ground set $\Omega$ 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. More in detail, we introduce the notion of {\em granular sub-bijection} to formalize in all these situations the undefined notion of {\em 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 $\Omega$. Next, through a property of adjunctivity, we see that in order to {\em generate} a dependence set relation it suffices to consider {\em pointed relations} on $\Omega$, namely collections of pairs in $\Omega \times \pset (\Omega)$. Because of that, we study order-theoretical properties of some relevant subclasses of pointed relations and analyze the granular maps on $\Omega$ which determine two non-trivial 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 $\textnormal{Dep}\mfp$, that is a dependence set relation induced by an information system $\mfP$ on $\Omega$ 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 $\mcD$ on $\Omega$, we consider the smallest dependence set relation $\mcD^+$ on $\Omega$ containing $\mcD$ and call it {\em dependence closure} of $\mcD$. Then, when $\Omega$ is a finite set, we show how to generate $\mcD^+$ in four different and recursive ways by starting from $\mcD$. Moreover, again in the finite case, given an information system $\mfP$ on $\Omega$, we also determine a binary set relation $\mcL\mfp$ on $\Omega$ for which $\mcL\mfp^+$ agrees with $\textnormal{Dep}\mfp$ and whose cardinality is minimum with respect to that of all binary set relations whose dependence closure agrees with $\textnormal{Dep}\mfp$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
