Affiliations: Department of Mathematics, Liaocheng University, Liaocheng, P.R. China
Correspondence:
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Corresponding author. Lingqiang Li, Department of Mathematics, Liaocheng University, Liaocheng 252059, P.R. China. E-mail: lilingqiang0614@126.com.
Abstract: The generalized neighborhood system-based rough set is an important extension of Pawlak’s rough set. The rough sets based on generalized neighborhood systems include two basic models: optimistic and pessimistic rough sets. In this paper, we give a further study on pessimistic rough sets. At first, to regain some properties of Pawlak’s rough sets that are lost in pessimistic rough sets, we introduce the mediate, transitive, positive (negative) alliance conditions for generalized neighborhood systems. At second, some approximation operators generated by special generalized neighborhood systems are characterized, which include serial, reflexive, symmetric, mediate, transitive, and negative alliance generalized neighborhood systems and their combinations (e.g. reflexive and transitive). At third, we discuss the topologies generated by the upper and lower approximation operators of the pessimistic rough sets. Finally, combining practical examples, we apply pessimistic rough sets to rule extraction of incomplete information systems. Particularly, we prove that different decision rules can be obtained when different neighborhood systems are chosen. This enables decision makers to choose decisions based on personal preferences.
Keywords: Rough set, multi-granulation rough
sets, neighborhood system, approximation operator, axiomatic
characterizations, information system
