Neighborhood operator systems and approximations Wei-Zhi Wu a,b, * , Wen-Xiu Zhang a a Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China b Information College, Zhejiang Ocean University, Zhoushan, Zhejiang 316004, People’s Republic of China Received 8 October 2000; received in revised form 30 July 2001; accepted 10 November 2001 Abstract This paper presents a framework for the study of generalizing the standard notion of equivalence relation in rough set approximation space with various categories of k-step neighborhood systems. Based on a binary relation on a finite universe, six families of binary relations are obtained, and the corresponding six classes of k-step neighborhood systems are derived. Extensions of Pawlak’s rough set approximation operators based on such neighborhood systems are proposed. Properties of neighborhood operator systems and rough set approximation operators are investigated, and their connections are examined. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Approximation operator; Binary relation; Neighborhood system; Rough set 1. Introduction The theory of rough set, proposed by Pawlak [11], is an extension of the classical set theory. This theory is motivated by practical needs in classification, data mining, and concept formation with insufficient and incomplete infor- mation [1–3,11,12,16,24] and has been found very successful applications in the Information Sciences 144 (2002) 201–217 www.elsevier.com/locate/ins * Corresponding author. E-mail addresses: wuwz8681@sina.com (W.-Z. Wu), wxzhang@xjtu.edu.cn (W.-X. Zhang). 0020-0255/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0020-0255(02)00180-9