Uncorrected Author Proof Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx DOI:10.3233/JIFS-179449 IOS Press 1 The topological properties of intuitionistic fuzzy rough sets 1 2 Zia Bashir a , M.G. Abbas Malik b , Saba Asif a and Tabasam Rashid c,∗ 3 a Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 4 b Universal College of Learning Palmerston North, New Zealand 5 c Department of Mathematics, University of Management and Technology, Lahore, Pakistan 6 Abstract. In this paper, an in depth study is done on topological properties of intuitionistic fuzzy rough sets in light of different conditions like serial, strongly serial, left continuity, transitivity on intuitionistic fuzzy relations, t-norms, implicators by adopting a axiomatic approach with the ingredients of intuitionistic fuzzy logic. Numerous intuitionistic fuzzy topologies based on many different kinds of intuitionistic fuzzy relations are explored. Also, a special class of intuitionistic fuzzy relations known as T -similarity class has been studied algebraically and found interesting lattices to model real life problems for better applications of intuitionistic fuzzy rough sets. 7 8 9 10 11 12 Keywords: Intuitionistic fuzzy rough sets, intuitionistic fuzzy topologies, intuition fuzzy logic, lattices 13 1. Introduction 14 The complexities of real life compel us to design 15 more sophisticated tools to comprehend its vague- 16 ness. In the attempt to model imprecise and unclear 17 scenarios where one cannot come up with concrete 18 answers, Pawlak [17] and Zadeh [27] initiated the 19 fuzzy set theory and the rough set theory, respec- 20 tively. The both theories then explored in various 21 dimensions and have been successfully applied to 22 solve problems in many fields like data mining, game 23 theory, decision analysis, pattern recognition, image 24 encryption, etc. 25 Dubois and Prade [12] unified these theories by 26 proposing rough fuzzy set and fuzzy rough set. This 27 unified approach is more useful to model problems 28 of high complexity level with uncertain data, there- 29 fore a number of researchers developed a rich theory 30 related to these sets by using combinations of fuzzy 31 ∗ Corresponding author. Tabasam Rashid, Department of Math- ematics, University of Management and Technology, Lahore- 54770, Pakistan E-mail: tabasam.rashid@umt.edu.pk. logic, fuzzy relations and approximation operators. 32 Qin and Pei [18] studied the topological properties of 33 fuzzy rough sets, Radzikowska and Kerre [19] did a 34 comparative study on fuzzy rough sets and general- 35 ized the notion with the help of fuzzy implicator and 36 t-norm, Wang and Hu [23] studied granular variable 37 precision fuzzy rough sets based on generalized fuzzy 38 relations, Wu et al. [22] characterized the various 39 classes of fuzzy approximation operators by differ- 40 ent set of axioms and with respect to different fuzzy 41 implicators, Liu and Zhu [14] studied fuzzy rough 42 sets with algebraic point of view, Li and Cui [15] and 43 Wang [24] explored topological characterization of 44 fuzzy rough sets and similarity of fuzzy relations. 45 Atannasov [1–3] come up with the observations 46 that in all real life scenarios, it was not realistic to 47 expect a strict relation between membership and non- 48 membership values and proposed intuitionistic fuzzy 49 set developing a more flexible tool to handle uncer- 50 tainty in more objective way. Topology is classical 51 and important concept in mathematics and also in 52 applications [4, 30]. Burillo and Bustince [5] stud- 53 ied intuitionistic fuzzy relations, Coker [7] introduced 54 ISSN 1064-1246/19/$35.00 © 2019 – IOS Press and the authors. All rights reserved