On Some Topological Properties of Pessimistic Multigranular Rough Sets

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1 I.J. Intelligent Systems Alications, 2012,, ublished Online July 2012 in MES (htt:// DOI: /ijisa On Some Toological roerties of essimistic Multigranular Rough Sets B.K.Triathy SSE, VIT University, Vellore , Tamil Nadu, INDIA M. Nagaraju SSE, VIT University, Vellore , Tamil Nadu, INDIA Abstract Rough set theory was introduced by awlak as a model to cature imreciseness in data since then it has been established to be a very efficient tool for this urose. The definition of basic rough sets deends uon a single equivalence relation defined on the universe or several equivalence relations taken one each at a time. There have been several extensions to the basic rough sets introduced since then in the literature. From the granular comuting oint of view, research in classical rough set theory is done by taking a single granulation. It has been extended to multigranular rough set (MGRS) model, where the set aroximations are defined by taking multile equivalence relations on the universe simultaneously. Multigranular rough sets are of two tyes; namely otimistic MGRS essimistic MGRS. Toological roerties of rough sets introduced by awlak in terms of their tyes were studied by Triathy Mitra to find the tyes of the union, intersection comlement of such sets. Triathy Raghavan have extended the toological roerties of basic single granular rough sets to the otimistic MGRS context. Incomlete information systems take care of missing values for items in data tables. MGRS has also been extended to such tye of incomlete information systems. In this aer we have carried out the study of toological roerties of essimistic MGRS by finding out the tyes of the union, intersection comlement of such sets. Also, we have rovided roofs examles to illustrate that the multile entries in the table can actually occur in ractice. Our results hold for both comlete incomlete information systems. The multile entries in the tables occur due to imreciseness ambiguity in the information. This is very common in many of the real life situations needed to be addressed to hle such situations in efficient manner. Index Terms Rough Sets, Equivalence Relations, Tolerance Relations, Tye of Rough Sets, Multi Granular Rough Sets I. Introduction The main observations of our traditional tools for formal modeling, reasoning comuting are cris, deterministic reciseness nature, which restrict their alicability in real life situations, led to the extension of the concet of cris sets so as to model imrecise data enhance their modeling ower. One such aroach to cature imreciseness was due to awlak [5, 6], who introduced the notion of Rough Sets, which is an excellent tool to cature imreciseness in data. The basic assumtion of rough set theory is that human knowledge about a universe deends uon their caability to classify its objects. lassifications of a universe equivalence relations defined on it are known to be interchangeable notions. So, for mathematical reasons equivalence relations were considered by awlak to define rough sets. A rough set is reresented by a air of cris sets, called the lower aroximation comrises of elements, which belong to it definitely uer aroximation comrises of elements, which are ossibly in the set with resect to the available information. To imrove the modeling caability of basic rough sets several extensions have been made in different directions. One such extension is the rough sets based uon tolerance relations instead of equivalence relations. These rough sets are sometimes called incomlete rough set models. In the view of granular comuting, classical rough set theory is researched by a single granulation. The basic rough set model has been extended to rough set model based on mu ltigranulations (MGRS) in [10], where the set aroximations are defined by using multi-equivalences on the universe. Using similar concets, that is taking multile tolerance relations instead of multile equivalence relations; incomlete rough set model based on multi-granulations was introduced in [11]. Several fundamental roerties of these tyes of rough sets have been studied [10, 11, 12]. Emloying the notions of lower uer aroximations of rough sets, an interesting

2 On Some Toological roerties of essimistic Multigranular Rough Sets 11 characterization of rough sets has been made by awlak in [6], where he introduced the tyes (originally called kinds) of rough sets. There are two different ways of characterizing rough sets; the accuracy coefficient the toological characterization introduced through the notion of tyes. As mentioned by awlak himself [6], in ractical alications of rough sets we combine both tyes of information about the borderline region, that is, of the accuracy of measure as well as the information about the toological classification of the set under consideration. Keeing this in mind, Triathy Mitra [16] have studied the tyes of rough sets by finding out the tyes of union intersection of rough sets of different tyes. These results were extended to the context of otimistic multi granular rough sets by Triathy et al [17]. In this aer, we study these results for the essimistic multi granular context, which also remains the same for both the comlete incomlete cases. II. Definitions And Notations Let U be a universe of discourse R be an equivalence relation over U. By U/R we denote the family of all equivalence classes of R, referred to as categories or concets of R the equivalence class of an element x U is denoted by [x] R. By a knowledge base, we underst a relational system K ( U, ), where U is as above relations over U. For any subset ( ), the intersection of all equivalence relations in is denoted by IND() is called the indiscernibility relation over. Given any X U R IND (K), we associate two subsets, RX { Y U / R : Y X} RX = { Y U / R : Y X }, called the R-lower R- uer aroximations of X resectively. The R- boundary of X is denoted by BN R (X) is given by BNR ( X ) RX RX. The elements of R X are those elements of U, which can certainly be classified as elements of X, the elements of R X are those elements of U, which can ossibly be classified as elements of X, emloying knowledge of R. We say that X is rough with resect to R if only if RX RX, equivalently BN ( X ). R X is said to be R- definable if only if RX RX BNR ( X )., or In the view of granular comuting (roosed by L. A. Zadeh), an equivalence relation on the universe can be regarded as a granulation, a artition on the universe can be regarded as a granulation sace [2, 3]. For an incomlete information system, similarly, a tolerance relation on the universe can be regard as a granulation, a cover induced by the relation can be regarded as a granulation sace. Several measures in knowledge base closely associated with granular comuting, such as knowledge granulation, granulation measure, information entroy rough entroy, were discussed in [2, 3, 4]. On research of rough set method based on multi-granulations, Y. H. ian J. Y. Liang brought forward a rough set model based on multigranulations [10], which is established by using multi equivalence relations. In [11] an extension of MGRS, rough set model based on multi tolerance relations in incomlete information systems is develoed. We define below the otimistic MGRS. Definition 2.1: Let K= (U, R) be a knowledge base, R be a family of equivalence relations, X U, R. We define the otimistic multi-granular lower aroximation uer aroximation of X in U as (2.1) X { x / [ x] X or [ x] X} (2.2) X ( ( X )) Another kind of multi-granular rough sets called essimistic multi-granular rough sets was introduced by uian et al [13]. Now, they call the above tye of multigranular rough sets as the otimistic mult i-granular rough sets (which was introduced as the multi-granular rough sets [10]). We define below the essimistic multi-granular rough sets (MGRS). Definition 2.2: Let K= (U, R) be a knowledge base, R be a family of equivalence relations, X U, R. We define the essimistic multi-granular lower aroximation uer aroximation of X in U as (2.3) ( ) X { x / [ x ] X [ x ] X } (2.4) ( ) X (( ) ( X )) We state below several roerties of essimistic multi-granular rough sets (MGRS) from [13]. roerty 2.1: Let K= (U, R) be a knowledge base, R be a family of equivalence relations, X U, R. The following roerties hold true. (2.4) ( ) ( X ) X ( ) ( X ) (2.5) ( ) ( ) ( ) ( ), ( ) ( U ) U ( ) ( U ) (2.6) ( ) ( X ) (( ) ( X )) (2.7) ( ) ( X ) X X (2.) ( ) ( X ) X X (2.9) ( ) ( X ) ( ) ( X ), ( ) ( X ) ( ) ( X )

3 12 On Some Toological roerties of essimistic Multigranular Rough Sets roerty 2.2: Let K= (U, R) be a knowledge base, R be a family of equivalence relations, X,Y U, R. The following roerties hold true. (2.10) ( ) ( X Y ) ( ) X ( ) Y (2.11) ( ) ( X Y ) ( ) X ( ) Y (2.12) ( ) ( X Y ) ( ) X ( ) Y (2.13) ( ) ( X Y ) ( ) X ( ) Y Next, we define MGRS in incomlete information systems. Definition 2.3: An information system is a air S = (U, A), where U is a non-emty finite set of objects, A is a non-emty finite set of attributes. For every a A, there is a maing a : U Va, where V a is called the value set of a. If V a contains a null value for at least one attribute a A, then S is called an incomlete information system. Otherwise, it is comlete. Definiti on 2.4: Let S = ( U, A) be an incomlete information system, A an attribute set. We define a binary relation on U as follows (2.14) SIM() = {(u,v) U X U a, a(u) = a(v) or a(u) = * or a(v) = *}. In fact, SIM() is a tolerance relation on U, the concet of a tolerance relation has a wide variety of alications in classifications [1, ]. It can be shown that SIM() = a SIM({A}). Let S (u) denote the set {v (u,v) SIM()}. S (u) is the maximal set of objects which are ossibly indistinguishable by with u. Let U/SIM() denote the family sets {S (u) u U}, the classification or the knowledge induced by. A member S (u) from U/SIM() will be called a tolerance class or an information granule. It should be noticed that the tolerance classes in U/SIM() do not constitute a artition of U in general. They constitute a cover of U, i.e., S (u) for every u U, u U S (u) = U. Definition 2.5: Let S = (U, A) be an incomlete information system,, A two attribute subsets, X U, we define a lower aroximation of x a uer aroximation of x in U by the following (2.15) ( ) SIM (x) X} X = {x SIM (x) X (2.15) ( ) (X) = (( ) ( X c )) c A Multi-granulation Rough Set can be classified into following four tyes: Definition 2.6: Let K= (U, R) be a knowledge base, R be a family of equivalence relations, X U, R. Then (2.16) If ( ) (X) ( ) U, then we say that X is essimistic roughly +-definable. (2.17) If ( ) (X) = ( ) U, then we say that X is essimistic internally +-undefinable. (2.1). If ( ) (X) ( ) = U, then we say that X is essimistic externally + - undefinable. (2.19) If ( ) (X) = ( ) = U, then we say that X is essimistic totally + undefinable. III. Results In this section we shall find out the tyes of essimistic multi granular rough sets (MGRS). There are four sets of results accumulated in four tables. The first rovides the tye of a + rough set from the tyes of its rough set tyes. The second table rovides the tyes of the comlement of a multi granular rough set. In the third table we obtain the tyes for the union of two multi g ranular rough sets of all ossible tyes. Similarly we establish the tyes of the intersection of two multi granular rough sets of all ossible tyes. These results will be useful for further studies in aroximation of classifications rule generation. 3.1 Table for tye of X with resect to ( ) This subsection rovides the tye of a + rough set from the tyes of its rough set tyes in the following table. roofs with examles for some of the entries in the table are then given. Tye of X with resect to Tye of X with resect to T-1 T-2 T-3 T-4 T-1 T-1 T-1 T-1 T-1 T-2 T-1 T-2 T-1 T-2 T-3 T-1 T-1 T-3 T-3 T-4 T-1 T-2 T-3 T-4

4 On Some Toological roerties of essimistic Multigranular Rough Sets Examle to rove entry (1,1) U / SIM ( ) {{ a, a },{ a, a, a, a, a, a },{ a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, Let X { a, a, a, a } X { a, a } X { a, a, a, a, a, a, a } U Thus X is of Tye 1 w. r. t.. X { a, a } X { a, a, a, a, a, a, a } U Thus X is of Tye 1 w. r. t.. ( ) X { a, a } 1 7 ( ) X { a, a, a, a, a, a, a } U Thus X is of Tye 1 w. r. t. ( ) Examle to rove entry (1,3) Let U { a, a, a, a, a, a, a, a }. U / SIM ( ) {{ a, a },{ a, a, a, a, a, a },{ a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, Let X { a, a, a }. 1 7 X { a, a, a } X { a, a, a } U. Thus X is of Tye 1 w. r. t.. X { a, a, a } X U. Thus X is of Tye 3 w. r. t.. ( ) ( X ) { a, a, a } ( ) X { a, a, a } U. Thus X is of Tye 1 w. r. t. ( ) Examle to rove entry of row2 Let U { a, a, a, a, a, a, a, a }. U / SIM ( ) {{ a, a },{ a, a, a, a, a, a },{ a }} Let X { a, a, a } X X { a, a, a, a, a } U Thus X is of Tye 2 w. r. t.. X { a, a, a, a, a }. X { a, a, a } X { a, a, a, a, a, a, a, a } U. c Thus X is of Tye 3 w. r. t. 3.3 Table for tye of X Y with resect to ( ) This subsection rovides the tyes of union of two essimistic multi granular rough sets with resect to + in the following table. roofs with examles for some of the entries in the table are then given. roof of entry (1, 1) Suose X Y are both of Tye-1. Then ( ) X,( ) Y, ( ) X U ( ) Y U. From (2.12) it follows that ( ) ( X Y ). But using (2.11) we see that ( ) ( X Y) has both the ossibilities of being equal or not equal to U. So, X Y can be of Tye 1or of Tye Table for tye of X with resect to ( ) This subsection rovides the tyes of comlement of essimistic multi granulation rough set in the following table. roofs with examles for some of the entries in the table are then given. X X T-1 T-1 T-2 T-3 T-3 T-2 T-4 T-4 Tye of X with resect to ( ) T-1 T-2 Tye of Y with resect to( ) T-1 T-2 T-3 T-4 T-1/ T-3 T-1/T-3 T-3 T-3 T-1/ T-1/T-2/ T-3/ T-3 T-3 T-3/T-4 T-4 T-3 T-3 T-3 T-3 T-3 T-4 T-3 T-3/T-4 T-3 T-3/ T-4

5 14 On Some Toological roerties of essimistic Multigranular Rough Sets Examles to rove entry (1,1) U / SIM ( ) {{ a, a },{ a, a, a, a, a, a }, { a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, ase Let X { a, a } Y { a }. Then XUY { a, a, a }. ( ) X { a, a } ( ) X { a, a } U. Thus X is of Tye ( ) Y { a } ( ) Y { a } U. ThusY is of Tye 1 ( ) ( XUY ) { a, a, a } ( ) ( XUY ) { a, a, a } U. Thus XUY is of Tye 1 ase 2 Let X { a, a, a, a } Y { a, a, a } Then XUY { a, a, a, a, a, a } ( ) X { a, a } 1 7 ( ) X { a, a, a, a, a, a, a } U. Thus X is of Tye ( ) Y { a } ( ) Y { a, a, a, a, a } U. ThusY is of Tye ( ) ( XUY ) { a, a, a } ( ) ( XUY ) { a, a, a, a, a, a, a, a } U. Thus XUY is of Tye 3 roof of entry (1, 3) Let both X Y be of Tye 1 Tye 3. Then from the roerties of tye 1 tye 3 ( ) (X), ( ) (Y), ( ) (X) U ( ) (Y) = U. So, using (2.11) (2.12) we get ( ) (XUY) ( ) (XUY) = U. Hence XUY is of tye 3 only. The other cases can be similarly established Examle to rove entry (1,3) in XUY table U / SIM ( ) {{ a, a },{ a, a, a, a, a, a },{ a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, 1 7 Let X { a, a, a, a } Y { a, a, a, a, a } ( ) X { a, a } ( ) X U. 1 7 Thus X is of Tye 1. ( ) Y { a, a, a } ( ) Y U. Thus Y is of Tye 3. XUY { a, a, a, a, a, a } ( ) ( XUY ) { a, a, a } ( ) ( XUY ) U. Thus XUY is of Tye Examles to rove entry (2,2) in XUY table Let us take the following examles rovide roofs for first two cases. U / SIM ( ) {{ a, a },{ a, a, a, a, a, a }, { a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, ase Let X { a, a, a } Y { a } ( ) X ( ) X { a, a, a, a, a, a, a } U. Thus X is of Tye ( ) Y ( ) Y { a, a, a, a, a, a, a } U. Thus Y is of Tye 2. XUY { a, a, a, a }. ( ) ( XUY ) { a, a } ( ) ( XUY ) { a, a, a, a, a, a } U Thus XUY is of Tye 1.

6 On Some Toological roerties of essimistic Multigranular Rough Sets 15 ase 2 Let X { a, a, a, a } Y { a, a, a } ( ) X ( ) X { a, a, a, a, a } U. Thus X is of Tye ( ) Y ( ) Y { a, a, a, a, a } U. Thus Y is of Tye 2. XUY { a, a, a, a } ( ) ( XUY ) ( ) ( XUY ) { a, a, a, a, a } U. Thus XUY is of Tye Then let us take the following examles rovide roofs for the next two cases. Let U { a, a, a, a a, a, a, a } , U / SIM ( ) {{ a, a },{ a, a, a, a },{ a, a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, ase { a, a, a },{ a, a }} Let X { a, a, a } Y { a, a } ( ) X ( ) X { a, a, a, a } U. Thus X is of Tye ( ) Y ( ) Y { a, a, a, a, a } U. Thus Y is of Tye XUY { a, a, a, a, a }.( ) ( XUY ) { a, a } ( ) ( XUY ) U. Thus XUY is of Tye 3. ase 4 Let X { a, a, a } Y { a, a } ( ) X ( ) X U. Thus X is of Tye 2. ( ) Y ( ) Y U. Thus Y is of Tye 2. XUY { a, a, a, a, a } ( ) ( XUY ) ( ) ( XUY ) U. Thus XUY is of Tye Table for tye of X Y with resect to ( ) This subsection rovides the tyes of intersection of two essimistic multi granular rough sets with resect to + in the following table. roofs with examles for some of the entries in the table are then given. Tye of X with resect to ( ) roof of entry (1, 3) Tye of Y with resect to ( T-1 T-2 T-3 T-4 T-1 T-1/T-2 T-2 T-1/T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-3 T-1/T-2 T-2 T-1/T-2/ T-2 T-3/T-4 /T-4 T-4 T-2 T-2 T-2/T-4 T-2 /T-4 Suose X is of Tye-1 Y is of Tye-3. Then ( ) X,( ) Y, ( ) X U ( ) Y U. From (2.13) it follows that ( ) ( X Y ) U. But using (2.10) we see that ( ) ( X Y) has both the ossibilities of being or not being equal to. So X Y can be of Tye 1orTye Examles to rove entry (1,3) U / SIM ( ) {{ a, a },{ a, a, a, a, a, a }, { a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, ase 1 ) 1 7 Let X { a, a a, a, a } Y { a, a a, a, a, a, a }. 2 3, , Then X Y { a, a a, a, a }. 2 3, 4 5 6

7 16 On Some Toological roerties of essimistic Multigranular Rough Sets ( ) X ( ) X { a, a a, a, a } U. Thus X is of Tye , ( ) Y ( ) Y { a, a, a a, a, a, a, a } U , ThusY is of Tye 3. ( ) ( X Y ) ( ) ( X Y ) U. Thus X Y is of Tye 1. ase 2 Let X { a a, a, a } Y { a, a, a a, a, a, a }. 3, , Then X Y { a a, a }. 3, 4 5 ( ) X { a } ( ) X { a, a a, a, a, a } U. Thus X is of Tye , ( ) Y { a } ( ) Y { a, a, a a, a, a, a, a } U , ThusY is of Tye 3. ( ) ( X Y ) ( ) ( X Y ) U. Thus X Y is of Tye 2. Let X Y be of Tye 2 Tye 1 resectively. Then from the roerties of tye 2 tye 1 multi granular rough sets we get ( (X) =, ( ) U. (Y) =, ( ) ) (X) U ( ) (Y) So using roerties (2.10) (2.13) we get ( ) (X Y) = ( ) (X Y) U. So, X Y is of tye 2. This comletes the roof. The other cases can be established similarly Examle to rove entry (2,1) U / SIM ( ) {{ a, a },{ a, a, a, a, a, a }, { a }} U / SIM ( ) {{ a, a },{ a, a, a, a, a, a, a, a }, 1 7 roof of entry (2, 1) Let X { a a, a, a } Y { a a, a }. 3, 4 5 3, 4 5 Then X Y { a a, a }. 3, 4 5 ( ) X { a } ( ) X { a, a a, a, a, a } U. Thus X is of Tye , ( ) Y ( ) Y { a, a a, a, a } U. 2 3, ThusY is of Tye 2. ( ) ( X Y ) ( ) ( X Y ) { a 2, a 3, a 4, a 5, a 6 } U. Thus X Y is of Tye 2 IV. onclusion Two tyes of multi-granular rough sets have been introduced in the literature ([10], [13]). Toological roerties of otimistic multigranular rough sets were studied by Triathy et al ([17]). In this aer we studied the toological roerties of essimistic multi granular rough sets with resect to the three set theoretic oerations of union, intersection comlementation. The tables show that there are multile answers to some of the cases as like as the case of basic rough sets. These multile answers secify imreciseness ambiguity in information that is available with the user to classify object of a universe. Thus the models discussed in this aer are rightly suitable for hling imreciseness in data in more effective elegant manner using both tyes of multi-granular rough sets. Also, we rovided examles in some cases to illustrate the fact that the multile answers can actually occur. These results can be used in aroximation of classifications rule induction. Also our results hold true for both comlete incomlete essimistic multigranulation systems References [1] Kryszkiewicz, K.: Rough set aroach to incomlete information systems, Information Sciences, vol.112, (199),

8 On Some Toological roerties of essimistic Multigranular Rough Sets 17 [2] Liang, J.Y Shi, Z.Z.: The information entroy, rough entroy knowledge granulation in rough set theory, International Journal of Uncertainty, Fuzziness Knowledge-Based Systems, vol.12(1),(2001), [3] Liang, J.Y, Shi, Z.Z., Li, D. Y. Wierman, M. J.: The information entroy, rough entroy knowledge granulation in incomlete information system, International Journal of general systems, vol.35(6), (2006), [4] Liang, J.Y Li, D. Y.: Uncertainty Knowledge acquisition in Information Systems, Science ress, Beijing, hina, (2005). [5] awlak, Z., Rough sets, Int. jour. of omuter Information Sciences,11, (192), [6] awlak, Z.: Rough sets: Theoretical asects of reasoning about data, Kluwer academic ublishers (London), (1991). [7] awlak, Z. Skowron, A., Rudiments of rough sets, Information Sciences-An International Journal, Elsevier ublications, 177(1), (2007), [] awlak, Z. Skowron, A., Rough sets: Some extensions, Information Sciences-An International Journal, Elsevier ublications, 177(1), (2007), [9] awlak, Z. Skowron, A., Rough sets Boolean reasoning,. Information Sciences-An International Journal, Elsevier ublications, 177(1), (2007), [10] ian, Y.H Liang, J.Y.: Rough set method based on Multi-granulations, roceedings of the 5 th IEEE onference on ognitive Informatics, vol.1, (2006), [11] ian, Y.H, Liang, J.Y. Dang,.Y.: MGRS in Incomlete Information Systems, IEEE onference on Granular omuting,(2007), [12] ian, Y.H, Liang, J.Y. Dang,.Y.: Incomlete Multigranulation Rough set, IEEE Transactions on Systems, Man ybernetics - art A: Systems Humans, Vol.40, No.2, March 2010, [13] ian, Y.H., Liang, J.Y Dang,.Y.: essimistic rough decision, roceedings of RST 2010, Zhoushan, hina, (2010), [14] Triathy, B.K.: On Aroximation of classifications, rough equalities rough equivalences, Studies in omutational Intelligence, vol.174, Rough Set Theory: A True Lmark in Data Analysis, Sringer Verlag, (2009), [15] Triathy, B.K.: Rough Sets on Fuzzy Aroximation Saces Intuitionistic Fuzzy Aroximation Saces, Studies in omutational Intelligence, vol.174, Rough Set Theory: A True Lmark in Data Analysis, Sringer Verlag, (2009), [16] Triathy, B.K. Mitra, A.: Toological roerties of Rough Sets their Alications, International Journal of Granular omuting, Rough Sets Intelligent Systems (IJGRSIS), (Switzerl),vol.1, no.4, (2010), [17] Triathy, B.K. Raghavan, R.: On Some Toological roerties of Multigranular Rough Sets, Journal of Advances in Alied science Research, Vol.2, no.3, (2011), Dr. B.K Triathy is a senior rofessor in the school of comuting sciences engineering, VIT University, at Vellore, India, has ublished more than 150 technical aers in international journals/ roceedings of international conferences/ edited book chaters of reuted ublications like Sringer guided 12 students for hd. so far. He is having more than 30 years of teaching exerience. He is a member of international rofessional associations like IEEE, AM, IRSS, SI, IMS, OITS, OMS, IASIT, IST, AEEE is a reviewer of around 20 international journals which include IEEE, World Scientific, Sringer Science Direct ublications. Also, he is in the editorial board of at least 10 international journals. His current research interest includes Fuzzy sets systems, Rough sets knowledge engineering, Granular comuting, soft comuting, bag theory, data clustering, database anonymisation, list theory social network analysis. M. Nagaraju is a assistant rofessor (senior) in the school of comuting sciences engineering, VIT University, Vellore, India. He is ursuing his h.d. Degree in comuter science in the same school. He is a life member in rofessional associations like ISTE SI. His current research interest includes Fuzzy sets systems, Rough sets knowledge engineering, Granular comuting, Database systems, Exert Systems, Data Mining Decision Suort Systems. How to cite this aer: B.K.Triathy,M. Nagaraju,"On Some Toological roerties of essimistic Multigranular Rough Sets", International Journal of Intelligent Systems Alications(IJISA), vol.4, no.,.10-17, DOI: /ijisa