Decomposition of Continuity and Separation Axioms Via Lower and Upper Approximation

In this paper we study the rough set theory by defined the concepts of rough regularity and rough normality in the topological spaces which we can consider them as results from the general relations on the approximation spaces. Keywords: Topologized approximation space, rough regularity, rough normality, rough continuity, rough homeomorphism

Covering rough sets based on neighborhoods: An approach without using neighborhoods

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin

The Relation Between Rough Sets And Fuzzy Sets Via Topological Spaces

Abstract: Theories of rough sets and fuzzy sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness, respectively. They are generalizations of classical set theory for modeling vagueness and uncertainty. A fundamental question concerning both theories is their connections and differences. There have been many studies on this topic. Topology is a branch of mathematics, whose ideas exist not only in almost all branches of mathematics but also in many real life applications. The topological structure on an abstract set is used as the base, which used to extract knowledge from data. In this paper: topological structure is used to study the relation between rough sets and fuzzy sets. Membership function is used to convert from rough set to fuzzy set and vice versa. This conversion will achieve the advantages of two theories. Some examples and theories are introduced to indicate the importance of using general binary relations in the construction of rough set concepts, and indicate the relation between rough sets and fuzzy sets according to the topological spaces

Matroidal and Lattices Structures of Rough Sets and Some of Their Topological Characterizations

Matroids, rough set theory and lattices are efficient tools of knowledge discovery. Lattices and matroids are studied on preapproximations spaces. Li et al. proved that a lattice is Boolean if it is clopen set lattice for matroids. In our study, a lattice is Boolean if it is closed for matroids. Moreover, a topological lattice is discussed using its matroidal structure. Atoms in a complete atomic Boolean lattice are completely determined through its topological structure. Finally, a necessary and sufficient condition for a predefinable set is proved in preapproximation spaces. The value k for a predefinable set in lattice of matroidal closed sets is determined

Matroidal and Lattices Structures of Rough Sets and Some of Their Topological Characterizations

Matroids, rough set theory and lattices are efficient tools of knowledge discovery. Lattices and matroids are studied on preapproximations spaces. Li et al. proved that a lattice is Boolean if it is clopen set lattice for matroids. In our study, a lattice is Boolean if it is closed for matroids. Moreover, a topological lattice is discussed using its matroidal structure. Atoms in a complete atomic Boolean lattice are completely determined through its topological structure. Finally, a necessary and sufficient condition for a predefinable set is proved in preapproximation spaces. The value k for a predefinable set in lattice of matroidal closed sets is determined

Rough action on topological rough groups

[EN] In this paper we explore the interrelations between rough set theory and group theory. To this end, we first define a topological rough group homomorphism and its kernel. Moreover, we introduce rough action and topological rough group homeomorphisms, providing several examples. Next, we combine these two notions in order to define topological rough homogeneous spaces, discussing results concerning open subsets in topological rough groups.The authors wish to thank the Deanship for Scientific Research (DSR) at King Abdulaziz University for financially funding this project under grant no. KEP-PhD-2-130-39. Also, we would like to thank the editor and referees for their valuable suggestions which have improved the presentation of the paper.Altassan, A.; Alharbi, N.; Aydi, H.; Özel, C. (2020). Rough action on topological rough groups. Applied General Topology. 21(2):295-304. https://doi.org/10.4995/agt.2020.13156OJS295304212S. Akduman, E. Zeliha, A. Zemci and S. 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Topological Interpretation of Rough Sets

Rough sets, developed by Pawlak, are an important model of incomplete or partially known information. In this article, which is essentially a continuation of [11], we characterize rough sets in terms of topological closure and interior, as the approximations have the properties of the Kuratowski operators. We decided to merge topological spaces with tolerance approximation spaces. As a testbed for our developed approach, we restated the results of Isomichi [13] (formalized in Mizar in [14]) and about fourteen sets of Kuratowski [17] (encoded with the help of Mizar adjectives and clusters’ registrations in [1]) in terms of rough approximations. The upper bounds which were 14 and 7 in the original paper of Kuratowski, in our case are six and three, respectively. It turns out that within the classification given by Isomichi, 1st class subsets are precisely crisp sets, 2nd class subsets are proper rough sets, and there are no 3rd class subsets in topological spaces generated by approximations. Also the important results about these spaces is that they are extremally disconnected [15], hence lattices of their domains are Boolean. Furthermore, we develop the theory of abstract spaces equipped with maps possessing characteristic properties of rough approximations which enables us to freely use the notions from the theory of rough sets and topological spaces formalized in the Mizar Mathematical Library [10].Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok PolandLilla Krystyna Baginska and Adam Grabowski. On the Kuratowski closure-complement problem. Formalized Mathematics, 11(3):323-329, 2003.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Adam Grabowski. Automated discovery of properties of rough sets. Fundamenta Informaticae, 128:65-79, 2013. doi:10.3233/FI-2013-933. http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000327181800006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21-28, 2004.Adam Grabowski. Relational formal characterization of rough sets. Formalized Mathematics, 21(1):55-64, 2013. doi:10.2478/forma-2013-0006.Yoshinori Isomichi. New concepts in the theory of topological space - supercondensed set, subcondensed set, and condensed set. Pacific Journal of Mathematics, 38(3):657-668, 1971.Magdalena Jastrzębska and Adam Grabowski. The properties of supercondensed sets, subcondensed sets and condensed sets. Formalized Mathematics, 13(2):353-359, 2005.Zbigniew Karno. The lattice of domains of an extremally disconnected space. Formalized Mathematics, 3(2):143-149, 1992.Artur Korniłowicz. On the topological properties of meet-continuous lattices. Formalized Mathematics, 6(2):269-277, 1997.Kazimierz Kuratowski. Sur l’opération A de l’analysis situs. Fundamenta Mathematicae, 3:182-199, 1922.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Bartłomiej Skorulski. First-countable, sequential, and Frechet spaces. Formalized Mathematics, 7(1):81-86, 1998.Bartłomiej Skorulski. The sequential closure operator in sequential and Frechet spaces. Formalized Mathematics, 8(1):47-54, 1999.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495-500, 1990.Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1 (1):187-190, 1990.Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990