Geometric lattice structure of covering and its application to attribute reduction through matroids

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice to study the attribute reduction issues of information systems

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. Narli, Rough topology on covering based rough sets, 1st International Eurasian Conference on Mathematical Sciences and Applications (IECMSA), Prishtine, Kosovo, 3ÔÇô7 September 2012.S. Akduman, A. Zemci and C. Özel, Rough topology on covering-based rough sets, Int. J. Computational Systems Engineering 2, no. 2 (2015),107-111. https://doi.org/10.1504/IJCSYSE.2015.077056N. Alharbi, H. Aydi and C. Özel, Rough spaces on covering based rough sets, European Journal of Pure And Applied Mathematics (EJPAM) 12, no. 2 (2019). https://doi.org/10.29020/nybg.ejpam.v12i2.3420N. Alharbi, H. Aydi, C. Park and C. Özel, On topological rough groups, J. Computational Analysis and Applications 29, no. 1 (2021), 117 -122.A. Arhangel'skii and M. Tkachenko, Topological groups and related structures, Atlantis press/ World Scientific, Amsterdam-Paris, 2008. https://doi.org/10.2991/978-94-91216-35-0N. Bagirmaz, I. Icen and A. F. Ozcan, Topological rough groups, Topol. Algebra Appl. 4 (2016), 31-38. https://doi.org/10.1515/taa-2016-0004R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42 (1994), 251-254.E. Brynairski, A calculus of rough sets of the first order, Bull. of the Polish Academy Sciences: Mathematics 37, no. 1-6 (1989), 71-78.G. Chiaselotti and F. Infusino, Some classes of abstract simplicial complexes motivated by module theory, Journal of Pure and Applied Algebra 225 (2020), 106471, https://doi.org/10.1016/j.jpaa.2020.106471G. Chiaselotti and F. Infusino, Alexandroff topologies and monoid actions, Forum Mathematicum 32, no. 3 (2020), 795-826. https://doi.org/10.1515/forum-2019-0283G. Chiaselotti, F. Infusino and P. A. Oliverio, Set relations and set systems induced by some families of integral domains, Advances in Mathematics 363 (2020), 106999, https://doi.org/10.1016/j.aim.2020.106999G. Chiaselotti, T. Gentile and F. Infusino, Lattice representation with algebraic granular computing methods, Electronic Journal of Combinatorics 27, no. 1 (2020), P1.19. https://doi.org/10.37236/8786S. Hallan, A. Asberg and T. H. Edna, Additional value of biochemical tests in suspected acute appendicitis, European Journal of Surgery 163, no. 7 (1997), 533-538.R. R. Hashemi, F. R. Jelovsek and M. Razzaghi, Developmental toxicity risk assessment: A rough sets approach, Methods of Information in Medicine 32, no. 1 (1993), 47-54. https://doi.org/10.1055/s-0038-1634890A. Huang, H. Zhao and W. Zhu, Nullity-based matroid of rough sets and its application to attribute reduction, Information Sciences 263 (2014), 153-165. https://doi.org/10.1016/j.ins.2013.11.014A. Kusiak, Decomposition in data mining: An industrial case study, IEEE Transactions on Electronics Packaging Manufacturing 23 (2000), 345-353. https://doi.org/10.1109/6104.895081A. Kusiak, Rough set theory: A data mining tool for semiconductor manufacturing, IEEE Transactions on Electronics Packaging Manufacturing 24, no. 1(2001), 44-50. https://doi.org/10.1109/6104.924792C. A. Neelima and P. Isaac, Rough anti-homomorphism on a rough group, Global Journal of Mathematical Sciences: Theory and Practical 6, no. 2, (2014), 79-80.M. Novotny and Z. Pawlak, On rough equalities, Bulletin of the Polish Academy of Sciences, Mathematics 33, no. 1-2 (1985), 99-104.N. Paul, Decision making in an information system via new topology, Annals of fuzzy Mathematics and Informatics 12, no. 5 (2016), 591-600.Z. Pawlak,Rough sets, Int. J. Comput. Inform. Sci. 11, no. 5 (1982), 341-356. https://doi.org/10.1007/BF01001956J. Pomykala, The stone algebra of rough sets, Bulletin of the Polish Academy of Sciences, Mathematics 36, no. 7-8 (1988), 495-508.J. Tanga, K. Shea, F. Min and W. Zhu, A matroidal approach to rough set theory, Theoretical Computer Science 471 (2013), 1-11. https://doi.org/10.1016/j.tcs.2012.10.060S. Wang, Q. Zhu, W. Zhu and F. Min, Graph and matrix approaches to rough sets through matroids, Information Sciences 288 (2014), 1-11. https://doi.org/10.1016/j.ins.2014.07.023S. Wang, Q. Zhu, W. Zhu and F. Min, Rough set characterization for 2-circuit matroid, Fundamenta Informaticae 129 (2014), 377-393. https://doi.org/10.3233/FI-2013-97

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Advances in Discrete Differential Geometry

Differential Geometr

Advances in Discrete Differential Geometry

Differential Geometr

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum