Combinatorial integers (

We discuss the calculation of integral cohomology ring of LG/T and ΩG. First we describe the root system and Weyl group of LG, then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures of LG/T and ΩG for affine group A^2. We introduce combinatorial integers (m,nj) which play a crucial role in our calculations and give some interesting identities among these integers. Last we calculate generators for ideals and rank of each module of graded integral cohomology algebra in the local coefficient ring ℤ[1/2]

Topological distances and geometry over the symmetrized Omega algebra

[EN] The aim of this paper is to study some topological distances properties, semidendrites and convexity on th symmetrized omega algebra. Furthermore, some properties and exponents on the symmetrized omega algebra are introduced.Alqahtani, M.; Özel, C.; Zekraoui, H. (2020). Topological distances and geometry over the symmetrized Omega algebra. Applied General Topology. 21(2):247-264. https://doi.org/10.4995/agt.2020.13049OJS247264212A. C. F. Bueno, On the exponential function of right circulant matrices, International Journal of Mathematics and Scientific Computing 3, no. 2 (2013).L. Hörmander, Notions of convexity, Progress in Mathematics 127, Birkh¨auser, Boston- Basel-Berlin (1994).S. Khalid Nauman, C. Ozel and H. Zekraoui, Abstract Omega algebra that subsumes min and max plus algebras, Turkish Journal of Mathematics and Computer Science 11 (2019) 1-10.G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: a brief introduction, Journal of Mathematical Sciences 140, no. 3 (2007), 426-444. https://doi.org/10.1007/s10958-007-0450-5D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, 2015. https://doi.org/10.1090/gsm/161C. Ozel, A. Piekosz, E. Wajch and H. Zekraoui, The minimizing vector theorem in symmetrized max-plus algebra, Journal of Convex Analysis 26, no. 2 (2019), 661-686.J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50-69, Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511662508.004I. Simon, Recognizable sets with multiplicities in the tropical semiring, in: Mathematical Foundations of Computer Science (Carlsbad, 1988), Lecture Notes in Computer Science, vol. 324, Springer, Berlin, 1988, pp. 107-120. https://doi.org/10.1007/BFb001713

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. 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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

New Topology on Symmetrized Omega Algebra

The purpose of this paper is to define a new topology called symmetrized omega algebra topology and discuss some of its topological properties. Two different examples from an ordered infinite set of symmetrized omega topology are introduced. Furthermore, we study the relationship between symmetrized omega topology and weaker kinds of normality

Some New Algebraic and Topological Properties of the Minkowski Inverse in the Minkowski Space

We introduce some new algebraic and topological properties of the Minkowski inverse A⊕ of an arbitrary matrix A∈Mm,n (including singular and rectangular) in a Minkowski space μ. Furthermore, we show that the Minkowski inverse A⊕ in a Minkowski space and the Moore-Penrose inverse A+ in a Hilbert space are different in many properties such as the existence, continuity, norm, and SVD. New conditions of the Minkowski inverse are also given. These conditions are related to the existence, continuity, and reverse order law. Finally, a new representation of the Minkowski inverse A⊕ is also derived