Some betweenness relation topologies induced by simplicial complexes

Year 2022, , 981 - 994, 01.08.2022

Abd El Fattah El Atik , Ashgan Wahba

https://doi.org/10.15672/hujms.787479

Cited By: 1

Abstract

This article aims to create an approximation space from any simplicial complex by representing a finite simplicial complex as a union of its components. These components are arranged into levels beginning with the highest-dimensional simplices. The universal set of the approximation space is comprised of a collection of all vertices, edges, faces, and tetrahedrons, and so on. Moreover, new types of upper and lower approximations in terms of a betweenness relation will be defined. A betweenness relation means that an element lies between two elements: an upper bound and a lower bound. In this work, based on Zhang et al.'s concept, a betweenness relation on any simplicial complex, which produces a set of order relations, is established and some of its topologies are studied.

Keywords

simplicial complexes, approximation spaces, order relation, topological spaces, betweenness relation

Supporting Institution

Tanta University

References

• [1] F.G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenian. (N.S.), 68 (1), 17–25, 1999.
• [2] L.M. Blumenthal and D.O. Ellis, Notes on lattices, Duke Math. J. 16, 585–590, 1949.
• [3] D. Cavaliere, S. Senatore and V. Loia, Context-a ware profiling of concepts from a semantic topological space, Knowledge-Based Systems, 130, 102–115, 2017.
• [4] V. Chvátal, Sylvester-Gallai theorem and metric betweenness, Discrete Comput. Geom. 31, 175–195, 2004.
• [5] N. Düvelmeyer and W, Wenzel, A characterization of ordered sets and lattices via betweenness relations, Results Math. 46, 237–250, 2004.
• [6] A. El Atik, Reduction based on similarity and decision-making, J. Egyptian Math. Soc. 28 (22), 1–12, 2020.
• [7] A. El Atik and H. Hassan, Some nano topological structures via ideals and graphs, J. Egyptian Math. Soc. 28 (41), 1–21, 2020.
• [8] A. El Atik and A. Nasef, Some topological structures of fractals and their related graphs, Filomat, 34 (1), 153–165, 2020.
• [9] E. Estrada and G.J. Ross, Centralities in simplicial complexes, Applications to protein interaction, J. Theoret. Biol. 438, 46–60, 2018.
• [10] E.V. Huntington, A new set of postulates for betweenness, with proof of complete independence, Trans. Amer. Math. Soc. 26, 257–282, 1924.
• [11] E.V. Huntington and J.R. Kline, Sets of independent postulates for betweenness, Trans. Amer. Math. Soc. 18, 301–325, 1917.
• [12] F. Klein, Vorlesungen äber höhere Geometrie 22, Springer-Verlag, 2013.
• [13] E.F. Lashin, A.M. Kozae, A.A. Khadra, and T. Medhat, Rough set theory for topological spaces, Internat. J. Approx. Reason. 40 (1-2), 35–43, 2005.
• [14] S.A. Morris, Topology without tears, University of New England, 1989.
• [15] M. Pasch, Vorlesunges über neuere geometrie, 23, Teubner, Leipzig, Berlin, 1882.
• [16] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5), 341–356, 1982.
• [17] Z. Pawlak, Rough set approach to knowledge-based decision support, European J. Oper. Res. 99 (1), 48–57, 1997.
• [18] R. Pérez-Fernández and B. De Baets, On the role of monometrics in penalty-based data aggregation, IEEE Trans. Fuzzy Sys. 27 (7), 1456–1468, 2018.
• [19] M.F. Smiley and W.R. Transue, Applications of transitivities of betweenness in lattice theory, Bull. Amer. Math. Soc. 49, 280–287, 1943.
• [20] B. Stolz, Computational topology in neuro science, M.Sc. thesis, University of Oxford, London-England, 2014.
• [21] H.P. Zhang, R.P. Fernandez and B.D. Baets, Topologies induced by the representation of a betweenness relation as a family of order relations, Topology Appl. 258, 100–114, 2019.
• [22] L. Zuffi, Simplicial complexes from graphs towards graph persistence, M.Sc. thesis, Universita di Bologna, 2015/2016.