On proximity spaces and topological hyper nearrings

Abstract

In 1934 the concept of algebraic hyperstructures was first introduced by a French mathematician, Marty. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the result of this composition is a set. In this paper, we prove some results in topological hyper nearring. Then we present a proximity relation on an arbitrary hyper nearring and show that every hyper nearring with a topology that is induced by this proximity is a topological hyper nearring. In the following, we prove that every topological hyper nearring can be a proximity space.

Keywords

• Heper nearring
• Topological hyper nearrings
• Complete part
• Proximity relation

References

1. Ameri, R., Topological (transposition) hypergroups, Italian Journal of Pure and Applied Mathematics, (13) (2003), 181-186.
2. Borhani-Nejad, S., Davvaz, B., Topological hyper nearrings, submitted.
3. Corsini, P., Prolegomena of Hypergroup Theory, Second ed., Aviani Editore, Tricesimo, Italy, 1993.
4. Corsini, P., Leoreanu, V., Applications of Hypergroup Theory, Kluwer Academic Publishers, 2003.
5. Dasic, V., Hypernearrings, Fourth Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990), World Scientific, (1991), 75-85.
6. Davvaz, B., Hypernearrings and weak hypernearrings, 11th Algebra Seminar of Iranian Math. Soc. Isfahan University of Technology, Isfahan, October 27-29, (1999), 68-78.
7. Davvaz, B., Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
8. Davvaz, B., Semihypergroup Theory, Elsevier, 2016.

9. Davvaz, B., Leoreanu-Fotea, V., Hyperring Theory and Applications, International Academic Press, 115, Palm Harber, USA, 2007.
10. Efremovc, V.A., The geometry of proximity, Mat. Sb., 31 (1952), 189-200.
11. Efremovic, V.A., Infinitesimal spaces, Doklady Akademii Nauk SSSR (N.S.), (in Russian), 76 (1951), 341-343.
12. Freni, D., A note on the core of a hypergroup and the transitive closure β^{∗} of β, Riv. Mat. Pura Appl., 8 (1991), 153-156.
13. Gontineac, V.M., On hypernear-rings and H-hypergroups, Fifth Int. Congress on Algebraic Hyperstructures and Applications (AHA 1993), Hadronic Press, Inc., USA, (1994), 171-179.
14. Heidari, D., Davvaz, B., Modarres, S.M.A., Topological hypergroups in the sense of Marty, Comm. algebra, 42 (2014), 4712-4721.
15. Heidari, D., Davvaz, B., Modarres, S.M.S., Topological polygroups, Bull. Malays. Math. Sci. Soc., 3942 (2016), 707-721.
16. Hosková-Mayerová, S., Topological hypergroupoids, Comput. Math. Appl., 64(9) (2012), 2845-2849.
17. Jancic-Rasovic S., Cristea, I., Division hypernear-rings, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat., 26(3) (2018), 109-126.
18. Jancic-Rasovic S., Cristea, I., Hypernear-rings with a defect of distributivity, Filomat, 32(4) (2018), 1133-1149.
19. Kim, K.H., Davvaz, B., Roh, E.H., On Hyper R-subgroups of hypernear-rings, Scientiae Mathematicae Japonicae, Online (2007), 649-656.
20. Koskas, M., Groupoides, demi-hypergroupes et hypergroupes, J. Math. Pure Appl., 49(9) (1970), 155-192.
21. Munkres, J.R., Topology, Prentice Hall, Upper Saddle River, NJ, 2000.
22. Marty, F., Sur une generalization de la notion de group, 8th congress Math. Scandinaves, Stockholm, (1934), 45-49.
23. Mittas, J., Hypergroups canoniques, Math. Balkanica, 2 (1972), 165-179.
24. Naimpally S.A., Warrack, B.D., Proximity Spaces, Cambridge Tract, 1970.
25. Ursul, M., Topological rings satisfying compactness conditions, Springer Science Business Media (2002), vol 549.