Theoretical models for constructing $H_v$-near fields

Shahrestani, Maryam; Davvaz, Bijan

doi:10.22103/jmmr.2026.26305.1903

Theoretical models for constructing $H_v$-near fields

Articles in Press

Document Type : Research Paper

Authors

Maryam Shahrestani

Bijan Davvaz

Department of Mathematical Sciences, Yazd University, Yazd, Iran

10.22103/jmmr.2026.26305.1903

Abstract

In this paper, we introduce the notion of $H_v$-near fields as a new generalization of classical near fields within the framework of hyperstructure theory. The concept of an $H_v$-near field is introduced and defined here for the first time. The main contribution of this paper is the development of the foundational theory of $H_v$-near fields, including the investigation of their basic algebraic properties, structural characteristics, and construction methods. Several examples and construction techniques are presented to illustrate the existence and diversity of these hyperalgebraic structures. Furthermore, we define and study the fundamental relation $\theta^{*}$, which plays a central role in describing equivalence classes and the internal structure of $H_v$-near fields. Our approach is based on methods and techniques from hyperstructure theory.

Keywords

$H_v$-nearring

near field

fundamental relation

Subjects

Algebra

[1] Al Tahan, M., & Davvaz, B. (2017). Hyperstructures associated to Biological inheritance. Mathematical Biosciences, 285, 112-118. https://doi.org/10.1016/j.mbs.2017.01.002

[2] Corsini, P. (1993). Prolegomena of hypergroup theory. Ssecond edition, Aviani editor.

[3] Davvaz, B. (2000). Hv-near rings. Math. Japonica, 52(2), 387-392.

[4] Davvaz, B., & Vougiouklis, T. (2019). A walk through weak hyperstructures: Hv-structures. World Scienti c Publishing Co. Pte. Ltd., Hackensack, NJ. https://doi.org/10.1142/11229

[5] Davvaz, B., Dehghan Nezad, A., & Heidari, M.M. (2013). Inheritance examples of algebraic hyperstructures. Information Sciences, 224, 180-187. https://doi.org/10.1016/j.ins.2012.10.023

[6] Davvaz, B., Dehghan Nezad, A., Mazloum-Ardakani, M., & Sheikh-Mohseneib, M.A. (2014). Describing the algebraic hyperstructure of all elements in radiolytic processes in cement medium. MATCH Commun. Math. Comput. Chem., 72(2), 375-388.

[7] Dehghan Nezhad, A., Moosavi Nejad, S.M., Nadja khah, M., & Davvaz, B. (2012). A physical example of algebraic hyperstructures: Leptons. Indian Journal of Physics, 86(11), 1027-1032. https://doi.org/10.1007/s12648-012-0151-x

[8] Dembrowski, P. (1968). Finite geometries. Springer, Berlin. https://doi.org/10.1007/BFb0094449

[9] Dickson, L.E. (1905). On nite algebras. Nachr. Gesellsch. Wissensch. Gott. Math.-Phys. Klasse, 358-393.

[10] Dramalidis, A. (1996). Dual Hv-rings. Rivista di Matematica Pura Appl., 17, 55-62.

[11] Farag, M. (2007). Hill Ciphers over near- elds. Mathematics and Computer Education, 41(1), 46-54.

[12] Krasner, M. (1983). A class of hyperrings and hyper elds. Int. J. Math. and Math. Sci., 2, 307-312. https://doi.org/10.1155/S0161171283000265

[13] Pilz, G. (1983). Near-rings: the theory and its applications. North-Holland, Amsterdam.

[14] Rota, R. (1996). Hyperane planes over hyperrings. Discrete Math., 155, 215-223. https://doi.org/10.1016/0012-365X(94)00385-V

[15] Spartaliies, S. (1996). On the number of Hv-rings with P-hyperoperations. Discrete Math., 155, 225-231. https://doi.org/10.1016/0012-365X(94)00386-W

[16] Spartalies, S., & Vongiouklis, T. (1994). The fundamental relation on Hv-rings. Rivista di Matematica Pura Appl., 7-20.

[17] Vougiouklis, T. (1994). Hyperstructures and their Representations. Monographs in Math., Hadronic, Palm Harbor, United States.

[18] Vougiouklis, T. (2020). Minimal Hv- elds. Ratio Mathematica, 38, 313-328.

[19] Vougiouklis, T. (2017). Hv- elds, h=v- elds. Ratio Mathematica, 33, 181-201.

[20] Vougiouklis, T. (1995). A new class of hyperstructures. J. Combin. Inform. System Sci., 20, 229-235.

21] Vougiouklis, T. (1997). Convolutions on WASS hyperstructures. Discrete Math., 174, 347-355. https://doi.org/10.1016/S0012-365X(96)00348-2

[22] Vougiouklis, T. (1991). The fundamental relation in hyperrings, The general hyper eld. 4th AHA, Xanthi (1990), World Scienti c, Singapore.

[23] Veblen, O., & Wedderburn, J.H. (1907). Non-desarguesian and non-pascalian geometrie. Trans. Amer. Math. Soc., 8, 379-388. https://doi.org/10.2307/1988781