Search Advanced

AIMS Mathematics

2020, Volume 5, Issue 6: 6552-6579. doi: 10.3934/math.2020422
Previous Article Next Article
Research article

Advanced results in enumeration of hyperfields

  • 1.
    School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, Iran
  • 2.
    Department of Mathematics and Physics, University of Defence in Brno, Czech Republic
  • Received: 04 June 2020 Accepted: 06 July 2020 Published: 24 August 2020

  • MSC : 20N20

  • The purpose of this paper is to introducing a computational method to construction and classification of finite hyperfields (in the sense of Krasner). In this regards first we introduce a mathematical method to produce hyperfields from a family of a non-empty subsets of a given multiplicative group under specific conditions and then we apply this method to enumerate all finite hyperfields of order less that 7, up to isomorphism by a computer programming. Of course this program can be used to produce hyperfields of finite higher orders, but it's commotional complexity is of high order and it must be used of very high-speed computers system.

    Citation: R. Ameri, M. Eyvazi, S. Hoskova-Mayerova. Advanced results in enumeration of hyperfields[J]. AIMS Mathematics, 2020, 5(6): 6552-6579. doi: 10.3934/math.2020422

    Related Papers:

  • The purpose of this paper is to introducing a computational method to construction and classification of finite hyperfields (in the sense of Krasner). In this regards first we introduce a mathematical method to produce hyperfields from a family of a non-empty subsets of a given multiplicative group under specific conditions and then we apply this method to enumerate all finite hyperfields of order less that 7, up to isomorphism by a computer programming. Of course this program can be used to produce hyperfields of finite higher orders, but it's commotional complexity is of high order and it must be used of very high-speed computers system.


    加载中


    [1] F. Marty, Sur une Generalization de la Notion de Groupe, 8th Congres Math., Scandinaves, Stockholm, (1934), 45-49.
    [2] R. Ameri, M. Aivazi, S. Hoskova-Mayerova, Superring of polynomials over a hyperring, Mathematics, 2019, 7, 902, doi:10.3390/math7100902.
    [3] R. Ameri, M. M. Zahedi, Hyperalgebraic systems, Ital. J. Pure Appl. Math., 6 (1999), 21-32.
    [4] R. Ameri, A. Kordi, S. Hoskova-Mayerova, Multiplicative hyperring of fractions and coprime hyperideals, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 25 (2017), 5-23.
    [5] D. Stratigopoulos, Sur les hypercorps et les hyperanneaux (Onhyperfields and hyperrings), in: Algebraic Hyperstructures and Applications, Xanthi, 1990, World Sci. Publ., Teaneck, NJ, 1991, 33-53 (in French).
    [6] A. Asadi, R. Ameri, A categorical connection between categories (m, n)-hyperrings and (mn)-rings via the fundamental relation, Kragujevac J. Math., 45 (2021), 361-377.
    [7] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publishers, 2002.
    [8] P. Corsini, Prolegomena of Hypergroup Theory, Second Edition, Aviani Editor, 1999.
    [9] S. Hoskova, Topological hypergroupoids, Comput. Math. Appl., 64 (2012), 2845-2849.
    [10] N. Ramaruban, Commutative hyperalgebra, Ph.D thesis, 2013.
    [11] S. Atamewoue, S. Ndjeya, C. Lele, Codes over hyperfields, Discuss. Mathematicae-General Algebra Appl., 37 (2017), 147-60. doi: 10.7151/dmgaa.1277
    [12] M. Krasner, A class of hyperrings and hyperfields, Int. J. Math. Math. Sci., 6 (1983), 1-5.
    [13] M. Krasner, Approximation des corps valués complets de caractéristique p10 par ceux de caractéristique 0. Colloque dé algébre supérieure, tenu á Bruxelles du 19 au 22 décembre 1956, in: Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain, Librairie Gauthier-Villars, Paris, 1957, 129-206.
    [14] Ch. G. Massouros, Constructions of hyperfields, Matematica Balkanica, 5 (1991), 250-257.
    [15] V. Vahedi, M. Jafarpour, H. Aghabozorgi, et al. Extension of elliptic curves on Krasner hyperfields, Commun. Algebra, 47 (2019), 4806-4823. doi: 10.1080/00927872.2019.1596279
    [16] O. Viro, Hyperfield for tropical geometry I: Hyperfields and dequantization, arXiv:1006.3034 (2010).
    [17] R. Ameri, N. Jafarzadeh, Exact category of (m, n)-ary hypermodules, Categories General Algebraic Structures Appl., 12 (2020), 69-88.
    [18] R. Ameri, I. G. Rosenberg, Congruences of multialgebras, Multivalued Logic Soft Comput., 15 (2009), 525-536.
    [19] R. Ameri, M. Norouzi, New fundamental relation of hyperrings, Eur. J. Combin. 34 (2013), 884-891. doi: 10.1016/j.ejc.2012.12.003
    [20] A. Connes, C. Consani, The hyperring of adele classes, J. Number Theory, 131 (2011), 159-194.
    [21] Ch. G. Massouros, Methods of constructing hyperfields, Internat. J. Math. Math. Sci., 8 (1985), 725-728. doi: 10.1155/S0161171285000813
    [22] R. Bayon, N. Lygeros, Advanced results in enumeration of hyperstructures, J. Algebra, 320 (2008), 821-835. doi: 10.1016/j.jalgebra.2007.11.010
    [23] H. Bordbar, I. Cristea, Height of Prime Hyperideals in Krasner Hyperrings, Filomat, 31 (2017), 6153-6163. doi: 10.2298/FIL1719153B
    [24] I. Cristea, M. Jafarpour, S. S. Mousavic, A. Soleymani, Enumeration of Rosenberg hypergroups, Comput. Math. Appl., 60 (2010), 2753-2763.
    [25] Ch. Tsitouras, Ch. G. Massouros, On enumeration of hypergroups of order 3, Comput. Math. Appl., 59 (2010), 519-523.
    [26] Ch. Tsitouras, Ch. G. Massouros, Enumeration of Rosenberg-type hypercompositional structures defined by binary relations, Eur. J. Combin., 33 (2012), 1777-1786.
    [27] C. G. Massouros, On the theory of hyperrings and hyperfields, Algebra and Logic, (Springer), 24 (1985), 728-742.
    [28] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc., 1994.
    [29] V. Vahedi, M. Jafarpour, S. Hoskova-Mayerova, et al. Derived Hyperstructures from Hyperconics, Mathematic, 8 (2020), 429, doi:10.3390/math8030429.
    [30] R. Ameri, M. Norouzi, Prime and primary hyperideales in Krasner (m, n)-hyperring, European J. Combin., 34 (2013), 379-390. doi: 10.1016/j.ejc.2012.08.002
    [31] H. Bordbar, I. Cristea, M. Novak, Height Of Hyperideals In Noetherian Krasner Hyperrings, University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 79 (2017), 31-42.
    [32] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, USA, (2007).
    [33] A. Connes, C. Consani, On the notion of geometry over $\mathbb{F_1}$, J. Algebraic Geom. 20 (2008). doi: 10.1090/S1056-3911-2010-00535-8.
    [34] Ch. G. Massouros, A field theory problem relating to questions in hyperfield theory, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2011 Halkidiki, American Institute of Physics (AIP) Conference Proceedings, 2011, 1852-1855.
    [35] Ch. G. Massouros, A Class of hyperfields and a problem in the theory of fields, Math. Montisnigri, (1993), 73-84.
    [36] A. Connes, C. Consani, From monoids to hyperstructures: In search of an absolute arithmetic, In: Casimir Force, Casimir Operators and the Riemann Hypothesis, de Gruyter, (2010), 147-198.
    [37] A. Connes, C. Consani, M. Marcolli, The Weil proof and thegeometry of the adéles class space, in Algebra, Arithmetic, and Geometry: in honor of Yu. I. Manin, Vol. I, Progr. Math. 269, Birkhauser Boston, Inc., Boston, MA, (2009), 339-405.
    [38] M. Stefanescu, Constructions of hyperfields and hyperrings, SStud. Cercet. Stiint. Ser. Math. Univ. Din Bacau, 16 (2006), 563-571.
    [39] V. Bergelson, D. B. Shapiro, Multiplicative subgroups of finite index in a ring, Proc. Amer. Math. Soc., 116 (1992), 885-896.
    [40] S. Atamewoue, S. Ndjeya, C. Lele, Codes over hyperfields, Discuss. Mathematicae-General Algebra Appl., 37 (2017), 147-60. doi: 10.7151/dmgaa.1277
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
2.2 3

Metrics

Article views(2874) PDF downloads(165) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog