On enumeration of $EL$-hyperstructures with $2$ elements

Document Type : Research Paper

Authors

1 Department of Mathematics, Payame Noor University, Tehran, Iran

2 Department of Mathematics, University of Shahid Ashrafi Esfahani, P.O.Box 81798-49999, Esfahan, Iran

10.22103/jmmrc.2021.18452.1173

Abstract

$EL$-hypergroups  were defined by Chvalina 1995. Till now, no exact statistics of $EL$-hypergroups have been done. Moreover, there is no classification of $EL$-hypergroups and $EL^2$-hypergroups even over small sets. In this paper we classify all $EL$-(semi)hypergroups over sets with two elements obtained from quasi ordered semigroups. Also, we characterize all quasi ordered $H_v$-group and   then we enumerate the number of $EL^2$-$H_v$-hypergroups and $EL^2$-hypergroups of order $2$.

Keywords


[1] F. Arabpur, M. Jafarpour, On strongly Hv-groups, J. Mahani Math. Res. Cent., 8(1)(2019) 13-21.
[2] R. Bayon, N. Lygeros, Advanced results in enumeration of hyperstructures, Journal of Algebra, 320(2)(2008), 821-835,
[3] J. Chvalina, Functional Graphs, Quasi-Ordered Sets and Commutative Hypergroups, Masaryk University, Brno, (1995), (in Czech).
[4] P. Corsini, Prolegomena of Hypergroup Theory, Aviani, Udine, 2003.
[5] P. Corsini, V. Leoreanu-Fotea, Applications of Hyperstruture Theory, Kluwer Academic Publishers, 2003.
[6] B. Davvaz, Semihypergroup Theory, Elsevier, 2016.
[7] B. Davvaz, Polygroup Theory and Related System, World Scienti c Publishing Co. Pte. Ltd., 2012.
[8] B. Davvaz and T. Vougiouklis, A Walk Through Weak Hyperstructures; Hv-Structure, World Scienti c Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.
[9] A. Distler, Classi cation and Enumeration of Finite Semigroups, Ph.D. Thesis, University of St Andrews, (2010).
[10] S. H. Ghazavi, S. M. Anvarieh and S. Mirvakili, EL2{hyperstructures derived from (partially) quasi ordered hyperstructures, Iranain Journal of Mathematical Sciences and Informatics, 10(2) (2015), 99-114.
[11] S. H. Ghazavi, S. M. Anvarieh and S. Mirvakili, Ideals in EL-semihypergroups associated to ordered semigroups, Journal of Algebraic Systems, 3(2) (2016), 109-125.
[12] S. H. Ghazavi and S. M. Anvarieh, EL-hyperstructures associated to n-ary relations, Soft Computing, 21(2013), 5841-5850.
[13] S. Hoskova, Binary Hyperstructures Determined by Relational and Transformation Systems, Habilitation Thesis, Faculty of Science, Universityof Ostrava, (2008).
[14] R. Migliorato, Ipergruppi di cardinalit 3 e isomor smi di ipergruppoidi commutativi totalmente regolari, Atti Convegno su Ipergruppi, Udine, (1985).
[15] G. Nordo, An algorithm on number of isomorphism classes of hypergroups of order 3, Italian J. Pure Appl. Math. 2(1997), 37-42.
[16] M. Novak, Potential of the Ends lemma to create ring-like hyperstructures from quasi-ordered (semi)groups, South Bohemia Mathem. Letters, 17(1)(2009), 39-50.
[17] M. Novak, The notion of subhyperstructure of"Ends lemma"-based hyperstructures, Journal of Applied Mathematics, 3(2) (2010), 237-247.
[18] M. Novak, Important elements of EL-hyperstructures, in APLIMAT:10th International Conference, STU in Bratislava, Bratislava, (2011), 151-158.
[19] M. Novak, El-hyperstructures: an overview, Ratio Mathematica, 23(2012), 65-80.
[20] M. Novak, Some basic properties of EL-hyperstructure, European Journal of Combinatorics, 34(2013), 446-459.
[21] M. Novak, On EL-semihypergroup, European Journal of Combinatorics, 44(2015), 274-286.
[22] P. Rackova, Hypergroups of symmetric matrices, 10th International Congress of Algebraic Hyperstructures and Applications, Proceeding of AHA, (2008).
[23] I. G. Rosenberg, Hypergroups and join spaces determined by relations, Italian Journal of Pure and Applied Mathematics, 4(1998), 93-101.
[24] Ch. Tsitouras and Ch. G. Massouros, On enumeration of hypergroups of order 3, Comput. Math. Appl. 59(2010) 519-523.
[25] T. Vougiouklis, The e-hyperstructures, J. Mahani Math. Res. Cent., 1(1)(2012) 13-28.
[26] T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Florida,1994.