Hypegroups obtained from $BCK$-Algebras with condition $(S)$

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematical Sciences, Yazd University, Yazd, Iran.

2 Department of Mathematics, Payame Noor University, Tehran, Iran.

Abstract

In this paper, we construct a novel hyperoperation derived from a ($BCI$-algebra)$BCK$-algebra with condition $(S)$ and present several significant findings. We establish that this construction leads to the formation of commutative hypergroups and join spaces. Furthermore, we demonstrate that when the $BCK$-algebra {satisfies} condition $(S)$, the resulting hyperstructure constitutes $H_b$-monoids. {Moreover}, we show that every proper subhypergroup of this hypergroup {cannot be} closed, invertible, ultraclosed and conjugable. Finally, we have derived all the hypergroups obtained from $BCK$-algebras with condition $(S)$ of order less than 5.

Keywords

Main Subjects

References

[1] R. Ameri, A. Radfar and R. A. Borzooei, On Hyper P-Semi Simple I-algebra, Int. J. Algebra, 5 No. 23 (2011) 1117-1133.
[2] J. Chvalina, Functional Graphs, Quasi-Ordered Sets and Commutative Hypergroups, Masaryk University, Brno, 1995.
[3] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, 2003.
[4] P. Corsini, Hyperstructures associated with ordered sets, Bull. Greek Math. Soc., 48 (2003) 7-18.
[5] B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., 2012.
[6] B. Davvaz, Semihypergroup Theory, Academic Press, 2016.
[7] B. Davvaz and T. Vougiouklis, A Walk Through Weak Hyperstructures; Hv-Structure, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.
[8] S. H. Ghazavi, S. M. Anvarieh and S. Mirvakili, EL2-hyperstructures derived from (partially) quasi-ordered hyperstructures, Iran. J. Math. Sci. Inform., 10 No. 2 (2015) 99-114.
[9] S. H. Ghazavi, S. M. Anvarieh and S. Mirvakili, Ideals in EL-semihypergroups associated with ordered semigroups, J. Algebr. Syst., 3 No. 2 (2016) 109-125.
[10] D. Heidari and B. Davvaz, On ordered hyperstructures, U. P. B. Sci. Bull. Series A, 73 No. 2 (2011) 85-96.
[11] S. Hoskova, Binary Hyperstructures Determined by Relational and Transformation Systems, Ph.D. thesis, Faculty of Science, University of Ostrava, 2008.
[12] Y. Huang, BCI-Algebra, Science Press, Beijing, 2006.
[13] K. Iski, An algebra related with a propositional calculus, Proc. Japan Acad., 42 No. 1 (1966) 26-29.
[14] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Company, 1994.
[15] H. Mirabdollahi, S. M. Anvariyeh and S. Mirvakili, The associated hyperringoid to a Krasner hyperring, J. Taibah Univ. Sci., 12 No. 3 (2018) 348-356.
[16] S. Mirvakili, H. Naraghi, M. A. Dehghanizadeh and H. Naraghi, On the equivalence of sequences dependent on fuzzy ideals in the BCI-algebra, J. algebr. hyperstrucres log. algebr., 3 No. 4 (2022) 65-73.
[17] M. Mohseni Takallo, R. A. Borzooei, M. Aaly Kologani and Y. B. Jun, Commutative true-false ideals in BCI/BCK-algebras, Alg. Struc. Appl., 10 No. 2 (2023) 155-172.
[18] M. Novak, The notion of subhyperstructure of “Ends lemma”-based hyperstructures, J. Appl. Math., 3 No. 2 (2010) 237-247.
[19] M. Novak, EL-hyperstructures: An overview, Ratio Math., 23 (2012) 65-80.
[20] M. Novak, Some basic properties of EL-hyperstructure, Eur. J. Comb., 34 No. 3 (2013) 446-459.
[21] M. Novak, On EL-semihypergroup, Eur. J. Comb., 44 No. 2 (2015) 274-286.
[22] I. G. Rosenberg, Hypergroups and join spaces determined by relations, Ital. J. Pure Appl. Math., 4 (1998) 93-101.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 20 August 2026

Files

Share

How to cite

Statistics