An overview of hyper logical algebras

Document Type : Original Article

Authors

1 Shahid Beheshti University

2 Hatef University

Abstract

Hyper logical algebras were first studied in 2000 by Borzooei et al. They applied the concept of hyperstructures to one of the logical algebraic structures known as the BCK-algebra, and introduced two generalizations of them called the hyper BCK-algebra and hyper K-algebra. Then many researchers in this field continued their research and used hyperstructures on other logical algebras and introduced the concepts of hyper residuated lattices, hyper BL-algebras, hyper MV-algebras, hyper EQ-algebras, hyper BE-algebras, hyper equality algebras, hyper hoops and etc. Moreover, they defined some new notions such as different kinds of hyper ideals, hyper filters and hyper congruence relations on these structures and studied some properties, the relation among them and the quotient structure. Now, in this paper, we review the definitions of all those hyper logical algebras and investigate relations among them.

Keywords

References

[1] A. Borumand Saeid, R.A. Borzooei, M.M. Zahedi, (Weak) Implicative hyper K-ideals, Bulletin
of the Korean Mathematical Society, 40(1) (2003), 123{137.
[2] R.A. Borzooei, M. Bakhshi, Some results on hyper BCK-algebras, Quasigroups and Related
Systems (Poland), 11 (2004), 9{24.
[3] R.A. Borzooei, M. Bakhshi, On positive implicative hyper BCK-ideals, Scientiae Mathematicae
Japonicae, 59(3) (2004), 505{516.
[4] R.A. Borzooei, P. Corsini, M.M. Zahedi, Some kinds of positive implicative hyper K-ideals,
Journal of Discrete Mathematical Sciences and Cryptography, 6(1) (2003), 97{108.
[5] R.A. Borzooei, W.A. Dudek, N. Koohestani, On hyper BCC-algebras, Hindawi Publishing
Corporation International Journal of Mathematics and Mathematical Sciences, 2006 (2006),
1{18.
[6] R.A. Borzooei, B. Ganji Sa ar, R. Ameri, On hyper EQ-algebras, Italian Journal of Pure and
Application Mathematics, 31 (2013), 77{96.
[7] R.A. Borzooei, A. Hasankhani, Hyper KH-algebra, Italian Journal of Pure and Applied Math-
ematics, 10 (2001), 207{212.
[8] R.A. Borzooei, A. Hasankhani, M.M. Zahedi, Y.B. Jun, On hyper K-algebras, Math Japonica,
1 (2000), 113{121.
[9] R.A. Borzooei, S. Niazian, Weak hyper residuated lattices, Quasigroups and Related Systems,
21 (2013), 29{42.
[10] R.A. Borzooei, H. Varasteh, K. Borna, On hyper hoop-algebras, Ratio Mathematica, 30 (2016),
67{81.
[11] R.A. Borzooei, M.M. Zahedi, Positive implicative hyper K-ideals, Scientiae Mathematicae
Japonicae, 53(3) (2001), 525{533.
[12] R.A. Borzooei, M.M. Zahedi, H. Rezaei, Classi cation of hyper BCK-algebras of order 3,
Italian Journal of Pure and Applied Mathematics, 12 (2002), 175{184.
[13] X.Y. Cheng, X.L. Xin, Y.B. Jun, Hyper equality algebras, Quantitative Logic and Soft Com-
puting, Advances in Intelligent Systems and Computing, 510 (2016), 415{428.
[14] P. Corsini, Hyperstructures associated with ordered sets, Bullten Greek Mathematics Society,
48 (2013), 7{18.
[15] P. Corsini, V. Leoreanu Fotea, Isomorphisms of hypergroups and of n-hypergroups with appli-
cations, Soft Computing, 13(10) (2009), 985{994.
[16] P. Corsini, G. Romeo, Hypergroupes complets et t-groupoids, Convegno su Sistemi Binari e
loro Applicazioni, Taormina, 1978.
[17] S. Ghorbani, A. Hasankhani, E. Eslami, Hyper MV-algebras, Set-Valued Mathematics and
Applications, 2 (2008), 205{222.
[18] M.A. Hashemi, R.A. Borzooei, Results on hyper equality algebras, Journal of Mathematical
Extension, to apear.
[19] Y.B. Jun, Scalar elements and hyperatoms of hyper BCK-algebras, Scientiae Mathematica,
2(3) (1999), 303-309.
[20] Y.B. Jun, On hyper BCK-algebra, Italian journal of pure and applied Mathematics, 10 (2000),
127{136.
[21] Y.B. Jun, W.H. Sim, Fuzzy implicative hyper BCK-ideals of hyper BCK-algebras, International
Journal of Mathematics and Mathematical Sciences, 29 (2002), https://doi.org/10.1155/S0161
17120201195X.
[22] Y.B. Jun, W.H. Sim, Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras, In-
formation Sciences, 170(2-4) (2005), 351{361.
[23] Y.B. Jun, X.L. Xin, Fuzzy hyper BCK-ideals of hyper BCK-algebras, Scientiae Mathematicae
Japonicae, 53(2) (2001), 353{360.
[24] Y.B. Jun, X.L. Xin, M.M. Zahedi, R.A. Borzooei, On a hyper BCK-algebra that satis es the
hyper condition, Mathematicae Japonicae, 52(1) (2000), 95{101.
[25] Y.B. Jun, M.M. Zahedi, X.L. Xin, R.A. Borzooei, On hyper BCK-algebras, Italian Journal of
Pure and Applied Mathematics, 8 (2000), 127{136.
[26] F. Marty, Sur une generalization de la notion de group, 8th congress Math. Scandinaves,
Stockholm, (1934), 45{49.
[27] A. Radfar, A. Rezaei, A. Borumand Saeid, Hyper BE-algebras, Search Novi Sad Journal of
Mathematics, 44(2) (2014), 137{147.
[28] Y.J. Seo, H.S. Kim, Y.B. Jun, S.S. Ahn, Multipolar intuitionistic fuzzy hyper BCK-ideals in
hyper BCK-algebras, to apear, DOI: 10.3390/math8081373.
[29] T. Vougiouklis, A new class of hyperstructure, Journal of Combinatorics Information System
Sciences.
[30] X.L. Xin, Hyper BCI-algebras, Discussiones Mathematicae General Algebra and Applications,
26 (2006), 5{19.
[31] X.L. Xin, M.M. Zahedi, R. Hwan, Strong hyper BCK-ideals of hyperBCK-algebras, Mathe-
matica japonicae, 51(3) (2000), 493{498.
[32] X.L. Xin, Y. X. Zou, J. M. Zhan, Hyper BL-algebras, Filomat, 32(19) (2018), 6675{6689.
[33] M.M. Zahedi, R.A. Borzooei, H. Hasankhani, Some results on hyper K-algebras, Scientiae
Mathematicae, 3(1) (2000), 53{59.
[34] M.M. Zahedi, R.A. Borzooei, H. Rezaei, Some classi cations of hyper K-algebras of order 3,
Scientiae Mathematicae Japonicae, 53(1) (2001), 133{142.
[35] M.M. Zahedi, L. Torkzadeh, R.A. Borzooei, Hyper I-algebras and polygroups, Quasigroups
and Related Systems (Poland), 11 (2004), 103{113.
[36] O. Zahiri, R.A. Borzooei, M. Bakhshi, Quotient hyper residuated lattices, Quasigroups and
Related Systems, 20 (2012), 125{138.
Journal of Algebraic Hyperstructures and Logical Algebras
Volume 1, Issue 3
August 2020
Pages 31-50

Files

Share

How to cite

Statistics