On hyper BI-algebras

Document Type : Original Article


Faculty of Medicine, Tehran Medical Sciences, Islamic Azad University, Tehran, Iran


In this paper, we introduce the notion of hyper BI-algebra and investigate some properties of it. Also, we state and prove some theorems which determine the relationship among $R/ C/ D/ T$ and V-hyper BI-algebras under some conditions. Then we study the relation among hyper BI-algebra with some of other hyper logical algebras such as hyper BCI/BCK/K/B/BCC-algebras and show that under which condition these hyper structures coincide. In addition, we define hyper subalgebra and (weak) ideal of a hyper BI-algebra and obtain some results and the relation between them. Finally, we construct the quotient structure of hyper BI-algebra and examine the isomorphism theorems.


[1] J.C. Abbott, Semi-Boolean algebras, Matematicki Vesnik, 4 (1967), 177{198.
[2] S.S. Ahn, J.S. Han, On BP-algebras, Hacettepe Journal of Mathematics and Statistics, 42
(2013), 551{557.
[3] A. Borumand Saeid, H.S. Kim, A. Rezaei, On BI-algebras, Analele Stiinti ce ale Universitatii
Ovidius Constanta, 25(1) (2017), 177{194.
[4] R.A. Borzooei, M. Aaly Kologani, An overview of hyper logical algebras, Journal of Algebraic
Hyperstructures and Logical Algebras, 1(3) (2020), 31{50.
[5] R.A. Borzooei, W.A. Dudek, N. Kouhestani, On hyper BCC-algebras, International Journal
of Mathematics and Mathematical Sciences, (2006), 1{18.
[6] R.A. Borzooei, A. Hasankhani, M.M. Zahedi, Y.B. Jun, On hyper K-algebras, Japanese Jour-
nal of Mathematics, 52(1) (2000), 113{121.
[7] W.Y. Chen, J.S. Oliveira, Implication algebras and the metropolis rota axioms for cubic lat-
tices, Journal of Algebra, 171 (1995), 383{396.
[8] P. Corsini, Prolegomena of hypergroup theory, (Second Edition), Aviani Editor, 1993.
[9] W.A. Dudek, On BCC-algebras, Logique et Analyse. Nouvelle Series, 33(129-130) (1990),
[10] J.C. Endam, On JB-Semigroups and hyper B-Algebras, Dissertation, MSU-IIT, 2014.
[11] K. Iseki, On BCI-algebras, Mathematics Seminar Notes, 8 (1980), 125{130.
[12] 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.
[13] C.B. Kim, H.S. Kim, On BM-algebras, Scientiae Mathematicae Japonicae, 63(3) (2006), 421{
[14] C.B. Kim, H.S. Kim, On BG-algebras, Demonstratio Mathematica, 41 (2008), 497{505.
[15] C.B. Kim, H.S. Kim, On BO-algebras, Mathematica Slovaca, 62 (2012), 855{864.
[16] C.B. Kim, H.S. Kim, On BN-algebras, Kyungpook Mathematical Journal, 53 (2013), 175{184.
[17] Y. Komori, The variety generated by BCC-algebras is  nitely based, Reports of Faculty of
Science, Shizuoka University, 17 (1983), 13{16.
[18] Y. Komori, The class of BCC-algebras is not a variety, Mathematica Japonica, 29(3) (1984),
[19] F. Marty, Sur une generalization de la notion de group, 8th Congress Math. Scandinaves,
Stockholm, (1934), 45{49.
[20] J. Neggers, H.S. Kim, On D-algebras, Mathematica Slovaca, 49 (1999), 19{26.
[21] J. Neggers, H.S. Kim, On B-algebras, Matematiki Vesnik, 54 (2002), 21{29.
[22] J. Neggers, H.S. Kim, A fundamental theorem of B-homomorphism for B-algebras, Interna-
tional Journal of Mathematics, 2 (2002), 215{219.
[23] A. Rezaei, F. Smarandache The neutrosophic triplet of BI-algebras, Neutrosophic Sets and
Systems, 33 (2020), 313{321.
[24] A.L.O. Vicedo, J.P. Vilela, Hyper B-ideals in hyper B-algebra, Journal of Ultra Scientist of
Physical Sciences, 29(8) (2017), 339{347.
[25] A. Walendziak, On BF-algebras, Mathematica Slovaca, 57 (2007), 119{128.
[26] A. Wronski, BCK-algebras do not form a variety, Mathematica Japonica, 28 (1983), 211{213.
[27] X.L. Xin, Hyper BCI-algebras, Discuss Mathematics Society, 26 (2006), 5{19.