Internal and External Cubic Subalgebras of BCK/BCI-Algebras

Hashem Bordbar, Mohammad Mehdi Zahedi, Young Bae Jun


A characterization of cubic subalgebra is established. The notion of internal and external cubic subalgebra in BCK/BCI-algebra and several properties are investigated. The R-union, R-intersection, P-union, and P-intersection of internal and external cubic subalgebras in BCK/BCI-algebra are discussed.


Cubic subalgebra, internal cubic subalgebra, external cubic subalgebra.

Full Text:



S. S. Ahn. Y. H. Kim and J. M. Ko, Cubic subalgebras and filters of CI-algebras, Honam Math. J. Vol. 36(1) (2014) 43-54.

M. Akram, N. Yaqoob and M. Gulistan, Cubic KU-subalgebras, Int. J. Pure Appl. Math. Vol. 89(5) (2013) 659-665.

H. Bordbar, M. R. Bordbar, Y. B. Jun, A Generalization of Semidetached Subalgebras in BCK/BCI-algebras, New Math. Nat. Comput., Vol.15 (03), (2019) 489-501.

H. Bordbar, R. A. Borzooei, and Y. B. Jun, Uni-Soft Commutative Ideals and Closed UniSoft Ideals in BCI-Algebras, New Math. Nat. Comput., Vol. 14(2), (2018) 1-13.

H. Bordbar, S. S. Ahn, M. M. Zahedi and Y. B. Jun, Semiring structures based on meet and plus ideals in lower BCK-semilattices, J. Comput. Anal. Appl., Vol 23(5), (2017) 945-954.

H. Bordbar, I. Cristea, Height of Prime Hyperideals in Krasner Hyperrings, Filomat, Vol. 19 (31), 6153-6163.

H. Bordbar, I. Cristea, Regular Parameter Elements and Regular Local Hyperrings, Mathematics 9 (3), 243.

H. Bordbar, M. Novak, I. Cristea, Properties of reduced meet ideals in lower BCKsemilattices, APLIMAT 2018, 97-109.

H. Bordbar, I. Cristea, M. Novak, Height of hyperideals in Noetherian Krasner hyperrings. Univ. Politeh. Buchar. Sci. Bull. Ser. A Appl. Math. Phys., Vol. 79, 2017 31-42.

H. Bordbar, Y. B. Jun, S.Z. Song, Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras, Mathematics, Vol. 8 (4), 2020, 567.

H. Bordbar, S. Khademan, R. A. Borzooei, M. M. Zahedi and Y. B. Jun, Double-framed soft set theory applied to hyper BCK-algebras,

H. Bordbar, G Muhiuddin, A. M Alanazi, Primeness of Relative Annihilators in BCKAlgebra, Symmetry, Vol. 12 (2), 2020, 286.

H. Bordbar and M. M. Zahedi, A Finite type Closure Operations on BCK-algebras, Applied Math. Inf. Sci. Lett., Vol.4(2), (2016) 1-9.

H. Bordbar and M. M. Zahedi, Semi-prime Closure Operations on BCK-algebra, Commun. Korean Math. Soc., Vol 30(4), (2015) 385-402.

H. Bordbar, M. M. Zahedi, S. S. Ahn and Y. B. Jun, Weak closure operations on ideals of BCK-algebras, J. Comput. Anal. Appl., Vol 23(2), (2017), 51-64.

H. Bordbar, M. M. Zahedi and Y. B. Jun, Relative annihilators in lower BCK-semilattices, Math. Sci. Lett., Vol. 6(2) (2017) 1-7.

Y. Huang, BCI-algebra, Science Press, Beijing, 2006.

K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica Vol. 23(1), (1978) 1-26.

Y. B. Jun and A. Khan, Cubic ideals in semigroups, Honam Math. J. Vol. 35(4), (2013) 607-623.

Y. B. Jun, C. S. Kim and J. G. Kang, Cubic q-ideals of BCI-algebras, Ann. Fuzzy Math. Infom. Vol. 1(1), (2011) 25-34.

Y. B. Jun, C. S. Kim and K. O. Yang, Cubic sets, Ann. Fuzzy Math. Infom. Vol. 4(1), (2012) 83-98.

Y. B. Jun, C. S. Kim and M. S. Kang, Cubic subalgebras and ideals of BCK/BCI-algebras, Far East J. Math. Sci. Vol. 44(2), (2010) 239-250.

Y. B. Jun and K. J. Lee, Closed cubic ideals and cubic o-subalgebras in BCK/BCI-algebras, Appl. Math. Sci. Vol. 4(68) (2010) 3395-3402.

Y. B. Jun, K. J. Lee and M. S. Kang, Cubic structures applied to ideals of BCI-algebras, Comput. Math. Appl. Vol. 62 (2011) 3334-3342.

M. Khan, Y. B. Jun, M. Gulistan and N. Yaqoob, The generalized version of Jun’s cubic sets in semigroups, J. Intell. Fuzzy Syst. 28 (2015) 947-960.

J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co. Seoul, 1994.

G. Muhiuddin and A. M. Al-roqi, Cubic soft sets with applications in BCK=BCI-algebras, Ann. Fuzzy Math. Inform. 8(2) (2014) 291-304.

T. Senapati, C. S. Kim, M. Bhowmik and M. Pal, Cubic subalgebras and cubic closed ideals of B-algebras, Fuzzy Inf. Eng. 7 (2015) 129-149.

N. Yaqoob, S. M. Mostafa and M. A. Ansari, On cubic KU-ideals of KU-algebras, ISRN Algebra 2013, Art. ID 935905, 10 pp.



  • There are currently no refbacks.

Journal of the Indonesian Mathematical Society
Mathematics Department, Universitas Gadjah Mada
Senolowo, Sinduadi, Mlati, Sleman Regency, Special Region of Yogyakarta 55281, Telp. (0274) 552243

p-ISSN: 2086-8952 | e-ISSN: 2460-0245

Journal of the Indonesian Mathematical Society is licensed under a Creative Commons Attribution 4.0 International License

web statistics
View My Stats