Missouri Journal of Mathematical Sciences

Cubic Implicative Ideals of $BCK$-algebras

Tapan Senapati and K. P. Shum

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we apply the concept of cubic sets to implicative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic implicative ideals, cubic subalgebras and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic implicative ideal. We define inverse images of cubic implicative ideals and establish how the inverse images of a cubic implicative ideal become a cubic implicative ideal. Finally we introduce products of cubic $BCK$-algebras.

Article information

Missouri J. Math. Sci., Volume 29, Issue 2 (2017), 125-138.

First available in Project Euclid: 15 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 06F35: BCK-algebras, BCI-algebras [See also 03G25]
Secondary: 03G25: Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35] 94D05: Fuzzy sets and logic (in connection with questions of Section 94) [See also 03B52, 03E72, 28E10]

$BCK$-algebra cubic set cubic subalgebra cubic ideal cubic implicative ideal


Senapati, Tapan; Shum, K. P. Cubic Implicative Ideals of $BCK$-algebras. Missouri J. Math. Sci. 29 (2017), no. 2, 125--138. doi:10.35834/mjms/1513306826. https://projecteuclid.org/euclid.mjms/1513306826

Export citation


  • R. Biswas, Rosenfeld's fuzzy subgroups with interval valued membership function, Fuzzy Sets and Systems, 63.1 (1994), 87–90.
  • K. Iseki and S. Tanaka, An introduction to the theory of $BCK$-algebras, Math. Japonica, 23 (1978), 1–26.
  • C. Jana and T. Senapati, Cubic $G$-subalgebras of $G$-algebras, Ann. Pure App. Math., 10.1 (2015), 105–115.
  • Y. B. Jun, C. S. Kim, and K. O. Yang, Cubic sets, Ann. Fuzzy Math. Inform., 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., 44 (2010), 239–250.
  • Y. B. Jun and K. J. Lee, Closed cubic ideals and cubic $o$-subalgebras in $BCK/BCI$-algebras, Applied Mathematical Sciences, 4 (2010), 3395–3402.
  • Y. B. Jun, K. J. Lee, and M. S. Kang, Cubic structures applied to ideals of $BCI$-algebras, Comput. Math. Appl., 62 (2011), 3334–3342.
  • Y. B. Jun, G. Muhiuddin, M. A. Ozturk, and E. H. Roh, Cubic soft ideals in $BCK/BCI$-algebras, J. Comput. Anal. Appl., 22 (2017), 929–940.
  • Y. B. Jun, G. Muhiuddin, and A. M. Al-roqi, Ideal theory of $BCK/BCI$-algebras based on double-framed soft sets, Appl. Math. Inf. Sci., 7 (2013), 1879–1887.
  • G. Muhiuddin and A. M. Al-roqi, Cubic soft sets with applications in $BCK/BCI$-algebras, Annals of Fuzzy Mathematics and Informatics, 8 (2014), 291–304.
  • G. Muhiuddin, F. Feng, and Y. B. Jun, Subalgebras of $BCK/BCI$-algebras based on cubic soft sets, The Scientific World Journal, 2014 (2014), Article ID 458638, 9 pages, \tt.
  • G. Muhiuddin and A. M. Al-roqi, Subalgebras of $BCK/BCI$-algebras based on $(\alpha, \beta)$-type fuzzy sets, J. Comput. Anal. Appl., 18.6 (2015), 1057–1064.
  • T. Senapati, C. S. Kim, M. Bhowmik, and M. Pal, Cubic subalgebras and cubic closed ideals of $B$-algebras, Fuzzy Inf. Eng., 7.2 (2015), 129–149.
  • T. Senapati, Cubic $BF$-subalgebras of $BF$-algebras, An. Univ. Oradea Fasc. Mat., 23.1 (2016), 97–105.
  • T. Senapati, Cubic structure of $BG$-subalgebras of $BG$-algebras, J. Fuzzy Math., 24.1 (2016), 151-162.
  • L. A. Zadeh, Fuzzy sets, Inform. and Control, 8.3 (1965), 338–353.
  • L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. I, Inform. Sci., 8 (1975), 199–249.