Positive implicative True-False ideals in BCK-algebras

Document Type : Research Paper


1 Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

2 Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Republic of Korea

3 Hatef Higher Education Institute, Zahedan, Iran


In BCK-algebra, the concept of  a positive implicative $T\&F$-ideal is introduced, and further several properties are  investigated. The relationship between  $T\&F$-ideals and  positive implicative $T\&F$-ideals is established, and an example is given to reveal that  a $T\&F$-ideal is not a positive implicative $T\&F$-ideal. Various conditions under which  a $T\&F$-ideal can be a positive implicative $T\&F$-ideal are explored and various characterizations of a positive  implicative $T\&F$-ideal are studied. The extended property of a positive implicative $T\&F$-ideal is constructed.


[1] H. Bordbar, R. A. Borzooei, Y. B. Jun, Uni-soft commutative ideals and closed uni-soft ideals in BCI-algebras, New Mathematics and Natural Computation, 14(2) (2018), 235{247.
[2] R. A Borzooei, M. Mohseni Takallo, F. Smarandache, Y. B. Jun, Positive implicative BMBJ-neutrosophic ideals in BCK-algebras, Neutrosophic Set and Systems, 23 (2018), 126{141.
[3] R. A. Borzooei, X. Zhang, F. Smarandache, Y. B. Jun, Commutative generalized neutrosophic ideals in BCK-algebras, Symmetry, 10(8) (2018), 350.
[4] M. B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, 21 (1987), 1{17.
[5] Y. Huang, BCI-algebra, Science Press: Beijing, China, (2006).
[6] K. Iseki, S. Tanaka, An introduction to the theory of BCK-algebras, Mathematics, 23 (1978), 1{26.
[7] Y. B. Jun, C. S. Kim, K. O. Yang, Cubic sets, Annals of Fuzzy Mathematics and Informatics, 4(1) (2012), 83{98.
[8] J. Meng, Y. B. Jun, BCK-algebras, Kyungmoon Sa Co.: Seoul, Korea, (1994).
[9] M. Mohseni Takallo, M. Aaly Kologani, MBJ-neutrosophic  lters of equality algebras, Journal of Algebraic Hyperstructures and Logical Algebras, 1(2) (2020), 57{75.
[10] M. Mohseni Takallo, R. A. Borzooei, Y. B. Jun, True-False structures and its applications in groups and BCK/BCI-algebras, Bulletin of the Section of Logic, to appear.
[11] M. Mohseni Takallo, R. A. Borzooei, G. R. Rezaei, Y. B. Jun, True-False ideals of BCK/BCI-algebras, submitted.
[12] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.
Volume 11, Issue 3 - Serial Number 23
Special Issue dedicated to Prof. Mashaallah Mashinchi.
November 2022
Pages 69-85
  • Receive Date: 07 March 2022
  • Revise Date: 04 May 2022
  • Accept Date: 30 May 2022
  • First Publish Date: 16 June 2022