Ultra and Involution Ideals in $BCK$-algebras

Document Type : Research Paper


1 Farhangian University

2 University of Beheshti


In this paper, we define the notions of ultra and involution ideals in $BCK$-algebras. Then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. Specially, we show that in a bounded implicative $BCK$-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower $BCK$-semilattice, the notions of prime ideals and ultra ideals are coincide.


[1] R. A. Borzooei, J. Shohani, Fraction structures on bounded implicative BCK-algebras, Word Academy of
Science, Engineering and Thechnology, 49, 1084-1090 (2009).
[2] O. Heubo-Kwegna and J. B. Nganou, A Global Local Principle for BCK-modules, International Journal of
Algebra, 5(14), 691-702 (2011).
[3] Y. Huang, BCI-algebra, Science Press, Beijing (2006).
[4] Y. Imai and K. Iseki, On axiom systems of propositional calculi, Proceedings of the Japan Academy, 42,
19-21 (1966).
[5] K. Iseki, On ideals in BCK-algebras, Mathematics Seminar Notes, 3, 1-12 (1975).
[6] K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Mathematica Japonica, 21, 351-366 (1976).
[7] P. Jiayin, Normed BCK-algebras, Advances in Mathematics, 4, 492-500 (2011).
[8] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co, seoul, (1994).
[9] Z. M. Samaei and M. A. N. Azadani, A Class of BCK-algebras, International Journal of Algebra, 28, 1379-1385 (2011).