Commutative ideals of BCI-algebras based on Łukasiewicz fuzzy sets

Document Type : Original Article


1 University of Hatef, Zahedan, Iran

2 Department of Mathematics, University of Sistan and Baluchestan, Zahedan 98131, Iran

3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea


With the aim of applying the Łukasiewicz fuzzy set to commutative ideal in BCI-algebras, the concept of Łukasiewicz fuzzy commutative ideal is introduced, and its properties are investigated. The relationship between a Łukasiewicz fuzzy ideal and a Łukasiewicz fuzzy commutative ideal are discussed. After providing an example of a Łukasiewicz fuzzy ideal, not a Łukasiewicz fuzzy commutative ideal, conditions under which a Łukasiewicz fuzzy ideal can be a Łukasiewicz fuzzy commutative ideal are explored. Characterizations of Łukasiewicz fuzzy commutative ideals are displayed. Conditions under which $\in$-set, q-set, and O-set can be commutative ideals are found.


[1] Y.S. Huang, BCI-algebra, Science Press: Beijing, China, 2006.
[2] K. Iséki, On BCI-algebras, Mathematics Seminar Notes, 8 (1980), 125–130.
[3] K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica, 23 (1978),
[4] Y.B. Jun, Closed fuzzy ideals in BCI-algebras, Mathematica Japonica, 38(1) (1993), 199–202.
[5] Y.B. Jun, Łukasiewicz fuzzy subalgebras in BCK-algebras and BCI-algebras, Annals of Fuzzy Mathematics
and Informatics, 23(2) (2022), 213–223.
[6] Y.B. Jun, Łukasiewicz fuzzy ideals in BCK-algebras and BCI-algebras, Journal of Algebra and Related
Topics, (under review).
[7] Y.B. Jun, S.M. Hong, S.J. Kim, S. Z. Song, Fuzzy ideals and fuzzy subalgebras of BCK-algebras, Journal
of Fuzzy Mathematics, 7 (1999), 411–418.
[8] Y.B. Jun, J. Meng, Fuzzy commutative ideals in BCI-algebras, Communications of the Korean Mathematical
Society, 9(1) (1994), 19–25.
[9] Y.B. Jun, E.H. Roh, Fuzzy commutative ideals of BCK-algebras, Fuzzy Sets and Systems, 64 (1994),
[10] Y.B. Jun, S.Z. Song, Crossing cubic ideals of BCK/BCI-algebras, Journal of Algebraic Hyperstructures
and Logical Algebras, 2(1) (2021), 17–31.
[11] J. Meng, An ideal characterization of commutative BCI-algebras, Pusan Kyongnam Mathematical
Journal, 9 (1993), 1–6.
[12] J. Meng, Y.B. Jun, BCK-algebras, Kyungmoonsa Co. Seoul, Korea, 1994.
[13] J. Meng, X.L. Xin, Commutative BCI-algebras, Mathematica Japonica, 37 (1992), 569–572.
[14] P.M. Pu, Y.M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence,
Journal of Mathematical Analysis and Applications, 76 (1980), 571–599.
[15] S.Z. Song, Y.B. Jun, Łukasiewicz fuzzy positive implicative ideals in BCK-algebras, Journal of Algebraic
Hyperstructures and Logical Algebras, 3(2) (2022), 47–58.
[16] O.G. Xi, Fuzzy BCK-algebras, Mathematica Japonica, 36 (1991), 935–942.