The concept of a commutative soju ideal in a BCK-algebra and a BCI-algebra is introduced, and their properties are investigated. The relationship between a soju ideal and a commutative soju ideal are discussed, and examples to show that any soju ideal may not be a commutative soju ideal are provided. Conditions for a soju ideal to be a commutative soju ideal are considered, and characterizations of a commutative soju ideal are studied. A new commutative soju ideal using the given commutative soju ideal is maded, and the extension property for a commutative soju ideal is established. A commutative soju ideal is established by using a commutative ideal of a BCI-algebra. The notion of a closed soju ideal in a BCI-algebra is also introduced, and it is used in studying the characterization of a commutative soju ideal.
Citation: Seok-Zun Song, Hee Sik Kim, Young Bae Jun. Commutative ideals of BCK-algebras and BCI-algebras based on soju structures[J]. AIMS Mathematics, 2021, 6(8): 8567-8584. doi: 10.3934/math.2021497
The concept of a commutative soju ideal in a BCK-algebra and a BCI-algebra is introduced, and their properties are investigated. The relationship between a soju ideal and a commutative soju ideal are discussed, and examples to show that any soju ideal may not be a commutative soju ideal are provided. Conditions for a soju ideal to be a commutative soju ideal are considered, and characterizations of a commutative soju ideal are studied. A new commutative soju ideal using the given commutative soju ideal is maded, and the extension property for a commutative soju ideal is established. A commutative soju ideal is established by using a commutative ideal of a BCI-algebra. The notion of a closed soju ideal in a BCI-algebra is also introduced, and it is used in studying the characterization of a commutative soju ideal.
[1] | U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Comput. Math. Appl., 59 (2010), 3458-3463. |
[2] | M. Agarwal, K. K. Biswasa, M. Hanmandlu, Generalized intuitionistic fuzzy soft sets with applications in decision-making, Appl. Soft Comput., 13 (2013), 3552-3566. doi: 10.1016/j.asoc.2013.03.015 |
[3] | H. Aktas, N. Cagman, Soft sets and soft groups, Inform. Sci., 177 (2007), 2726-2735. |
[4] | K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87-96. |
[5] | K. T. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets Syst., 61 (1994), 137-142. doi: 10.1016/0165-0114(94)90229-1 |
[6] | Z. Bashir, J. Wątróbski, T. Rashid, W. Salabun, J. Ali, Intuitionistic-fuzzy goals in Zero-Sum multi criteria matrix games, Symmetry, 9 (2017), 158. doi: 10.3390/sym9080158 |
[7] | S. K. De, R. Biswas, A. R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets Syst., 117 (2001), 209-213. doi: 10.1016/S0165-0114(98)00235-8 |
[8] | F. Feng, Y. B. Jun, X. Zho, Soft semirings, Comput. Math. Appl., 56 (2008), 2621-2628. |
[9] | Y. Huang, BCI-algebra, Science Press, Beijing, 2006. |
[10] | Y. Jiang, Y. Tang, Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Appl. Math. Modell., 35 (2011), 824-836. doi: 10.1016/j.apm.2010.07.038 |
[11] | Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl., 56 (2008), 1408-1413. |
[12] | Y. B. Jun, Intuitionistic fuzzy finite switchboard state machines, J. Appl. Math. Comput., 20 (2006), 315-325. doi: 10.1007/BF02831941 |
[13] | Y. B. Jun, K. H. Kim, Intuitionistic fuzzy ideals of BCK-algebras, Internat. J. Math. Math. Sci., 24 (2000), 839-849. doi: 10.1155/S0161171200004610 |
[14] | Y. B. Jun, K. J. Lee, J. Zhan, Soft p-ideals of soft BCI-algebras, Comput. Math. Appl., 58 (2009), 2060-2068. doi: 10.1016/j.camwa.2009.07.072 |
[15] | Y. B. Jun, C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci., 178 (2008), 2466-2475. |
[16] | Y. B. Jun, S. Z. Song, E. H. Roh, Soju structures with applications in BCK/BCI-algebras, Appl. Math. J. Chinese Univ. Ser. B, submitted. |
[17] | P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077-1083. doi: 10.1016/S0898-1221(02)00216-X |
[18] | J. Meng, Commutative ideals in BCK-algebras, Pure Appl. Math., (in Chinese) 9 (1991), 49-53. |
[19] | J. Meng, An ideal characterization of commutative BCI-algebras, Pusan Kyongnam Math. J., 9 (1993), 1-6. |
[20] | J. Meng, On ideals in BCK-algebras, Math. Japon., 40 (1994), 143-154. |
[21] | J. Meng, Y. B. Jun, BCK-algebras, Kyungmoonsa Co. Seoul, Korea, 1994. |
[22] | J. Meng, X. L. Xin, Commutatitie BCI-algebras, Math. Japon., 37 (1992), 569-572. |
[23] | D. Molodtsov, Soft set theory - First results, Comput. Math. Appl., 37 (1999), 19-31. |
[24] | A. K. Srivastava, S. P. Tiwari, IF-topologies and IF-automata, Soft Comput., 14 (2010), 571-578. doi: 10.1007/s00500-009-0427-z |
[25] | M. Touqeer, N. Cagman, On some properties of $p$-ideals based on intuitionistic fuzzy sets, Cogent Math., 3 (2016), 1210001. doi: 10.1080/23311835.2016.1210001 |
[26] | Z. Zhang, A rough set approach to intuitionistic fuzzy soft set based decision making, Appl. Math. Modell., 36 (2012), 4605-4633. doi: 10.1016/j.apm.2011.11.071 |