Document Type : Original Manuscript

Authors

1 Department of Mathematics, Ege University, Bornova, Izmir, Turkey.

2 Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.

10.22044/jas.2022.11512.1582

Abstract

The aim of this study is to introduce (anti) fuzzy ideals of a Sheffer stroke BCK-algebra. After describing an anti fuzzy subalgebra and an anti fuzzy (sub-implicative) ideal of a Sheffer stroke BCK-algebra, the relationships of these structures are demonstrated. Also, a t-level cut and a complement of a fuzzy subset are defined and some properties are investigated. An implicative Sheffer stroke BCK-algebra is defined and it is proved that a fuzzy subset of an implicative Sheffer stroke BCK-algebra is an anti fuzzy ideal if and only if it is an anti fuzzy sub-implicative ideal of this algebraic structure. A fuzzy congruence and a fuzzy quotient set of a Sheffer stroke BCK-algebra are studied in details and it is shown that there is a bijection between the set of fuzzy ideals and the set of fuzzy congruences on this algebraic structure. Finally, Cartesian product of fuzzy subsets of a Sheffer stroke BCK-algebra is determined and it is expressed that the Cartesian product of two anti fuzzy ideals of this algebraic structure is anti fuzzy ideal.

Keywords

###### ##### References
1. M. Akram, Spherical fuzzy K-algebras, J. algebr. hyperstrucres log. algebr., 2(3) (2021), 85–98.
2. M. Akram, and N. O. Alshehri, Fuzzy K-ideals of K-algebras, Ars Combin., 99 (2011), 399–413.
3. M. Akram, B. Davvaz, and F. Feng, Intuitionistic fuzzy soft K-algebras, Math. Comput. Sci., 7(3) (2013), 353–365.
4. R. Biswas, Fuzzy subgroups and anti fuzzy subgroups, Fuzzy Sets Syst. 35 (1990), 121–124.
5. I. Chajda, Sheffer operation in ortholattices, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 44(1) (2005), 19–23.
6. S. M. Hong and Y. B. Jun, Anti fuzzy ideals in BCK-algebras, Kyungpook Math. J., 38 (1998), 145–150.
7. Y. Imai, K. Ise´ki, On axiom systems of proposional calculi, XIV. Proc. Jpn. Acad., Ser. A, Math. Sci., 42 (1966), 19–22.
8. W. McCune, et.al. Short single axioms for Boolean algebra, J. Autom. Reason., 29(1) (2002), 1–16.
9. T. Oner, T. Katican, A. Borumand Saeid, (Fuzzy) Filters of Sheffer stroke BL Algebras, Kragujev. J. Math., 47(1) (2023), 39–55.
10. T. Oner, T. Katican, A. Borumand Saeid, Relation between Sheffer stroke operation and Hilbert algebras, Categories Gen. Algebraic Struct. with Appl., 14(1) (2021), 245–268.
11. T. Oner, T. Katican, A. Borumand Saeid, M. Terziler, Filters of strong Sheffer stroke non-associative MV-algebras, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat., 29(1) (2021), 143–164.
12. T. Oner, T. Katican, A. Ulker, Interval Sheffer stroke Basic algebras, TWMS J. of Apl. & Eng. Math., 9(1) (2019), 134–141.
13. T. Oner, T. Kalkan, A. Borumand Saeid, Class of Sheffer stroke BCK-Algebras, Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 30(1) (2022), 247–269.
14. T. Oner, T. Kalkan, N. Kircali Gursoy, Sheffer stroke BG-algebras, Int. J. Maps Math., 4(1) (2021), 27–39.
15. H. M. Sheffer, A set of five independent postulates for Boolean algebras, with application to logical constants, Trans. Am. Math. Soc., 14(4) (1913), 481–488.
16. O. G. Xi, Fuzzy BCK-algebras, Math. Japon., 36 (1991), 935–942.
17. L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353