Fuzzy Implicative Ideals in KU-algebras

Year 2018, Issue: 22, 82 - 91, 26.03.2018

Samy Mohammed Mostafa , Ola Wageeh Abd El-baseer

Abstract

In this paper, we consider KU-implicative ideal (briefly
implicative ideal) in KU-algebras. The notion of fuzzy implicative ideals in KU-algebras
are introduced, several appropriate examples are provided and their some
properties are investigated. The image and the inverse image of fuzzy
implicative ideals in KU-algebras are defined and how the image and the inverse
image of fuzzy implicative
ideals in KU-algebras become fuzzy implicative ideals are studied. Moreover,
the Cartesian product of fuzzy implicative ideals in Cartesian product of KU-algebras
are given.

Keywords

Fuzzy implicative ideal , image (inverse image ) of fuzzy implicative ideals , Cartesian product of fuzzy implicative ideals

References

• [1] P. Bhattacharye and N. P. Mukheriee, Fuzzy relations and fuzzy group inform, Sci, 36(1985), 267-282.
• [2] W. A. Dudek, The number of sub-algebras of finite BCC algebras, Bull. Inst. Math. Academia Sinica, 20(1992), 129-136.
• [3] W. A. Dudek, On proper BCC-algebras, Bull. Inst. Math. Academia Sinica, 20 (1992), 137-150.
• [4] W. A. Dudek and Y. B. Jun, Fuzzy BCC-ideals in BCC-algebra, Math. Montisnigri, 10 (1999), 21-30.
• [5] W. A. Dudek and Y. B. Jun, Normalizations of fuzzy BCC-ideals in BCC-algebras, Math Moravica, 3(1999),17-24.
• [6] W. A. Dudek, Y. B. Jun and S. M. Hong, On fuzzy topological BCC-algebras, discussions math (algebra and stochastic methods), 20 (2000), (in print).
• [7] W. A. Dudek, Y. B. Jun and Z. Stojakovic, On fuzzy ideals in BCC-algebras, fuzzy sets and systems (in print).
• [8] W. A. Dudek and X. Zhang, On ideals and congurences in BCC-algebras, Czechoslovak Math. Journal, 48 (1998), no. 123,12-29.
• [9] W. A. Dudek, On proper BCC-algebras, Bull. Inst. Math. Academia Sinica 20(1992), 137-150.
• [10] S. M. Hong and Y. B. Jun, Fuzzy and level subalgebras of BCK (BCI) algebras Pusan Kyongnam Math. J.(Presently, East Asain Math. J.),7(2) (1991), 185-190.
• [11] K. Iseki. On ideals in BCK-algebras Math. Sem. Notes. 3 (1975) 1-12
• [12] K. Iseki, On BCI-algebras. Math.Sem. Notes 8 (1980) 125–130
• [13] K. Iseki. K. and S. Tanaka, An introduction to the theory of BCK-algebras. Math. Japon 23(1978) 1- 26.
• [14] Y. B. Jun, A. M. Hong, A. T. Kim and S. Z. Song, Fuzzy ideals and fuzzy subalgebras of BCK-algebras, J.fuzzy Math,7(2) (1999), 411-418.
• [15] Y. B. Jun, S. M. Hong and E. H. Roh, Fuzzy characteristic subalgebras/ideals of a Bck-algebra, Pusan Kyongnam Math J. (presently East Asian Math. J.), 9(1) (1993), 127-132.
• [16] Y. Komori, The class of BCC-algebras is not a variety, Math. Japan, 29(1984), 391-394.
• [17 ] D. S. Malik and J. N. Mordeson, Fuzzy relation on rings and groups, fuzzy sets and systems, 43(1991),117-123.
• [18] L. Liu Y, J. Meng. And Y. Xu, BCI-implicative ideals of BCI-algebras, Information Sciences(2007) 1 - 10.
• [19] J. J. Meng. And L. Xin, Positive implicative BCI-algebras, Pure Appl. Math. 9(1)(1993)-1922.
• [20] J. Meng. On ideals in BCK.algebras_Math. Japon, 40(1994) 143-154
• [21] J. Meng, An ideal Characterizaton of commutative BCI-algebars. Pusan Kyongnam Math J. 9:1(1993) 1-6
• [22] J. Meng. andL. Xin, Commutative BCI-algebras, Japonica 37 (1992), 569-572.
• [23] J. Meng and L. Xin, Implicative BCI-algebras, Selected papers on BCK-and BCI-algebras, Vol.1 (1992), pp. 50-53.
• [24] J. Meng. And L. Xin, Positive implicative BCI-algebras, Selected papers on BCK-and BCI-algebras, Vol.1 (1992), pp. 45-49.
• [25] S. M. Mostafa, M. Abd-Elnaby and M. Yousef. "Fuzzy ideals of KU-Algebras" International Math. Forum, Vol. 6, (2011), no.63, 3139-3149.
• [26] S. M., Mostafa., F.F.Kareem.,N-Fold Commutative KU-Algebras International Journal of Algebra, Vol. 8, 2014, no. 6, 267 – 275
• [27] A. E. Radwan, S. M. Mostafa., F. A. Ibrahem & F. F. Kareem,Topology spectrum of a KU-Algebra. Journal of New Theory 5 (2015) Year: 2015, Number: 8, Pages: 78-91
• [28] S. M. Mostafa, fuzzy implicative ideal of BCK-algebras Fuzzy Sets and Systems 87 (1997) 361-368
• [29] C. Prabpayak and U. Leerawat , On ideals and congruence in KU-algebras, Scientia Magna Journal, Vol. 5(2009), No.1, 54-57.
• [30] C. Prabpayak and U. Leerawat, On isomorphisms of KU-algebras, Scientia Magna journal, 5(2009), no.3, 25-31
• [31] A. Rosenfeld, Fuzzy Group. J. Math. Anal. Appl., 35(1971), 512-517.
• [32] O. G Xi, fuzzy BCK-algebras, Math. Japan, 36(1991).935-942.
• [33] L. A. Zadeh, Fuzzy sets, inform. and [1] K. Iseki, On ideals in BCK-algebras Math. Sem. Notes. 3 (1975) 1-12.