ON ($\epsilon, \epsilon \vee q$)-FUZZY IDEALS OF BCI-ALGEBRAS

Document Type : Research Paper

Authors

1 Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei Province,445000, P. R. China

2 Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Korea

3 Department of Mathematics, Yazd University, Yazd, Iran

Abstract

The aim of this paper is to introduce the notions of ($\epsilon, \epsilon \vee q$)-
fuzzy p-ideals, ($\epsilon, \epsilon \vee q$)-fuzzy q-ideals and ($\epsilon, \epsilon \vee q$)-fuzzy a-ideals in BCIalgebras and to investigate some of their properties. Several characterization
theorems for these generalized fuzzy ideals are proved and the relationship
among these generalized fuzzy ideals of BCI-algebras is discussed. It is shown
that a fuzzy set of a BCI-algebra is an ($\epsilon, \epsilon \vee q$)-fuzzy a-ideal if and only if it
is both an ($\epsilon, \epsilon \vee q$)-fuzzy p-ideal and an ($\epsilon, \epsilon \vee q$)-fuzzy q-ideal. Finally, the concept of implication-based fuzzy a-ideals in BCI-algebras is introduced and,
in particular, the implication operators in Lukasiewicz system of continuousvalued
logic are discussed.

Keywords

References

[1] S. K. Bhakat, (2, 2_ q)-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and
Systems, 112 (2000), 299-312.
[2] S. K. Bhakat and P. Das, (2, 2 _ q)-fuzzy subgroups, Fuzzy Sets and Systems, 80 (1996),
359-368.
[3] C. C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc., 88 (1958),
467-490.
[4] B. Davvaz, (2, 2 _q)-fuzzy subnear-rings and ideals, Soft Computing, 10 (2006), 206-211.
[5] B. Davvaz and P. Corsini, Redefined fuzzy Hv-submodules and many valued implications,
Inform. Sci., 177 (2007), 865-875.
[6] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous
t-norms, Fuzzy Sets and Systems, 124 (2001), 271-288.
[7] P. H´ajek, Metamathematics of fuzzy logic, Kluwer Academic Press, Dordrecht, 1998.
[8] Y. Imai and K. Iseki, On axiom system of propositional calculus, Proc. Japan Acad., 42
(1966), 19-22.
[9] A. Iorgulescu, Some direct ascendents of wajsberg and MV algebras, Sci. Math. Japon., 57
(2003), 583-647.
[10] A. Iorgulescu, Pseudo-Iseki algebras. connection with pseudo-BL algebras, Multiple-Valued
Logic and Soft Computing, 11 (2005), 263-308.
[11] K. Iseki, An algebra related with a propositional calculus, Proc. Japan Acad., 42 (1966),
26-29.
[12] K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon., 21 (1966), 351-366.
[13] Y. B. Jun, Closed fuzzy ideals in BCI-algebras, Math. Japon., 38 (1993), 199-202.
[14] Y. B. Jun, On ( , )-fuzzy ideals of BCK/BCI-algebras, Sci. Math. Japon., 60 (2004), 613-
617.
[15] Y. B. Jun, On ( , )-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc., 42
(2005), 703-711.
[16] Y. B. Jun and J. Meng, Fuzzy p-ideals in BCI-algebras, Math. Japon, 40 (1994), 271-282.
[17] Y. B. Jun and J. Meng, Fuzzy commutative ideals in BCI-algebras, Comm. Korean Math.
Soc., 9 (1994), 19-25.
[18] Y. B. Jun and W. H. Shim, Fuzzy strong implicative hyper BCK-ideals of hyper BCK-algebras,
Inform. Sci., 170 (2005), 351-361.
[19] Y. B. Jun, Y. Xu and J. Ma, Redefined fuzzy implicative filters, Inform. Sci., 177 (2007),
1422-1429.
[20] T. D. Lei and C. C. Xi, p-radical in BCI-algebras, Math. Japon., 30 (1995), 511-517.
[21] Y. L. Liu, Some results on p-semisimple BCI-algebras, Math. Japon, 30 (1985), 511-517.
[22] Y. L. Liu, S. Y. Liu and J. Meng, FSI-ideals and FSC-ideals of BCI-algebras, Bull. Korean
Math. Soc., 41 (2004), 167-179.
[23] Y. L. Liu and J. Meng, Fuzzy q-ideals of BCI-algebras, J. Fuzzy Math., 8 (2000), 873-881.
[24] Y. L. Liu and J. Meng, Fuzzy ideals in BCI-algebras, Fuzzy Sets and Systems, 123 (2001),
227-237.
[25] Y. L. Liu, J. Meng, X. H. Zhang and Z. C. Yue, q-ideals and a-ideals in BCI-algebras, SEA
Bull. Math., 24 (2000), 243-253.
[26] Y. L. Liu, Y. Xu and J. Meng, BCI-implicative ideals of BCI-algebras, Inform. Sci., 177
(2007), 4987-4996.
[27] Y. L. Liu and X. H. Zhang, Fuzzy a-ideals in BCI-algebras, Adv. in Math.(China), 31 (2002),
65-73.
[28] J. Meng and X. Guo, On fuzzy ideals in BCK-algebras, Fuzzy Sets and Systems, 149 (2005),
509-525.
[29] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co., Seoul, Korean, 1994.
[30] D. Mundici, MV algebras are categorically equivalent to bounded commutative BCK-algebras,
Math. Japon., 31 (1986), 889-894.
[31] P. M. Pu and Y. M. Liu, Fuzzy topology I: Neighourhood structure of a fuzzy point and
Moore-Smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599.
[32] X. H. Yuan, C. Zhang and Y. H. Ren, Generalized fuzzy groups and many valued applications,
Fuzzy Sets and Systems, 138 (2003), 205-211.
[33] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
[34] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outine, Inform. Sci., 172
(2005), 1-40.
[35] J. Zhan and Y. L. Liu, On f-derivation of BCI-algebras, Int. J. Math. Math. Sci., 2005,
1675-1684.
[36] J. Zhan and Z. Tan, Fuzzy a-ideals of IS-algebras, Sci. Math. Japon, 58 (2003), 85-87.
[37] J. Zhan and Z. Tan, Intuitionistic fuzzy a-ideals in BCI-algebras, Soochow Math. J., 30
(2004), 207-216.
[38] X. H. Zhang, H. Jiang and S. A. Bhatti, On p-ideals of BCI-algebras, Punjab Univ. J. Math.,
27 (1994), 121-128.
Iranian Journal of Fuzzy Systems
Volume 6, Issue 1 - Serial Number 1
January and February 2009
Pages 81-94

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