Some types of $(\in,\ivq)$-interval-valued 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

Abstract

In this paper, we introduce  the notions of   interval-valued and $(\in,\ivq)$-interval-valued fuzzy ($p$-,$q$- and $a$-) ideals    of   BCI algebras   and investigate some of their properties.   We then derive characterization theorems for these generalized interval-valued fuzzy ideals  and discuss their relationship.

[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] R. Biswas, Rosenfeld’s fuzzy subgroups with interval- valued membership functions, Fuzzy
Sets and Systems, 63 (1994), 87-90.
[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] B. Davvaz and P. Corsini, On ( , )-fuzzy Hv-ideals of Hv-rings, Iranian J. Fuzzy Systems,
5(2) (2008), 35-48.
[7] W. A. Dudek, On group-like BCI-algebras, Demonstratio Math., 21 (1998), 369-376.
[8] W. A. Dudek and J. Thomys, On decompositions of BCH-algebras, Math. Japon., 35 (1990),
1131-1138.
[9] G. Deschrijver, Arithmetric operators in interval-valued fuzzy theory, Inform. Sci., 177
(2007), 2906-2924.
[10] 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.
[11] P. H´ajek, Metamathematics of fuzzy logic, Kluwer Academic Press, Dordrecht, 1998.
[12] Y. Imai and K. Is´eki, On axiom system of propositional calculus, Proc. Japan Acad., 42
(1966), 19-22.
[13] A. Iorgulescu, Some direct ascendents of Wajsberg and MV algebras, Sci. Math. Japon., 57
(2003), 583-647.
[14] A. Iorgulescu, Iseki algebras, connection with BL algebras, Soft Computing, 8 (2004), 449-
463.
[15] K. Is´eki, An algebra related with a propositional calculus, Proc. Japan Acad., 42 (1966),
26-29.
[16] K. Is´eki, On BCI-algebras, Math. Seminar Notes (now Kobe Math J.), 8 (1980), 125-130.
[17] K. Is´eki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon., 21 (1966), 351-366.
[18] K. Is´eki and S. Tanaka, An introduction to the theory of BCK- algebras, Math. Japon., 23
(1978), 1-26.

[19] Y. B. Jun, Interval-valued fuzzy subalgebras/ideals in BCK-algebras, Sci. Math., 3 (2000),
435-444.
[20] Y. B. Jun, Interval-valued fuzzy ideals in BCI- algebras, J. Fuzzy Math., 9 (2001), 807-814.
[21] Y. B. Jun, On ( , )-fuzzy ideals of BCK/BCI- algebras, Sci. Math. Japon., 60 (2004),
613-617.
[22] Y. B. Jun, On ( , )-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc., 42
(2005), 703-711.
[23] Y. B. Jun and J. Meng, Fuzzy p-ideals in BCI-algebras, Math. Japon, 40 (1994), 271-282.
[24] Y. B. Jun and J. Meng, Fuzzy commutative ideals in BCI-algebras, Comm. Korean Math.
Soc., 9 (1994), 19-25.
[25] Y. L. Liu and J. Meng, Fuzzy q-ideals of BCI-algebras, J. Fuzzy Math., 8 (2000), 873-881.
[26] Y. L. Liu and J. Meng, Fuzzy ideals in BCI-algebras, Fuzzy Sets and Systems, 123 (2001),
227-237.
[27] 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.
[28] Y. L. Liu, Y. Xu and J. Meng, BCI-implicative ideals of BCI-algebras, Inform. Sci., 177
(2007), 4987-4996.
[29] Y. L. Liu and X. H. Zhang, Fuzzy a-ideals in BCI-algebras, Adv. in Math. (China), 31 (2002),
65-73.
[30] X. Ma, J. Zhan, B. Davvaz and Y. B. Jun, Some kinds of (2, 2 _ q)-interval-valued fuzzy
ideals of BCI-algebras, Inform. Sci., 178 (2008), 3738-3754.
[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] A. B. Saeid and Y. B. Jun, Redefined fuzzy subalgebras of BCK/BCI-algebras, Iranian J.
Fuzzy Systems, 5(2) (2008), 63-70.
[33] L. Torkzadeh, M. Abbasi and M. M. Zahedi, Some results of intuitionistic fuzzy weak dual
hyper K-ideals, Iranian J. Fuzzy Systems, 5(1) (2008), 65-78.
[34] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
[35] J. Zhan and Z. Tan, Intuitionistic fuzzy a-ideals in BCI- algebras, Soochow Math. J., 30
(2004), 207-216.