CHAIN CONDITIONS ON FUZZY POSITIVE IMPLICATIVE FILTERS OF BL-ALGEBRAS

Year 2011, Volume: 40 Issue: 6, 819 - 828, 01.06.2011

H. Hedayati  y.b. Jun

Abstract

In this paper, we discuss chain conditions of fuzzy positive implicative filters of BL-algebras. Specially, by using the notions of maximal and normal fuzzy positive implicative filters, we show that under certain conditions a fuzzy positive implicative filter is two-valued and takes the values 0 and 1. 

Keywords

BL-algebra , Positive implicative filter , Normal fuzzy positive implicative filter , Maximal fuzzy positive implicative filter

CHAIN CONDITIONS ON FUZZY POSITIVE IMPLICATIVE FILTERS OF BL-ALGEBRAS

Abstract

References

• Akram, M and Dudek, W. A. Intuitionistic fuzzy left k-ideals of semirings, Soft Comput. , 881–890, 2008.
• Chang, C. C. Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88, 467–490, Dudek, W. A. Special types of intuitionistic fuzzy h-ideals of hemirings, Soft Comput 12, –364, 2008.
• H´ajek, P. Metamathematics of Fuzzy Logic (Kluwer Academic Publishers, Dordrecht, 1998).
• Hedayati, H. Connections between generalized fuzzy ideals and sub-implicative ideals of BCI-algebras, IAENG Int. J. Appl. Math. 41 (1), 17–22, 2011.
• Hedayati, H. t-implication-based fuzzy interior hyperideals of semihypergroups, J. Discrete Math. Sci. Crypth. 13 (2), 123–140, 2010.
• Hedayati, H and Jafari, Z. Generalized fuzzy implicative ideals in pseudo-MV algebras, Int. J. Appl. Math. Stat. 18 (10), 24–35, 2010.
• Hoo, C. S. Fuzzy implicative and Boolean ideals of MV-algebras, Fuzzy Sets and Systems , 315–327, 1994.
• Hoo, C. S. and Sessa, S. Implicative and Boolean ideals of MV-algebras, Math. Japon 39, –219, 1994.
• Iorgulescu, A. Some direct ascendents of Wajsberg and MV-algebras, Sci. Math. Japon 57, –647, 2003.
• Is´eki, K. and Tanaka, S. Ideal theory of BCK-algebras, Math. Japon 21, 351–366, 1976.
• Is´eki, K. and Tanaka, S. An introduction to the theory of BCK-algebras, Math. Japon 23, –26, 1978.
• Jun, Y. B. Fuzzy positive implicative and fuzzy associative filters of lattice implication alge- bras, Fuzzy Sets and Systems 121, 353–357, 2001.
• Jun, Y. B. and Song, S. Z. On fuzzy implicative filters of lattice implication algebras, J. Fuzzy Math. 10, 893–900, 2002.
• Kondo, M. Fuzzy congruence on BCI-algebras, Sci. Math. Japon 57, 191–196, 2003.
• Liu, L. and Li, K. Fuzzy Boolean and positive implicative filters of BL-algebras, Fuzzy Sets and Systems 152, 333–348, 2005.
• Mundici, D. MV-algebras are categorically equivalent to bounded commutative BCK- algebras, Math. Japon 31, 889–894, 1986.
• Rasouli, S., Heidari, D. and Davvaz, B. β-relations on implicative bounded hyper BCK- algebras, Hacet. J. Math. Stat. 39 (4), 461–469, 2010.
• Rom, E. H., Kim, S. Y., Xu, Y. and Jun, Y. B. Some operations on lattice implication algebras, IJMMS 1, 45–52, 2001.
• Turunen, E. Mathematics Behind Fuzzy Logic (Physica-Verlag, Heidelberg, 1999).
• Turunen, E. Boolean deductive systems of BL-algebras, Arch. Math. Logic 40, 467–473, Turunen, E. and Sessa, S. Local BL-algebras, Mult-Valued Logic 6, 229–249, 2001.
• Wang, G. J. MV-algebras, BL-algebras, R0algebras and multiple-valued logic, Fuzzy Sys- tems Math. 3, 1–15, 2002.
• Xi, O. Fuzzy BCK-algebras, Math. Japon 36, 935–942, 1991.
• Xu, Y. Lattice implication algebras, J. Southwest Jiaotong Univ. 1, 20–27, 1993.
• Xu, Y. and Qin, K. Y. On filters of lattice implication algebras, J. Fuzzy Math. 1, 251–260, Xu, Y. and Qin, K. Y. Fuzzy lattice implication algebras, J. Southwest Jiaotong Univ. 2, –127, 1995.
• Zadeh, L. A. Fuzzy sets, Inform and Control 8, 338–353, 1965.