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Fuzzy Boolean and positive implicative filters of BL-algebras. (English) Zbl 1072.03037
Summary: The aim of this paper is to introduce the notions of fuzzy Boolean filters and fuzzy positive implicative filters in BL-algebras and to investigate their properties. Several characterizations of fuzzy Boolean filters and fuzzy positive implicative filters are derived. Extension theorems of fuzzy Boolean filters and fuzzy positive implicative filters are obtained. The relation between fuzzy Boolean filters and fuzzy positive implicative filters is investigated and it is proved that every fuzzy Boolean filter is a fuzzy positive implicative filter, but the converse may not be true. Furthermore, conditions under which a fuzzy positive implicative filter is a fuzzy Boolean filter are established.

MSC:
03G25 Other algebras related to logic
03B52 Fuzzy logic; logic of vagueness
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[1] Chang, C.C., Algebraic analysis of many valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704
[2] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[3] Hoo, C.S., Fuzzy implicative and Boolean ideals of \(\mathit{MV}\)-algebras, Fuzzy sets and systems, 66, 315-327, (1994) · Zbl 0844.06007
[4] Hoo, C.S.; Sessa, S., Implicative and Boolean ideals of \(\mathit{MV}\)-algebras, Math. japon., 39, 215-219, (1994) · Zbl 0798.06020
[5] Iorgulescu, A., Some direct ascendents of wajsberg and \(\mathit{MV}\)-algebras, Sci. math. japon., 57, 583-647, (2003) · Zbl 1050.06006
[6] Iséki, K.; Tanaka, S., Ideal theory of \(\mathit{BCK}\)-algebras, Math. japon., 21, 351-366, (1976) · Zbl 0355.02041
[7] Iséki, K.; Tanaka, S., An introduction to the theory of \(\mathit{BCK}\)-algebras, Math. japon., 23, 1-26, (1978)
[8] Jun, Y.B., Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, Fuzzy sets and systems, 121, 353-357, (2001) · Zbl 0981.03066
[9] Jun, Y.B.; Song, S.Z., On fuzzy implicative filters of lattice implication algebras, J. fuzzy math., 10, 893-900, (2002) · Zbl 1029.03512
[10] Kondo, M., Fuzzy congruence on \(\mathit{BCI}\)-algebras, Sci. math. japon., 57, 191-196, (2003) · Zbl 1033.06013
[11] L.Z. Liu, K.T. Li, Fuzzy filters of \(\mathit{BL}\)-algebras, Inform. Sci., in press.
[12] Mundici, M., \(\mathit{MV}\)-algebras are categorically equivalent to bounded commutative \(\mathit{BCK}\)-algebras, Math. japon., 31, 889-894, (1986) · Zbl 0633.03066
[13] Rom, E.H.; Kim, S.Y.; Xu, Y.; Jun, Y.B., Some operations on lattice implication algebras, Ijmms, 1, 45-52, (2001) · Zbl 0998.03509
[14] Turunen, E., Mathematics behind fuzzy logic, (1999), Physica-Verlag Heidelberg · Zbl 0940.03029
[15] Turunen, E., Boolean deductive systems of \(\mathit{BL}\)-algebras, Arch. math. logic, 40, 467-473, (2001) · Zbl 1030.03048
[16] Turunen, E.; Sessa, S., Local \(\mathit{BL}\)-algebras, Mult-valued logic, 6, 229-249, (2001) · Zbl 1049.03045
[17] Wang, G.J., \(\mathit{MV}\)-algebras \(\mathit{BL}\)-algebras, \(R_0\) algebras and multiple-valued logic, Fuzzy systems math., 3, 1-15, (2002)
[18] Xi, O., Fuzzy \(\mathit{BCK}\)-algebras, Math. japon., 36, 935-942, (1991) · Zbl 0744.06010
[19] Xu, Y., Lattice implication algebras, J. southwest jiaotong univ., 1, 20-27, (1993) · Zbl 0784.03035
[20] Xu, Y.; Qin, K.Y., On filters of lattice implication algebras[J], J. fuzzy math., 1, 251-260, (1993) · Zbl 0787.06009
[21] Xu, Y.; Qin, K.Y., Fuzzy lattice implication algebras, J. southwest jiaotong univ., 2, 121-127, (1995) · Zbl 0830.03030
[22] Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606
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