American Institute of Mathematical Sciences

Advanced
December  2011, 4(6): 1611-1619. doi: 10.3934/dcdss.2011.4.1611

On fuzzy filters of Heyting-algebras

Wei Wang 1, and  Xiao-Long Xin 2,

1. 

Department of Mathematics, Northwest University, Xi'an 710069, China
2. 

Department of Mathematics, Northwest University, Xi'an, 710069, China

Received  March 2009 Revised  November 2009 Published  December 2010

The concept of fuzzy filter of Heyting-algebras was introduced and some important properties were discussed. Some special kinds of fuzzy filters were defined and we prove that fuzzy Boolean filter is equivelent to fuzzy implicative filter in Heyting-algebras. And the relation among the fuzzy filters were proposed.
Keywords: Heyting-algebras, positive implicative, implicative, fuzzy filter, ultra, fuzzy (prime, obstinate) filters., Boolean.
Mathematics Subject Classification: Primary: 03G10; Secondary: 03B5.
Citation: Wei Wang, Xiao-Long Xin. On fuzzy filters of Heyting-algebras. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1611-1619. doi: 10.3934/dcdss.2011.4.1611
References:
[1]

A. Micali and F. Zitan, On homogeneous weighted algebras, Communications in Algebra, 35 (2007), 2371-2385. doi: 10.1080/00927870701325801.

[2]

E. Turunen, "Mathematics Behind Fuzzy Logic," Physica-Verlag, Heidelberg, 1999.

[3]

K. Iséki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon., 21 (1976), 351-366.

[4]

C. S. Hoo and S. Sessa, Implicative and Boolean ideals of MV-algebras, Math. Japon., 39 (1994), 215-219.

[5]

J. D. Bashford and P. D.Jarvis, The genetic code as a peridic table: algebraic aspects, BioSystems, 57 (2000), 147-161. doi: 10.1016/S0303-2647(00)00097-6.

[6]

E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic, 40 (2001), 467-473. doi: 10.1007/s001530100088.

[7]

M. K. Kinyon and A. A. Sagle, Quadratic dynamical systems and algebras, J. Differential Equations, 117 (1995), 67-126. doi: 10.1006/jdeq.1995.1049.

[8]

L. Frappat, A. Sciarrino and P.Sorba, Crystalizing the genetic code, J. Biological Physics, 27 (2001), 1-34, arXiv:physics/9812041. doi: 10.1023/A:1011874407742.

[9]

J. J. Tian and B. L. Li, Coalgebraic structure of genetics inheritance, Mathematical Biosciences and Engineering, 1 (2004), 243-266.

[10]

L. Z. Liu and K. T. Li, Fuzzy Boolean and positive implicative filters of BL-algebras, Fuzzy Sets and Systems, 152 (2005), 333-348. doi: 10.1016/j.fss.2004.10.005.

[11]

R. Sánchez, R. Grau and E. Morgado, A novel Lie algebra of the genetic code over the Galois field of four DNA bases, Mathematical Biosciences, 202 (2006), 156-174. doi: 10.1016/j.mbs.2006.03.017.

[12]

L. Z. Liu and K. T. Li, Fuzzy filters of BL-algebras, Information Sciences, 173 (2005), 141-154. doi: 10.1016/j.ins.2004.07.009.

[13]

J. J. Tian, "Evolution Algebras and their Applications," Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-74284-5.

[14]

O. Xi, Fuzzy $BCK$-algebras, Math. Japon., 36 (1991), 935-942.

[15]

Xiao-Hong Zhang, Young Bae Jun and Myung Im Doh, On fuzzy filters and fuzzy ideals of BL-algebras, Information Sciences, 20 (2006), 8-20.

[16]

C. S. Hoo, Fuzzy implicative and Boolean ideals of MV-algebras, Fuzzy Sets and Systems, 66 (1994), 315-327. doi: 10.1016/0165-0114(94)90099-X.

[17]

Y. Xu and K. Y. Qin, On filters of lattice implication algebras, J. Fuzzy Math., 1 (1993), 251-260.

[18]

Y. Xu and K. Y. Qin, Fuzzy lattice implication algebras, J. Southwest Jiaotong Univ., 2 (1995), 121-127.

[19]

Y. B. Jun, Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, Fuzzy Sets and Systems, 121 (2001), 353-357. doi: 10.1016/S0165-0114(00)00030-0.

[20]

Y. B. Jun and S. Z. Song, On fuzzy implicative filters of lattice implication algebras, J. Fuzzy Math., 10 (2002), 893-900.

[21]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X.

show all references

References:
[1]

A. Micali and F. Zitan, On homogeneous weighted algebras, Communications in Algebra, 35 (2007), 2371-2385. doi: 10.1080/00927870701325801.

[2]

E. Turunen, "Mathematics Behind Fuzzy Logic," Physica-Verlag, Heidelberg, 1999.

[3]

K. Iséki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon., 21 (1976), 351-366.

[4]

C. S. Hoo and S. Sessa, Implicative and Boolean ideals of MV-algebras, Math. Japon., 39 (1994), 215-219.

[5]

J. D. Bashford and P. D.Jarvis, The genetic code as a peridic table: algebraic aspects, BioSystems, 57 (2000), 147-161. doi: 10.1016/S0303-2647(00)00097-6.

[6]

E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic, 40 (2001), 467-473. doi: 10.1007/s001530100088.

[7]

M. K. Kinyon and A. A. Sagle, Quadratic dynamical systems and algebras, J. Differential Equations, 117 (1995), 67-126. doi: 10.1006/jdeq.1995.1049.

[8]

L. Frappat, A. Sciarrino and P.Sorba, Crystalizing the genetic code, J. Biological Physics, 27 (2001), 1-34, arXiv:physics/9812041. doi: 10.1023/A:1011874407742.

[9]

J. J. Tian and B. L. Li, Coalgebraic structure of genetics inheritance, Mathematical Biosciences and Engineering, 1 (2004), 243-266.

[10]

L. Z. Liu and K. T. Li, Fuzzy Boolean and positive implicative filters of BL-algebras, Fuzzy Sets and Systems, 152 (2005), 333-348. doi: 10.1016/j.fss.2004.10.005.

[11]

R. Sánchez, R. Grau and E. Morgado, A novel Lie algebra of the genetic code over the Galois field of four DNA bases, Mathematical Biosciences, 202 (2006), 156-174. doi: 10.1016/j.mbs.2006.03.017.

[12]

L. Z. Liu and K. T. Li, Fuzzy filters of BL-algebras, Information Sciences, 173 (2005), 141-154. doi: 10.1016/j.ins.2004.07.009.

[13]

J. J. Tian, "Evolution Algebras and their Applications," Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-74284-5.

[14]

O. Xi, Fuzzy $BCK$-algebras, Math. Japon., 36 (1991), 935-942.

[15]

Xiao-Hong Zhang, Young Bae Jun and Myung Im Doh, On fuzzy filters and fuzzy ideals of BL-algebras, Information Sciences, 20 (2006), 8-20.

[16]

C. S. Hoo, Fuzzy implicative and Boolean ideals of MV-algebras, Fuzzy Sets and Systems, 66 (1994), 315-327. doi: 10.1016/0165-0114(94)90099-X.

[17]

Y. Xu and K. Y. Qin, On filters of lattice implication algebras, J. Fuzzy Math., 1 (1993), 251-260.

[18]

Y. Xu and K. Y. Qin, Fuzzy lattice implication algebras, J. Southwest Jiaotong Univ., 2 (1995), 121-127.

[19]

Y. B. Jun, Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, Fuzzy Sets and Systems, 121 (2001), 353-357. doi: 10.1016/S0165-0114(00)00030-0.

[20]

Y. B. Jun and S. Z. Song, On fuzzy implicative filters of lattice implication algebras, J. Fuzzy Math., 10 (2002), 893-900.

[21]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X.

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