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 & 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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

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.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

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