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.

[1]

Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
[2]

Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial and Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581
[3]

Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699
[4]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118
[5]

Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619
[6]

İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics and Games, 2021, 8 (3) : 267-275. doi: 10.3934/jdg.2021010
[7]

Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008
[8]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[9]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[10]

Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861
[11]

George A. Anastassiou. Fractional Ostrowski-Sugeno Fuzzy univariate inequalities. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3305-3317. doi: 10.3934/dcdss.2020111
[12]

Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial and Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158
[13]

Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial and Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157
[14]

Guojun Gan, Qiujun Lan, Shiyang Sima. Scalable clustering by truncated fuzzy $c$-means. Big Data & Information Analytics, 2016, 1 (2&3) : 247-259. doi: 10.3934/bdia.2016007
[15]

Jian Luo, Xueqi Yang, Ye Tian, Wenwen Yu. Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2743-2756. doi: 10.3934/jimo.2019078
[16]

Seiyed Hadi Abtahi, Hamidreza Rahimi, Maryam Mosleh. Solving fuzzy volterra-fredholm integral equation by fuzzy artificial neural network. Mathematical Foundations of Computing, 2021, 4 (3) : 209-219. doi: 10.3934/mfc.2021013
[17]

Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457
[18]

Lisha Wang, Huaming Song, Ding Zhang, Hui Yang. Pricing decisions for complementary products in a fuzzy dual-channel supply chain. Journal of Industrial and Management Optimization, 2019, 15 (1) : 343-364. doi: 10.3934/jimo.2018046
[19]

Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263
[20]

Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, 2018  doi: 10.3934/bdia.2017018

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (260)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy