Fuzzy n-fold obstinate and maximal (pre)filters of EQ-algebras

Document Type : Original Article

Author

Department of Mathematics Faculty of Mathematical Sciences Alzahra University Tehran Iran

Abstract

In this paper, we defined the concepts of fuzzy $n$-fold obstinate (pre)filter and maximal fuzzy (pre)filter of $EQ$-algebras and discussed the properties of them. We show that every maximal fuzzy (pre)filter of $\mathcal{\LomE}$ is normalized and takes only the values $\{0, 1\}$. Also we show that in good $EQ$-algebra, if $\m$ is a normalized fuzzy (pre)filter of $\mathcal{\LomE}$, then $\m$ is a fuzzy $n$-fold obstinate (pre)filter of $\mathcal{\LomE}$ if and only if every normalized fuzzy (pre)filter of quotient algebra $\mathcal{\LomE}/\m$ is a fuzzy $n$-fold obstinate (pre)filter of $\mathcal{\LomE}/\m$.
Also, we verify relation between fuzzy obstinate $n$-fold (pre)filters and other fuzzy (pre)filters of $EQ$-algebras. 

Keywords


[1] N. Akhlaghinia, M. Aaly Kologani, R.A. Borzooei, X.L. Xin, The category of EQ-algebras,
Bulletin of the Section of Logic, to appear, doi: 10.18778/0138-0680.2021.01.
[2] M. Behzadi, L. Torkzadeh, Some pre- lters in EQ-algebras, Applications and Applied Mathe-
matics, 12(2) (2017), 1057{1071.
[3] R.A. Borzooei, B. Ganji Sa ar, States on EQ-algebras, Journal of Intelligent and Fuzzy Sys-
tems, 29 (2015), 209{221.
[4] R.A. Borzooei, A. Namdar, M. Aaly Kologani, n-fold obstinate  lters in pseudo-hoop algebras,
International Journal of Industrial Mathematics, 12(2) (2020), 11 pages.
[5] M. El-Zekey, Representable good EQ-algebra, Soft Computing, 14 (2010), 1011{1023.
[6] M. El-Zekey, V. Novak, R. Mesiar, On good EQ-algebras, Fuzzy Sets and Systems, 178 (2011),
1{23.
[7] B. Ganji Sa ar, M. Aaly Kologani, R. A. Borzooei, n-fold  lters of EQ-algebras, submitted.
[8] B. Ganji Sa ar, G. Muhiuddin, M. Aaly Kologani, R.A. Borzooei, Construction of (n-fold)
EQ-algebras by using fuzzy n-fold  lters, New Mathematics and Natural Computation, to
appear.
[9] P. Hajek, Fuzzy logics with noncommutative conjuctions, Journal of Logic and Computation,
13 (2003), 469{479.
[10] L.Z. Liu, X.Y. Zhang, Implicative and positive implicative pre lters of EQ-algebras, Journal
of Intelligent and Fuzzy Systems, 26 (2014), 2087{2097.
[11] C. Luo, X. Xin, P. He, n-Fold (positive) implicative  lters of hoops, Italian Journal of Pure
and Applied Mathematics, 38 (2017), 631{642.
[12] Z.M. Ma, B.Q. Hu, Fuzzy EQ- lters of EQ-algebras, World Scienti c Proceedings Series
on Computer Engineering and Information Science: Quantitative Logic and Soft Computing,
(2012), 528{535.
[13] M. Mohseni Takallo, M. Aaly Kologani, MBJ-neutrosophic  lters of equality algebras, Journal
of Algebraic Hyperstructures and Logical Algebras, 1(2) (2020), 57{75.
[14] N. Mohtashamniya, L. Torkzadeh, A pre lter generated by a set in EQ-algebras, 46th Annual
Iranian Mathematics Conference Yazd University, (2015), 1{4.
[15] S. Motamed, A. Borumand Saeid, n-fold obstinate  lters in BL-algebras, Neural Comput and
Applic, 20 (2011), 461{472.
[16] V. Novak, EQ-algebra, primary concepts and properties, International Joint, Czech Republic-
Japan and Taiwan-Japan Symposium, Kitakyushu, Japan, (2006), 219{223.
[17] V. Novak, B. De Baets, EQ-algebras, Fuzzy Sets and Systems, 160 (2009), 2956{2978.
[18] A. Paad, On fuzzy implicative ideals in BL-algebras, Journal of Algebraic Hyperstructures
and Logical Algebras, 1(4) (2020), 83{94
[19] A. Rosenfeld, Fuzzy groups, Journal of Mathematics Annals and Application, 35(3) (1971),
512{517.
[20] W. Wang, X.L. Xin, J.T. Wang, EQ-algebras with internal states, Soft Computing, 22 (2018),
2825{2841.
[21] X.L. Xin, P.F. He, Y.W. Yang, Characterizations of some fuzzy pre lters ( lters) in EQ-
algebras, The Scienti c World Journal, (2014), 13 pages.
[22] X.L. Xin, M. Khan, Y.B. Jun, Generalized states on EQ-algebras, Iranian Journal of Fuzzy
Systems, 16(1) (2019), 159{172.
[23] L.A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.
[24] F. Zebardast, R.A. Borzooei, M. Aaly Kologhani, Results on equality algebras, Information
Sciences, 381 (2017), 270{282.