## Abstract and Applied Analysis

### Some Types of Generalized Fuzzy $n$-Fold Filters in Residuated Lattices

Zhen Ming Ma

#### Abstract

Fuzzy filters and their generalized types have been extensively studied in the literature. In this paper, a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences is established, a quotient residuated lattice with respect to generalized fuzzy filter is induced, and several types of generalized fuzzy $n$-fold filters such as generalized fuzzy $n$-fold positive implicative (fantastic and Boolean) filters are introduced; examples and results are provided to demonstrate the relations among these filters.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 736872, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512241

Digital Object Identifier
doi:10.1155/2013/736872

Mathematical Reviews number (MathSciNet)
MR3147858

#### Citation

Ma, Zhen Ming. Some Types of Generalized Fuzzy $n$ -Fold Filters in Residuated Lattices. Abstr. Appl. Anal. 2013 (2013), Article ID 736872, 8 pages. doi:10.1155/2013/736872. https://projecteuclid.org/euclid.aaa/1393512241

#### References

• M. Ward and P. R. Dilworth, “Residuated lattice,” Transactions of the American Mathematical Society, vol. 45, pp. 335–354, 1939.
• E. Turunen, “Boolean deductive systems of BL-algebras,” Archive for Mathematical Logic, vol. 40, no. 6, pp. 467–473, 2001.
• P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
• M. Haveshki, A. B. Saeid, and E. Eslami, “Some types of filters in BL algebras,” Soft Computing, vol. 10, no. 8, pp. 657–664, 2006.
• M. Kondo and W. A. Dudek, “Filter theory of BL algebras,” Soft Computing, vol. 12, no. 5, pp. 419–423, 2008.
• Y. Zhu and Y. Xu, “On filter theory of residuated lattices,” Information Sciences, vol. 180, no. 19, pp. 3614–3632, 2010.
• R. A. Borzooei, S. Khosravi Shoar, and R. Ameri, “Some types of filters in MTL-algebras,” Fuzzy Sets and Systems, vol. 187, no. 1, pp. 92–102, 2012.
• M. Haveshki, “A note on some types of filters in MTL-algebras,” Fuzzy Sets and Systems, 2013.
• M. Haveshki and E. Eslami, “n-fold filters in BL-algebras,” Mathematical Logic Quarterly, vol. 54, no. 2, pp. 176–186, 2008.
• E. Turunen, N. Tchikapa, and C. Lele, “n-Fold implicative basic logic is Gödel logic,” Soft Computing, vol. 16, no. 1, pp. 177–181, 2012.
• E. Turunen, N. Tchikapa, and C. Lele, “Erratum to: n-Fold implicative basic logic is Gödel logic, Soft Comput,” Soft Computing, vol. 16, no. 1, p. 183, 2012.
• O. Zahiri and H. Farahani, “n-Fold filters of MTL-algebras,” Afrika Matematika, 2013.
• Y. B. Jun, Y. Xu, and X. H. Zhang, “Fuzzy filters of MTL-algebras,” Information Sciences, vol. 175, no. 1-2, pp. 120–138, 2005.
• L. Liu and K. Li, “Fuzzy filters of BL-algebras,” Information Sciences, vol. 173, no. 1–3, pp. 141–154, 2005.
• L. Lianzhen and L. Kaitai, “Fuzzy Boolean and positive implicative filters of BL-algebras,” Fuzzy Sets and Systems, vol. 152, no. 2, pp. 333–348, 2005.
• X. Ma, J. Zhan, and W. A. Dudek, “Some kinds of $(\overline{\in },\overline{\in } \vee \overline{q})$-fuzzy filters of B L-algebras,” Computers and Mathematics with Applications, vol. 58, no. 2, pp. 248–256, 2009.
• Y. B. Jun, Y. U. Cho, E. H. Roh, and J. Zhan, “General types of $(\in ,\in \vee q)$-fuzzy filters in BL-algebras,” Neural Computing and Applications, vol. 20, no. 3, pp. 335–343, 2011.
• X. Ma and J. Zhan, “On $(\in ,\in \vee q)$-fuzzy filters of BL-algebras,” Journal of Systems Science and Complexity, vol. 21, no. 1, pp. 144–158, 2008.
• J. Zhan and Y. Xu, “Some types of generalized fuzzy filters of BL-algebras,” Computers and Mathematics with Applications, vol. 56, no. 6, pp. 1604–1616, 2008.
• T. Mahmood, “Hemirings Characterized by the properties of their $(\overline{\in },\overline{\in }\vee \overline{q}\kappa )$-fuzzy ideals,” Iranian Journal of Science and Technology, vol. 37, pp. 265–275, 2013.
• M. Shabir and T. Mahmood, “Characterizations of hemirings by $(\in ,\in \vee q\kappa )$-fuzzy ideals,” Computers and Mathematics with Applications, vol. 61, no. 4, pp. 1059–1078, 2011.
• M. Shabir and T. Mahmood, “Hemirings characterized by the properties of their fuzzy ideals with thresholds,” Quasigroups and Related Systems, vol. 18, pp. 195–212, 2010. \endinput