The present paper investigates the hesitant fuzzy filters on residuated lattices. A one-to-one correspondence between the set of all hesitant fuzzy filters and the set of all hesitant fuzzy congruences is established and a quotient residuated lattice with respect to a hesitant fuzzy filter is induced. Furthermore, several special types of hesitant fuzzy filters such as hesitant fuzzy implicative, regular and Boolean filters are introduced, and some alternative definitions of them are obtained, then some typical logical algebras are characterized by these identity forms.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
2.
AtanassovK.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.
3.
ZadehL.A., The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences8 (1975), 199–249.
4.
KarnikN.N. and MendelJ.M., Operations on type-2 fuzzy sets, Fuzzy Sets and Systems122 (2001), 327–348.
5.
DuboisD. and PradeH., Fuzzy sets and systems: theory and applications, In Academic Press, Publishing House: New York, (1980).
6.
TorraV. and NarukawaY., On hesitant fuzzy sets and decision, The 18th IEEE international Conference on Fuzzy Systems, Jeju Island, Korea (2009), 1378–1382.
7.
TorraV., Hesitant Fuzzy Sets, International Journal of Intelligent Systems25 (2010), 529–539.
8.
ZhuY.Q. and XuY., On filter theory of residuatedlattices, Information Sciences180 (2010), 3614–3632.
9.
JunY.B., Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, Fuzzy Sets and Systems121 (2001), 353–357.
10.
JunY.B., XuY. and ZhangX.H., Fuzzy filters of MTL-algebras, Information Sciences175 (2005), 120–138.
11.
LiuL. and LiK., Fuzzy filters of BL-algebras, Information Sciences173 (2005), 141–154.
12.
LiuY.L., An answer to the Jun-Shim-Lele’s open problem on the fuzzy filters, Journal of Applied Mathematics and Computing21 (2006), 325–329.
13.
RachunekJ. and S̆alounováD., Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo-BL-algebras, Information Sciences178 (2008), 3474–3481.
14.
ZhangX.H., Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras, Journal of Intelligent and Fuzzy Systems33 (2017), 1767–1774.
15.
BorzooeiR.A., KologaniM.A., KishM.S. and JunY.B., Fuzzy positive implicative filters of Hoops based on guzzy points, Mathematics7 (2019), 566–580.
16.
GhorbaniS., Intuitionstic fuzzy filters of residuated lattices, New Mathematics and Natural Computation7 (2011), 499–513.
17.
HedayatiH., On filters of R0-algebras connected to intuitionistic fuzzy sets, Journal of Intelligent and Fuzzy Systems25 (2013), 587–593.
18.
JunY.B. and SongS.Z., Hesitant fuzzy prefilters and filters of EQ-algebras, Applied Mathematical Sciences9 (2015), 515–532.
19.
MuhiuddinG., Hesitant fuzzy filters and hesitant fuzzy Gfilters in residuated lattices, Journal of Computational Analysis and Applications21 (2016), 394–404.
20.
ShaoS., ZhangX., BoC. and SmarandacheF., Neutrosophic hesitant fuzzy subalgebras and filters in pseudo-BCI algebras, Symmetry10 (2018), 19–38.
21.
BĕlohlávekR. and VychodilV., Fuzzy Equational Logic, In Springer, Publishing House: New York, (2005).
22.
HájekP., Metamathematics of Fuzzy Logic, In Kluwer, Publishing House: Dordrecht, (1998).
23.
TurunenE., Mathematics behind Fuzzy Logic, In Physica-Verlag, Publishing House: New York, (1999).
24.
EstevaF. and GodoL., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems124 (2001), 271–288.
25.
XiaM. and XuZ., Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52 (2011), 395–407.
26.
VermaR. and Dev SharmaB., New operations over hesitant fuzzy sets, Fuzzy Information and Engineering2 (2013), 129–146.
