The notion of hesitant fuzzy set is introduced by V. Torra, which is a very useful tool to express peoples’ hesitancy in daily life. The notion of pseudo-BCI algebra is introduced by W. A. Dudek and Y. B. Jun, which is a kind of nonclassical logic algebra and close connection with various non-commutative fuzzy logic algebras. In this paper, hesitant fuzzy theory is applied to pseudo-BCI algebras. The new concepts of hesitant fuzzy filter and anti-grouped hesitant fuzzy filter in pseudo-BCI algebras are proposed, and their characterizations are presented. Also, the relationships between fuzzy filters and hesitant fuzzy filters are discussed. Moreover, by introducing the notion of tip-extended pair of hesitant fuzzy filters, a new union operation (generated by the union of two hesitant fuzzy filters) is defined and it is proved that the set of all hesitant fuzzy filters in pseudo-BCI algebras forms a bounded distributive lattice about intersection and the new union.
AtanassovK.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.
2.
DaiJ.H., HanH.F. and ZhangX.H., Catoptrical rough set model on two universes using granule-based definition and its variable-precision extensions, Information Sciences390 (2017), 70–81.
3.
DudekW.A. and JunY.B., Pseudo-BCI algebras, East Asian Mathematical Journal24(2) (2008), 187–190.
4.
GeorgescuG., IorgulescuA., Pseudo-BCK algebras: An extension of BCK algebras, Combinatorics, Computability and Logic. Springer, 2001, 97–114.
5.
HeadT., A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems73(3) (1995), 349–358.
6.
HeadT., Erratum to ‘A metatheorem for deriving fuzzy theorems from crisp versions’, Fuzzy Sets and Systems79(2) (1996), 277–278.
7.
JunY.B., Soft BCK/BCI-algebras, , Computers and Mathematics with Applications56 (2008), 1408–1413.
8.
JunY.B., KimH.S. and NeggersJ., On pseudo-BCI ideals of pseudo-BCI algebras, Matematički Vesnik58, (1-2) (2006), 39–46.
9.
JunY.B. and SongS.Z., Hesitant fuzzy set theory applied to filters in MTL-algebras, Honam Math J36(4) (2014), 813–830.
10.
KabzinskiJ.K., BCI-algebras from the point of view of logic, Bulletin of the Section of Logic12(3) (1983), 126–128.
11.
LeeK.J. and ParkC.H., Some ideals of pseudo BCI-algebras, Journal of Applied Mathematics and Informatics27(1-2) (2009), 217–231.
12.
MaX., ZhanJ., AliM.I. and MehmoodN., A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review49(4) (2018), 511–529.
13.
MengB.L. and XinX.L., Falling fuzzy Gödel ideals of BL-algebras, Italian Journal of Pure and Applied Mathematics37 (2017), 237–258.
14.
MolodtsovD.A., Soft set theory-first results, Computers and Mathematics with Applications37 (1999), 19–31.
15.
PawlakZ., Rough sets, International Journal of Computer and Information Sciences11 (1982), 11, 341–356.
16.
QinK.Y. and HongZ.Y., On soft equality, Journal of Computational and Applied Mathematics234 (2010), 1347–1355.
17.
RosaM.R., LuisM. and FranciscoH., Hesitant fuzzy linguistic term sets for decision making, (1), IEEE Transactions on Fuzzy Systems20 (2012), 109–119.
18.
ShakibaA., HooshmandaslM.R., DavvazB. and FazeliS.A.S., S-approximation spaces: A fuzzy approach, Iranian Journal of Fuzzy Systems14 (2017), 127–154.
19.
SongS.Z. and JunY.B., Intuitionistic permeable values in BCK/BCI-algebras, Italian Journal of Pure and Applied Mathematics36 (2016), 791–800.
20.
TorraV., Hesitant fuzzy sets, International Journal of Intelligent Systems25 (2010), 529–539.
21.
VermaR. and DevB., Sharma, New Operations over hesitant fuzzy sets, Fuzzy Information and Engineering2 (2013), 129–146.
22.
WangC.Z., ShaoM.W. and HeQ., Feature subset selection based on fuzzy neighborhood rough sets, Knowledge-Based Systems111 (2016), 173–179.
23.
WangF.Q., LiX. and ChenX.H., Hesitant fuzzy soft set and its applications in multicriteria decision making, Journal of Applied Mathematics (2014). 10.1155/2014/643785.
24.
WeiG., Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge-Based Systems31 (2012), 176–182.
25.
XiaM. and XuZ.S., Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52(3) (2011), 395–407.
26.
XuZ.S. and XiaM., Distance and similarity measures for hesitant fuzzy sets, Information Sciences181(11) (2011), 2128–2138.
27.
XuZ.S. and XiaM., On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligent Systems26(5) (2011), 410–425.
28.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
29.
ZhanJ. and AlcantudJ.C.R., A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018). 10.1007/s10462-018-9617-3.
30.
ZhanJ.M. and DavvazB., A new rough set theory: Rough soft hemirings, Journal of Intelligent and Fuzzy Systems28(4) (2015), 1687–1697.
31.
ZhanJ.M. and JunY.B., Soft BL-algebras based on fuzzy sets, Applications59 (2010), 2037–2046.
32.
ZhangX.H., Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras, Journal of Intelligent and Fuzzy Systems33 (2017), 1767–1774.
33.
ZhangX.H., Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras, Journal of Intelligent and Fuzzy Systems28(5) (2015), 2309–2317.
34.
ZhangX.H., BoC.X., SmarandacheF., . New inclusion relation of neutrosophic sets with applications and related lattice structure, International Journal of Machine Learning and Cybernetics (2018), 1–11.
35.
ZhangX.H. and JunY.B., Anti-grouped pseudo-BCI algebras and anti-grouped pseudo-BCI filters, Fuzzy Systems and Mathematics28(2) (2014), 21–33.
36.
ZhangX.H. and LiW.H., On pseudo-BL algebras and BCC-algebra, Soft Computing10(10) (2006), 941–952.
37.
ZhangX.H., ParkC. and WuS.P., Soft set theoretical approach to pseudo-BCI algebras, Journal of Intelligent and Fuzzy Systems34(1) (2018), 559–568.
38.
ZhangX.H., PeiD.W., DaiJ.H., Fuzzy Mathematics and Rough Set Theory, Tsinghua University Press: Beijing, China, 2013.
39.
ZhangX.H., SmarandacheF. and XingL.L., Neutrosophic duplet semi-Group and cancellable neutrosophic triplet groups, Symmetry9(11) (2017), 275. 10.3390/sym9110275.
40.
ZhangX.H., ZhouB. and LiP., A general frame for intuitionistic fuzzy rough sets, Information Sciences216 (2012), 34–49.
41.
ZhangX.H., ZhouH.J. and MaoX.Y., IMTL(MV)-filters and fuzzy IMTL(MV)-filters of residuated lattices, Journal of Intelligent and Fuzzy Systems26(2) (2014), 589–596.
