The notion of pseudo-BCI algebra is introduced by W.A. Dudek and Y.B. Jun, it is a kind of non-classical logic algebra and a generalization of pseudo-BCK algebra which is close connection with various non-commutative fuzzy logic algebras. The concept of soft set is introduced by Molodtsov, it can be seen as a new mathematical tool for dealing with uncertainty. In this paper, soft set theory is applied to pseudo-BCI algebras, the new notions of soft pseudo-BCI algebras and filteristic soft pseudo-BCI algebras are introduced. The relationships between soft pseudo-BCI algebras and soft non-commutative residuated lattices are presented. The union, intersection, int-product, uni-product and difference operations of (filteristic) soft pseudo-BCI algebras are investigated. Finally, another application of soft set to pseudo-BCI algebras is discussed, the new concept of int-soft filters in pseudo-BCI algebras is introduced, and related properties are proved.
FengFeng,
JunY.B. and ZhaoX.Z., Soft semirings, Computers & Mathematics with Applications56(10) (2008), 2621–2628.
8.
GeorgescuG. and IorgulescuA., Pseudo-BCK algebras: An extension of BCK algebras, in: Combinatorics, Comutability and Logic. Springer Ser Discrete Math Theor Comput Sci (2001), 97–114.
9.
HanJ.S. and AhnS.S., Applications of soft sets in BF-algebras, J Computational Analysis and Applications22(7) (2017), 1323–1331.
10.
HájekP., Observations on non-commutative fuzzy logic, Soft Computing8(1) (2003), 38–43.
11.
IorgulescuA., Pseudo-Iséki algebras. Connection with pseudo-BL algebras, J Multiple-Valued Logic and Soft Computing11(3-4) (2005), 263–308.
12.
IsékiK., An algebra related with a propositional calculus, Proc Japan Acad42(1) (1966), 26–29.
13.
JunY.B., KimH.S. and NeggersJ., On pseudo-BCI ideals of pseudo-BCI algebras, Matematicki Vesnik58(1-2) (2006), 39–46.
14.
JunY.B., Soft BCK/BCI-algebras, Comput Math Appl56 (2008), 1408–1413.
15.
JunY.B. and ParkC.H., Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform Sci178 (2008), 2466–2475.
16.
JunY.B., Union-soft sets with applications in BCK/BCI- algebras, Bull Korean Math Soc50(6) (2013), 1937–1956.
17.
JunY.B., XuY. and ZhangX.H., Int-soft filters in lattice implication algebras, Bull Korean Math Soc53(5) (2016), 1483–1495.
18.
JunY.B. and ZhangX.H., Soft set theoretical approach to residuated lattices, Quasigroups and Related Systems24 (2016), 231–246.
19.
KabzinskiJ.K., BCI-algebras from the point of view of logic, Bulletin of the Section of Logic12(3) (1983), 126–128.
20.
KaraaslanF., Possibility neutrosophic soft sets and PNS – decision making method, Applied Soft Computing54 (2017), 403–414.
21.
LeeK.J. and ParkC.H., Some ideals of pseudo BCI-algebras, Journal of Applied Mathematics and Informatics27(1-2) (2009), 217–231.
22.
LiZ.W. and XieT.S., Roughness of fuzzy soft sets and related results, International Journal of Computational Intelligence Systems8(2) (2015), 278–296.
23.
LiuY.L. and KimH.S., Non-commutative residuated lattices based on soft sets, Journal of Intelligent & Fuzzy Systems29 (2015), 2271–2278.
24.
MaX.L., LiuQ. and ZhanJ.M., A survey of decision making methods based on certain hybrid soft set models, Artificial Intelligence Review47(4) (2017), 507–530.
25.
MolodtsovD., Soft set theory - First results, Comput Math Appl37 (1999), 19–31.
26.
QinK.Y. and HongZ.Y., On soft equality, Journal of Computational and Applied Mathematics234(5) (2010), 1347–1355.
27.
SezginA. and AtagünA.O., On operations of soft sets, Computers & Mathematics with Applications61(5) (2011), 1457–1467.
28.
XinX.L., ChengX.Y. and ZhangX.H., Generalized state operators on BCI-algebras, Journal of Intelligent & Fuzzy Systems32 (2017), 2591–2602.
29.
ZhanJ.M. and JunY.B., Soft BL-algebras based on fuzzy sets, Comput Math Appl59(6) (2010), 2037–2046.
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.,
LiuQ. and HerawanT., A novel soft rough set: Soft rough hemirings and corresponding multicriteria group decision making, Applied Soft Computing54 (2017), 393–402.
32.
ZhangX.H., JiangH. and BhattiS.A., On p-ideal of a BCI-algebra, Punjab Univ J Math27 (1994), 121–128.
33.
ZhangX.H. and YeR.F., BZ-algebra and group, J Math Phys Sci29 (1995), 223–233.
34.
ZhangX.H., WangY.Q. and DudekW.A., T-ideals in BZ-algebras and T-type BZ-algebras, Indian Journal Pure and Applied Mathematics34(11) (2003), 1559–1570.
35.
ZhangX.H. and LiW.H., On pseudo-BL algebras and BCC-algebra, Soft Computing10 (2006), 941–952.
36.
ZhangX.H., Fuzzy Logics and Algebraic Analysis, Science Press, Beijing, 2008.
37.
ZhangX.H. and DudekW.A., BIK+-logic and non-commutative fuzzy logics, Fuzzy Systems and Mathematics23(4) (2009), 8–20.
38.
ZhangX.H., BCC-algebras and residuated partially-ordered groupoid, Mathematica Slovaca63(3) (2013), 397–410.
39.
ZhangX.H., On interval soft sets with applications, Intelligence Systems7(1) (2014), 186–196.
40.
ZhangX.H. and JunY.B., Anti-grouped pseudo-BCI algebras and anti-grouped pseudo-BCI filters, Fuzzy Systems and Mathematics28(2) (2014), 21–33.
41.
ZhangX.H., ZhouH.J. and MaoX.Y., IMTL(MV)-filters and fuzzy IMTL(MV)-filters of residuated lattices, Journal of Intelligent & Fuzzy Systems26(2) (2014), 589–596.
42.
ZhangX.H., Fuzzy commutative filters and fuzzy closed filters in pseudo-BCI algebras, Journal of Computational Information Systems10(9) (2014), 3577–3584.
43.
ZhangX.H., Fuzzy 1-type and 2-type positive implicative filters of pseudo-BCK algebras, Journal of Intelligent & Fuzzy Systems28(5) (2015), 2309–2317.
44.
ZhangX.H., On regular filters and well filters of pseudo-BCI algebras, The 13th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery, 2017, in press.
45.
ZhangX.H., Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras, Journal of Intelligent & Fuzzy Systems33(3) (2017), 1767–1774.
46.
ZhangX.H., MaY.C. and SmarandacheF., Neutrosophic regular filters and fuzzy regular filters in pseudo-BCI algebras, Neutrosophic Sets and Systems17 (2017), 10–15.
