# Hesitant Fuzzy Soft Subalgebras and Ideals inBCK/BCI-Algebras

Authors: Young Bae Jun; Sun Shin Ahn; G. Muhiuddin
Journal: The Scientific World Journal
 (2014)

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## Abstract

As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in the papers by Babitha and John (2013) and Wang et al. (2014). The aim of this paper is to apply hesitant fuzzy soft set for dealing with several kinds of theories inBCK/BCI-algebras. The notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals are introduced, and related properties are investigated. Relations between a hesitant fuzzy soft subalgebra and a (closed) hesitant fuzzy soft ideal are discussed. Conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra are given, and conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal are provided. Characterizations of a (closed) hesitant fuzzy soft ideal are considered.


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## Body

## 1. Introduction

Nowadays, even if Molodtsov’s soft set theory (see [1]) is a newly emerging mathematical tool to handle uncertainty, the classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In order to overcome this situation, Maji et al. [2] introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem. The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets, fuzzy multisets, and so forth are a generalization of fuzzy sets. As another generalization of fuzzy sets, Torra [3] introduced the notion of hesitant fuzzy sets which are very useful to express peoples hesitancy in daily life. The hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. Xu and Xia [4] proposed a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity measures can be obtained. They investigated the connections of the aforementioned distance measures and further develop a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. Also, hesitant fuzzy set theory is used in decision making problem, and so forth (see [5–9]), and is applied to residuated lattices andMTL-algebras (see [10, 11]). In soft set theory membership is decided by adequate parameters, and hesitant fuzzy set employs all possible values for the membership of an element. The soft set model has been combined with other mathematical models, for example, fuzzy soft sets by combining fuzzy sets and soft sets (see [2]), intuitionistic fuzzy soft sets which are based on a combination of intuitionistic fuzzy sets and soft sets (see [12, 13]), interval-valued fuzzy soft sets (see [14]), trapezoidal fuzzy soft sets (see [15]), and forth. As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in [6, 16].In this paper, we apply the notion of hesitant fuzzy soft sets to subalgebras and ideals inBCK/BCI-algebras. We introduce the notion of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals and investigate related properties. We consider relations between a hesitant fuzzy soft subalgebra and a hesitant fuzzy soft ideal. We provide conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra. We also give conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal. We discuss characterizations of a (closed) hesitant fuzzy soft ideal.


## 2. Preliminaries

ABCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers.An algebra(X;*,0) of type (2,0) is called aBCI-algebra if it satisfies the following conditions:(I)
(∀x,y,z∈X)(((x*y)*(x*z))*(z*y)=0),(II)
(∀x,y∈X)((x*(x*y))*y=0),(III)
(∀x∈X)(x*x=0),(IV)
(∀x,y∈X)(x*y=0,y*x=0⇒x=y).If aBCI-algebra X satisfies the following identity:(V)
(∀x∈X)(0*x=0),thenX is called aBCK-algebra; anyBCK-algebra X satisfies the following axioms:
(a1)
(∀x∈X)(x*0=x),(a2)
(∀x,y,z∈X)(x≤y⇒x*z≤y*z,z*y≤z*x),(a3)
(∀x,y,z∈X)((x*y)*z=(x*z)*y),(a4)
(∀x,y,z∈X)((x*z)*(y*z)≤x*y),wherex≤y if and only if x*y=0;AnyBCI-algebra X satisfies the following axioms:(a5)
(∀x,y,z∈X)(0*(0*((x*z)*(y*z)))=(0*y)*(0*x)).(
∀
x
,
y
∈
X
)
(
0
*
(
0
*
(
x
*
y
)
)
=
(
0
*
y
)
*
(
0
*
x
)
).A nonempty subsetS of aBCK/BCI-algebra X is called asubalgebra of X if x*y∈S for all x,y∈S.A subsetI of aBCK/BCI-algebra X is called anideal of X if it satisfies
(1)0∈I,(∀x,y∈X)(x*y∈I,y∈I⟹x∈I).An idealI of aBCI-algebra X is said to beclosed if 0*x∈I for every x∈I.We refer the reader to the books [17, 18] for further information regardingBCK/BCI-algebras.Definition 1 (see [3]).
Ahesitant fuzzy set on a reference set (or an initial universe set) U is defined in terms of a function that when applied to U returns a subset of [0,1], which can be viewed as the following mathematical representation:
(2)H≔{(u,hH(u))∣u∈U},
where hH:U→P([0,1]).Denote byHF(U) the set of all hesitant fuzzy sets on a reference set (or an initial universe set) U.Definition 2 (see [19]).
LetX be aBCK/BCI-algebra. A hesitant fuzzy set,
(3)H≔{(x,hH(x))∣x∈X},
on X is called ahesitant fuzzy subalgebra of X if it satisfies
(4)(∀x,y∈X)(hH(x*y)⊇hH(x)∩hH(y)).Definition 3 (see [19]).
LetX be aBCK/BCI-algebra. A hesitant fuzzy set,
(5)H≔{(x,hH(x))∣x∈X},
on X is called ahesitant fuzzy ideal of X if it satisfies
(6)(∀x,y∈X)(hH(x*y)∩hH(y)⊆hH(x)⊆hH(0)).Definition 4 (see [6, 16]).
A pair(H~,A) is called ahesitant fuzzy soft set over a reference set U, where H~ is a mapping given by
(7)H~:A⟶HF(U).


## 3. Hesitant Fuzzy Soft Subalgebras

In what follows letE be a set of parameters and we take aBCK/BCI-algebra X as a reference set unless otherwise specified.Definition 5.
For a subsetA of E, a hesitant fuzzy soft set (H~,A) over X is called ahesitant fuzzy soft subalgebra based on e∈A (briefly, e-hesitant fuzzy soft subalgebra) over X if the hesitant fuzzy set,
(8)H~[e]≔{(x,hH~[e](x))∣x∈X},
on X is a hesitant fuzzy subalgebra of X. If (H~,A) is an e-hesitant fuzzy soft subalgebra over X for all e∈A, we say that (H~,A) is ahesitant fuzzy soft subalgebra.Example 6.
LetX={0,a,b,c,d} be aBCK-algebra with the following Cayley table.
(9)*0abcd00000daa0a0dbbb0bdcccc0ddddcbd
Consider a set of parameters E≔{e1,e2,e3,e4,e5}.
(1) Let(H~,A) be a hesitant fuzzy soft set over X where A={e1,e2,e3} which is given in Table 1.
It is routine to verify thatH~[e1], H~[e2], and H~[e3] are hesitant fuzzy subalgebras of X; that is, (H~,A) is a hesitant fuzzy soft subalgebra over X based on parameters “e1,” “e2,” and “e3.” Therefore (H~,A) is a hesitant fuzzy soft subalgebra over X.
(2) Let(G~,B) be a hesitant fuzzy soft set over X where B={e3,e4,e5} which is defined in Table 2.
It is easily checked that(G~,B) is a hesitant fuzzy soft subalgebra based on parameters “e3” and “e5” over X. But it is not an e4-hesitant fuzzy soft subalgebra over X since the hesitant fuzzy set,
(10)G~[e4]≔{(x,hG~[e4](x))∣x∈X},
is not a hesitant fuzzy subalgebra of X. In fact,
(11)hG~[e4](a*c)=hG~[e4](0)={0.3}⊉[0.3,0.4]=hG~[e4](a)∩hG~[e4](c).Table 1
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
0
a
b
c
d
e
1
[
0.2,0.8
]
[
0.2,0.8
]
[
0.2,0.8
]
(
0.2,0.3
)
(
0.2,0.3
)
e
2
[
0.1,0.9
]
[
0.1,0.2
)
∪
(
0.2,0.8
]
{
0.3
}
[
0.3,0.6
]
{
0.3
}
e
3
[
0.3,0.7
]
{
0.4
}
[
0.3,0.5
]
{
0.4
}
{
0.4
}Table 2
Tabular representation of the hesitant fuzzy soft set(G~,B).
G
~
0
a
b
c
d
e
3
[
0.3,0.7
]
[
0.3,0.7
]
[
0.3,0.7
]
{
0.3,0.7
}
{
0.3,0.7
}
e
4
{
0.3
}
[
0.2,0.6
)
[
0.3,0.4
]
[
0.3,0.4
]
{
0.3,0.4
}
e
5
[
0.2,0.5
]
(
0.2,0.3
)
[
0.2,0.5
)
(
0.2,0.3
)
(
0.2,0.3
)Example 7.
LetX={0,1,2,a,b} be aBCI-algebra with the following Cayley table.
(12)*012ab0000aa1101ba2220aaaaaa00bbab10
Given a set E≔{e1,e2,e3} of parameters, let (H~,E) be a hesitant fuzzy soft set over X which is described in Table 3.
Then(H~,E) is a hesitant fuzzy soft subalgebra over X.Table 3
Tabular representation of the hesitant fuzzy soft set(H~,E).
H
~
0
1
2
a
b
e
1
[
0.2,0.8
]
[
0.2,0.5
]
[
0.2,0.3
]
{
0.2
}
{
0.2
}
e
2
[
0.3,0.5
)
{
0.3
}
[
0.3,0.5
)
[
0.3,0.4
]
{
0.3
}
e
3
[
0.2,0.6
]
{
0.2
}
[
0.2,0.4
]
[
0.2,0.3
]
{
0.2
}Proposition 8.
If(H~,A) is a hesitant fuzzy soft subalgebra over X, then
(13)(∀x∈X)(hH~[e](x)⊆hH~[e](0)),
where e is any parameter in A.Proof.
For anyx∈X and e∈A, we have
(14)hH~[e](0)=hH~[e](x*x)⊇hH~[e](x)∩hH~[e](x)=hH~[e](x).
This completes the proof.Theorem 9.
Let(H~,A) be a hesitant fuzzy soft subalgebra over X. If B is a subset of A, then (H~|B,B) is a hesitant fuzzy soft subalgebra over X.Proof.
Straightforward.The following example shows that there exists a hesitant fuzzy soft set(H~,A) over X such that(i)
(H~,A) is not a hesitant fuzzy soft subalgebra over X,(ii)
there exists a subsetB of A such that (H~|B,B) is a hesitant fuzzy soft subalgebra over X.Example 10.
LetX={0,a,b,c,d} be aBCK-algebra as in Example 6. Consider a set A={ei∣i=1,2,3,4,5} of parameters. Let (H~,A) be a hesitant fuzzy soft set over X which is described in Table 4.
Then two hesitant fuzzy sets,(15)H~[e4]≔{(x,hH~[e4](x))∣x∈X},H~[e5]≔{(x,hH~[e5](x))∣x∈X},
on X are not hesitant fuzzy subalgebras of X since
(16)hH~[e4](a*c)=hH~[e4](0)={0,1}⊉[0.1,0.2]=hH~[e4](a)∩hH~[e4](c),hH~[e5](b*b)=hH~[e5](0)=[0.1,0.2]⊉[0.1,0.5]=hH~[e5](b)∩hH~[e5](b),
respectively. Therefore (H~,A) is not a hesitant fuzzy soft subalgebra over X. But if we take B={e1,e2,e3}, then (H~|B,B) is a hesitant fuzzy soft subalgebra over X.Table 4
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
0
a
b
c
d
e
1
[
0.3,0.8
]
[
0.3,0.8
]
[
0.3,0.8
]
[
0.3,0.5
]
[
0.3,0.5
]
e
2
[
0.3,0.9
)
[
0.3,0.8
)
{
0.3
}
[
0.3,0.6
]
{
0.3
}
e
3
[
0.1,0.7
]
{
0.1
}
[
0.1,0.5
]
{
0.1
}
{
0.1
}
e
4
{
0.1
}
[
0.1,0.2
]
[
0.1,0.3
]
[
0.1,0.4
]
[
0.1,0.5
)
e
5
[
0.1,0.2
]
{
0.1
}
[
0.1,0.5
]
[
0.1,0.6
)
{
0.1
}


## 4. Hesitant Fuzzy Soft Ideals

Definition 11.
Let(H~,A) be a hesitant fuzzy soft set over X where A is a subset of E. Given e∈A, (H~,A) is called ahesitant fuzzy soft ideal based on e (briefly, e-hesitant fuzzy soft ideal) over X if the hesitant fuzzy set,
(17)H~[e]≔{(x,hhH~[e](x))∣x∈X},
on X is a hesitant fuzzy ideal of X. If (H~,A) is an e-hesitant fuzzy soft ideal over X for all e∈A, we say that (H~,A) is ahesitant fuzzy soft ideal over X.Example 12.
LetX≔{apple,banana,grape,melon,peach} be a reference set, and consider an operation ⊗ which produces the following products:
(18)apple⊗x=apple∀x∈X,banana⊗y={appleify∈{banana,peaqch,melon},bananaify∈{apple,grape},grape⊗z={grapeifz∈{apple,banana,melon},appleifz∈{grape,peach},peach⊗u={peachifu=apple,bananaifu=grape,appleifu=peach,grapeifu∈{banana,melon},melon⊗v={appleifv=melon,melonifv∈{apple,grape},bananaifv∈{banana,peach}.
Then (X,⊗,apple) is aBCK-algebra. Consider a set of parameters
(19)E≔{cat,cow,dog,duck,horse,pig}.
Let (H~,E) be a hesitant fuzzy soft set over X which is described in Table 5.
Then(H~,E) is a hesitant fuzzy soft ideal over X based on parameters “cat,” “cow,” “dog,” “duck,” and “pig.” But (H~,E) is not a hesitant fuzzy soft ideal of X based on parameter “horse” since
(20)H~[horse](melon)={0.3}⊉[0.3,0.4)=H~[horse](melon⊗peach)∩H~[horse](peach).Table 5
Tabular representation of the hesitant fuzzy soft set(H~,E).
H
~
Apple
Banana
Carrot
Melon
Peach
Cat
[
0.2,0.5
]
[
0.2,0.3
]
[
0.2,0.3
]
[
0.2,0.3
]
[
0.2,0.3
]
Cow
[
0.3,0.6
)
[
0.3,0.4
)
[
0.3,0.6
)
[
0.3,0.4
)
[
0.3,0.4
)
Dog
[
0.2,0.7
]
[
0.2,0.7
]
{
0.2
}
[
0.2,0.7
]
{
0.2
}
Duck
[
0.1,0.3
]
{
0.1
}
[
0.1,0.3
]
{
0.1
}
{
0.1
}
Horse
[
0.3,0.4
)
[
0.3,0.4
)
[
0.3,0.9
)
{
0.3
}
[
0.3,0.6
)
Pig
[
0.1,0.5
)
[
0.1,0.4
]
{
0.1
}
[
0.1,0.4
]
{
0.1
}In general, we know that horses like grapes best of all. In the above example, we know that(H~,E) is not a hesitant fuzzy soft ideal of X based on parameter “horse.” This means that if a horse likes a grape better than the others, then (H~,E) cannot be a hesitant fuzzy ideal over X.Proposition 13.
Every hesitant fuzzy soft ideal(H~,A) over X satisfies the following condition:
(21)00000000000000000000(∀e∈A)(∀x,y,z∈X)(x*y≤z⟹hH~[e](x)⊇hH~[e](y)∩hH~[e](z)).Proof.
Lete∈A and x,y,z∈X be such that x*y≤z. Then (x*y)*z=0, and so
(22)hH~[e](x*y)⊇hH~[e]((x*y)*z)∩hH~[e](z)=hH~[e](0)∩hH~[e](z)=hH~[e](z).
It follows that hH~[e](x)⊇hH~[e](x*y)∩hH~[e](y)⊇hH~[e](y)∩hH~[e](z). This completes the proof.Theorem 14.
Let(H~,A) be a hesitant fuzzy soft set over X which satisfies the condition (13) and (21). Then (H~,A) is a hesitant fuzzy soft ideal over X.Proof.
Lete∈A. Since x*(x*y)≤y for all x,y∈X, it follows from (21) that
(23)hH~[e](x)⊇hH~[e](x*y)∩hH~[e](y).
Hence (H~,A) is an e-hesitant fuzzy soft ideal over X. Since e is arbitrary, we know that (H~,A) is a hesitant fuzzy soft ideal over X.Theorem 15.
In aBCK-algebra X, every hesitant fuzzy soft ideal (based on a parameter) over X is a hesitant fuzzy soft subalgebra (based on the same parameter) over X.Proof.
For anye∈A, let (H~,A) be an e-hesitant fuzzy soft ideal over X. Then
(24)hH~[e](x*y)⊇hH~[e]((x*y)*x)∩hH~[e](x)=hH~[e]((x*x)*y)∩hH~[e](x)=hH~[e](0*y)∩hH~[e](x)=hH~[e](0)∩hH~[e](x)⊇hH~[e](x)∩hH~[e](y),
for all x,y∈X, and so (H~,A) is an e-hesitant fuzzy soft subalgebra over X.The following example shows that the converse of Theorem15 is not true in general.Example 16.
LetX≔{apple,banana,grape,melon,peach} be a universe, and consider an operation ⊟ which produces the following products:
(25)apple⊟x={appleifx∈{grape,peach,melon},grapeifx∈{apple,banana},banana⊟y={bananaify∈{apple,grape,peach,melon},grapeifx=banana,grape⊟z=grape∀z∈U,melon⊟v={melonifv∈{grape,peach},grapeifv∈{apple,banana,melon},peach⊟u={peachifu∈{grape,melon},grapeifu∈{apple,peach,banana}.
Then (X,⊟,grape) is aBCK-algebra. Consider a set of parameters:
(26)B≔{duck,horse,pig}.
Let (H~,B) be a hesitant fuzzy soft set over X which is given in Table 6.
Then(H~,B) is a hesitant fuzzy soft subalgebra over X, but it is not a hesitant fuzzy soft ideal over X based on parameter “horse” since
(27)H~[horse](peach)={0.2}⊉[0.2,0.5)=H~[horse](peach⊗apple)∩H~[horse](apple).Table 6
Tabular representation of the hesitant fuzzy soft set(H~,B).
H
~
Apple
Banana
Grape
Melon
Peach
Duck
{
0.3
}
{
0.3
}
[
0.3,0.7
]
{
0.3
}
[
0.3,0.5
)
Horse
[
0.2,0.5
)
{
0.2
}
[
0.2,0.6
)
{
0.2
}
{
0.2
}
Pig
[
0.3,0.4
)
[
0.3,0.4
]
[
0.3,0.8
)
[
0.3,0.6
]
[
0.3,0.7
)We provide a condition for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal.Theorem 17.
Let(H~,A) be a hesitant fuzzy soft subalgebra over X. Then (H~,A) is a hesitant fuzzy soft ideal over X if and only if the condition (21) is valid.Proof.
Necessity is by Proposition13.
Conversely, assume that the condition (21) is valid. Since x*(x*y)≤y for all x,y∈X, it follows that hH~[e](x)⊇hH~[e](x*y)∩hH~[e](y) for all e∈A and x,y∈X. Combining this and (13), we know that (H~,A) is a hesitant fuzzy soft ideal over X.Proposition 18.
Every hesitant fuzzy soft ideal(H~,A) over aBCI-algebra X satisfies the following inequality:
(28)(∀e∈A)(∀x∈X)(hH~[e](0*(0*x))⊇hH~[e](x)).Proof.
Let(H~,A) be a hesitant fuzzy soft ideal of aBCI-algebra X. Then
(29)hH~[e](0*(0*x))⊇hH~[e]((0*(0*x))*x)∩hH~[e](x)=hH~[e](0)∩hH~[e](x)=hH~[e](x),
for all e∈A and x∈X.The following example shows that a hesitant fuzzy soft ideal over aBCI-algebra (based on a parameter) may not be a hesitant fuzzy soft subalgebra (based on the same parameter).Example 19.
LetX be a reference set which consists of all nonzero rational numbers. Let ÷ be a binary operation which is defined as division in general. Then (X,÷,1) is aBCI-algebra. For a subset A of E, let (H~,A) be a hesitant fuzzy soft set over X in which hH~[e] is defined as follows:
(30)hH~[e](x)≔{[0.2,0.7]ifx∈Z*,[0.5,0.7]otherwise,
for all e∈A and x∈X, where Z* is the set of all nonzero integers. Then (H~,A) is a hesitant fuzzy soft ideal over X, but it is not a hesitant fuzzy soft subalgebra over X since
(31)hH~[e](3÷2)=[0.5,0.7]⊉[0.2,0.7]=hH~[e](3)∩hH~[e](2).Definition 20.
A hesitant fuzzy idealH≔{(x,hH(x))∣x∈X} of a BCI-algebra X is said to beclosed if hH(x)⊆hH(0*x) for all x∈X.Definition 21.
A hesitant fuzzy soft ideal(H~,A) over a BCI-algebra X based on a parameter e∈A is said to beclosed if the hesitant fuzzy set,
(32)H~[e]≔{(x,hH~[e](x))∣x∈X},
on X is a closed hesitant fuzzy ideal of X.Example 22.
The hesitant fuzzy soft subalgebra(H~,E) over X which is described in Example 7 is a closed hesitant fuzzy soft ideal over X.Theorem 23.
In aBCI-algebra X, every closed hesitant fuzzy soft ideal over X based on a parameter is a hesitant fuzzy soft subalgerba over X based on the same parameter.Proof.
Let(H~,A) be a closed hesitant fuzzy soft ideal over X based on a parameter e∈A. Then hH~[e](x)⊆hH~[e](0*x) for all x∈X. It follows that
(33)hH~[e](x*y)⊇hH~[e]((x*y)*x)∩hH~[e](x)=hH~[e](0*y)∩hH~[e](x)⊇hH~[e](x)∩hH~[e](y),
for all x,y∈X. Therefore (H~,A) is a hesitant fuzzy soft subalgebra over X based on the parameter e∈A.Theorem 24.
Let(H~,A) be a hesitant fuzzy soft ideal over a BCI-algebra X based on a parameter e∈A. Then it is closed if and only if it satisfies
(34)(∀x,y∈X)(hH~[e](x*y)⊇hH~[e](x)∩hH~[e](y)).Proof.
Assume that(H~,A) is a closed hesitant fuzzy soft ideal over aBCI-algebra X based on a parameter e∈A. Since (x*y)*x≤0*y for all x,y∈X, it follows from (21) that
(35)hH~[e](x*y)⊇hH~[e](x)∩hH~[e](0*y)⊇hH~[e](x)∩hH~[e](y),
for all x,y∈X.
Conversely, let(H~,A) be a hesitant fuzzy soft ideal over aBCI-algebra X based on a parameter e∈A that satisfies the condition (34). Since hH~[e](x)⊆hH~[e](0) for all x∈X, we have
(36)hH~[e](0*x)⊇hH~[e](0)∩hH~[e](x)=hH~[e](x),
for all x∈X. Therefore (H~,A) is a closed hesitant fuzzy soft ideal over aBCI-algebra X based on a parameter e∈A.


## 5. Conclusion

As another generalization of the standard soft sets, the concept of hesitant fuzzy soft sets has been introduced in [6, 16]. They have presented an application of hesitant fuzzy soft soft sets in a decision making problem. The first author [20] has applied the notion of soft sets by Molodtsov to the theory ofBCK/BCI-algebras. Also, Jun et al. [21] have discussed soft set theory which is applied to ideals in d-algebras. In this paper, we applied the notion of hesitant fuzzy soft sets to the theory ofBCK/BCI-algebras. We introduced the concepts of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals, and then we investigated related properties. We provided relations between a hesitant fuzzy soft subalgebra and a hesitant fuzzy soft ideal. We gave conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra. We also provided conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal. We discussed characterizations of a (closed) hesitant fuzzy soft ideal. On the basis of these results, we will study applications of hesitant fuzzy soft sets to several ideals and filters ofBCK/BCI-algebras,MTL-algebras, MV-algebras, R0-algebras, and so forth.



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*Source: 763929-2014-10-14.xml*