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Journal of Intelligent & Fuzzy Systems 33 (2017) 2391–2402
DOI:10.3233/JIFS-17520
IOS Press
2391
A new study on soft rough fuzzy lattices
(ideals, filters) over lattices
Kuan Yun Zhua,band Bao Qing Hua,b,∗
aSchool of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China
bComputational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, P.R. China
Abstract. In this paper, we investigate the relationship among soft sets, rough sets, fuzzy sets and lattices. The notion of
soft rough fuzzy lattices (ideals, filters) over lattices is introduced, which is an extended notion of soft rough lattices (ideals,
filters) and rough fuzzy lattices (ideals, filters) over lattices. Moreover, we study roughness in lattices with respects to a soft
approximation space. Some new soft rough fuzzy operations over lattices are explored. In particular, lower and upper soft
rough fuzzy lattices (ideals, filters) over lattices with respect to another fuzzy soft set are investigated.
Keywords: Soft rough set, fuzzy sublattice (ideal, filter), soft rough fuzzy lattice (ideal, filter)
1. Introduction
Rough set theory was first introduced by Pawlak
[20], a mathematical tool for dealing with uncer-
tainty. It follows from the definition of rough sets
that any subset of a universe can be characterized by
equivalence relations. As far as known, an equiva-
lence relation on a set partitions the set into disjoint
classes and vice versa. We know that a subset can
be written as union of these classes, which is called
definable, otherwise it is not definable. In this case, it
can be approximated by two definable subsets called
lower and upper approximations of the set. How-
ever, these equivalence relations in Pawlak rough sets
are restrictive to some areas of applications. Thus,
some more general models have been proposed, such
as [27–29]. Nowadays, rough set theory has been
applied to many areas, such as knowledge discovery,
machine learning, data analysis, approximate classi-
fication, conflict analysis, and so on, see [4, 8, 24]. On
the other hand, some researchers applied this theory
∗Corresponding author. Bao Qing Hu, School of Mathematics
and Statistics, Wuhan University, Wuhan 430072, P.R. China. Tel.:
+86 027 68775350; E-mail: bqhu@whu.edu.cn.
to algebraic structures, such as Kuroki [14] proposed
the concept of rough ideals in a semigroup, Davvaz
[6, 7] applied this theory to rings. Further, Yamak
and Kazanc¸i et al. [23] discussed generalized lower
and upper approximations in a ring. Moreover, some
authors also apply rough set theory to other algebraic
structures, see [2, 4, 15, 17].
It is worth noting that the mathematical modelling
and manipulating of various types of uncertainties
has become an increasingly important issue in solv-
ing complicated problems arisings in a wide range of
areas such as economy, engineering, environmental
science, medicine and social science. We known that
fuzzy set theory [25], rough set theory [20] are all
effective tools for dealing with vagueness and uncer-
tainty. However, each of them has certain inherent
limitations. Based on this reason, Molodtsov [19] pro-
posed soft set theory, as a new mathematical tool
for dealing with uncertainties, which is free from
the difficulty affecting the above mentioned meth-
ods. Since then there has been a rapid growth of
interest in soft sets and their various applications.
In 2011, Ali [1] studied another view on reduction
of parameters in soft sets. Afterwards, a wide range
of applications of soft sets have been studied in
1064-1246/17/$35.00 © 2017 – IOS Press and the authors. All rights reserved
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2392 K.Y. Zhu and B.Q. Hu/Anewstudy on soft rough fuzzy lattices (ideals, filters) over lattices
many different fields including game theory, prob-
ability theory, smoothness of functions, operation
researches, Riemann integrations and measurement
theory and so on. Recently, there has been a rapid
growth of interest in soft set theory and its applica-
tions, such as [3, 22]. In particular, Jun and Park [13]
applied this theory to algebraic structures.
Recently, Feng and Li et al. [11, 12] provided
a framework to combine rough sets with soft sets,
which gives rise to some interesting new concepts
such as rough soft sets, soft rough sets and soft rough
fuzzy sets. In particular, in 2013, Shabir and Ali et al.
[21] pointed out that there exist some problems on
Feng’s soft rough set as follows: (1) An upper approx-
imation of a non-empty set may be empty. (2) The
upper approximation of a subset Xmay not contain
the set X. Based on these reasons, Shabir modified
the concept of soft rough set, which is called an
MSR-set. The underlying concepts are very close to
Pawlak rough sets. Further, Li and Xie [16] investi-
gated the relationship among soft sets, soft rough sets
and topologies.
In particular, Feng and Li et al. [11] proposed a
novel concept of soft rough fuzzy sets by combining
rough sets, soft sets with fuzzy sets, we call it Feng-
soft rough fuzzy set. In 2011, Meng and Zhang et al.
[18] further discussed the Feng-soft rough fuzzy sets
and put forward another kind of soft rough fuzzy sets,
we call it Meng-soft rough fuzzy set. However, Feng-
soft rough fuzzy sets and Meng-soft rough fuzzy sets
are all limited on full soft sets which is a rigorous
restrictive condition. Based on the abovereason, Zhan
and Zhu [26] established a novel concept of soft rough
fuzzy sets, called a Z-soft rough fuzzy set. Under this
definition, the restrictive condition, full soft set, can
be removed. Furthermore, it is worth nothing that Z-
soft rough fuzzy sets are more precise than Feng-soft
rough fuzzy sets and Meng-soft rough fuzzy sets. This
means that in the practical application, more precise
information can be obtained.
Based on the above idea, in this paper, we apply this
novel soft rough fuzzy set theory to lattices, which
propose the concept of soft rough fuzzy lattices (ide-
als, filters) over lattices. This paper is organized as
follows: In Section 2, we recall some concepts and
results on lattices, soft sets, fuzzy set and rough sets.
In Section 3, some new soft rough fuzzy operations
over lattices are explored. Further, lower and upper
soft rough fuzzy lattices (ideals, filters) over lattices
are investigated in Section 4. In particular, in Sec-
tion 5, we discuss soft rough fuzzy lattices (ideals,
filters) over lattices based on another fuzzy soft set.
2. Preliminaries
In this section, we recall some basic definitions and
results about lattices, soft sets, fuzzy sets and rough
sets.
A poset (L, ≤) is called a lattice if it satisfies the
condition that for any x, y ∈Lboth x∨yand x∧y
exist, where x∨y= sup{x, y}and x∧y= inf {x, y},
respectively. Throughout this paper, Lis always a
lattice.
Definition 2.1. [5] Let Lbe a lattice and ∅(X⊆L.
Then Xis a sublattice over Lif x, y ∈L,x∨y∈X
and x∧y∈X.
Definition 2.2. [5] Let ∅(I⊆L. Then Iis called
an ideal over Lif
(1) a, b ∈Iimplies x∨y∈I,
(2) a∈L, b ∈Iand a≤bimply a∈I.
Definition 2.3 [5] Let ∅(F⊆L. Then Fis called
a filter over Lif
(1) a, b ∈Fimplies x∧y∈F,
(2) a∈L, b ∈Fand a≥bimply a∈F.
Definition 2.4. [19] A pair S=(F, A) is called a
soft set over U, where A⊆Eand F:A→P(U)is
a set-valued mapping.
Definition 2.5. [11] A soft set S=(F, A) over Uis
called a full soft set if
a∈A
F(a)=U.
Definition 2.6. [9] Let μbe a fuzzy set over L. Then
μis called a fuzzy sublattice over Lif μ(x∧y)∧
μ(x∨y)≥μ(x)∧μ(y) for all x, y ∈L.
Definition 2.7. [9] Let μbe a fuzzy sublattice over
L. Then
(i) μis called a fuzzy ideal over Lif μ(x∨y)=
μ(x)∧μ(y) for all x, y ∈L.
(ii) μis called a fuzzy filter over Lif μ(x∧y)=
μ(x)∧μ(y) for all x, y ∈L.
Definition 2.8. [9] Let μbe a fuzzy sublattice over
L. Then
(i) μis called a fuzzy ideal over Lif and only if
x≤yimplies that μ(x)≥μ(y) for all x, y ∈L.
(ii) μis called a fuzzy filter over Lif and only if
x≤yimplies that μ(x)≤μ(y) for all x, y ∈L.
Definition 2.9. Let (
F,A) be a fuzzy soft set over L.
Then (
F,A) is called a fuzzy soft lattice (ideal, filter)
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if
F(x) is a fuzzy sublattice (ideal, filter) over Lfor
all x∈A.
Remark 2.10. Let μbe a fuzzy set over L. Then μt=
{x∈L:μ(x)≥t},t∈[0,1]. μis a fuzzy sublattice
(ideal, filter) over Lif and only if every non-empty set
μtis a sublattice (ideal, filter) over Lfor all t∈[0,1].
Definition 2.11. [20] Let Rbe an equivalence relation
on the universe Uand (U, R) be a Pawlak approxi-
mation space. A subset X⊆Uis called definable if
R(X)=R(X); otherwise, i.e., R(X)−R(X)/=∅,X
is said to be a rough set, where the two operators are
defined as:
R(X)={x∈U:[x]R⊆X},
R(X)={x∈U:[x]R∩X/=∅}.
Definition 2.12. [10] Let S=(F, A) be a soft set over
U. Then the pair P=(U, S) is called a soft approx-
imation space. Based on P, we define the following
two operators:
ap r P(X)={u∈U:∃a∈A, u ∈F(a)⊆X},
ap r P(X)={u∈U:∃a∈A, u ∈F(a),F(a)
∩X/=∅},
assigning to every subset X⊆U.
Two sets ap r P(X) and ap r P(X) are called the lower
and upper soft rough approximations of Xin P,
respectively. If ap r P(X)=ap r P(X), Xis said to be
soft definable; otherwise, Xis called a soft rough set.
Definition 2.13. [10] Let S=(F, A) be a full soft
set over Uand S=(U, S) be a soft approximation
space. For a fuzzy set μ∈F(U), the lower and upper
soft rough approximations of μwith respect to Sare
denoted by ap r S(μ) and apr S(μ), respectively, which
are fuzzy sets in Ugiven by
ap r S(μ)(x)={μ(y):∃a∈A, {x, y}⊆F(a)},
ap r S(μ)(x)={μ(y):∃a∈A, {x, y}⊆F(a)}
for all x∈U.
The operators ap r Sand apr Sare called the lower
and upper soft rough approximation operators on
fuzzy set μ, respectively. If ap r S(μ)=ap r S(μ), μ
is said to be soft definable; otherwise, μis called
a soft rough fuzzy set. In what follows, to facil-
itate the expression, we call it Feng-soft rough
fuzzy set.
In [18], Meng and Zhang et al. further discussed
Feng-soft rough fuzzy sets and defined a new kind of
lower and upper soft rough approximation operators
as follows.
Definition 2.14. [18] Let S=(F, A) be a full soft
set over Uand S=(U, S) be a soft approximation
space. For a fuzzy set μ∈F(U), the lower and upper
soft rough approximations of μwith respect to Sare
denoted by ap r
S(μ) and ap r
S(μ), respectively, which
are fuzzy sets in Ugiven by
ap r
S(μ)(x)=
x∈F(a)
y∈F(a)
μ(y),
ap r
S(μ)(x)=
x∈F(a)
y∈F(a)
μ(y)
for all x∈U.
The operators ap r Sand apr
Sare called the lower
and upper soft rough approximation operators on
fuzzy set μ, respectively. If ap r
S(μ)=ap r
S(μ), μ
is said to be soft definable; otherwise, μis called
a soft rough fuzzy set. In what follows, to facil-
itate the expression, we call it Meng-soft rough
fuzzy set.
In the above definitions, Feng-soft rough fuzzy sets
and Meng-soft rough fuzzy sets are all limited on
full soft sets which is a rigorous restrictive condition.
Based on the above reason, Zhan and Zhu [26] estab-
lished a novel concept of soft rough fuzzy sets. Under
this definitions, the restrictive condition, full soft set,
can be removed.
Definition 2.15. [26] Let (F, A) be a soft set over U
and δ:U→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then the pair (U, δ) is called a soft
approximation space. For any fuzzy set μ∈F(U),
the lower and upper soft rough approximations of μ
are denoted by μδand μδ, respectively, which are
fuzzy sets in Ugiven by
μδ(x)={μ(y):y∈U, δ(y)=δ(x)},
μδ(x)={μ(y):y∈U, δ(y)=δ(x)}
for every x∈U.
The operators μδand μδare called the lower and
upper soft rough approximation operators on a fuzzy
set, respectively. In particular, if μδ=μδ,μis said to
be soft definable; otherwise, μis called a soft rough
fuzzy set.
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3. Some operations of soft rough fuzzy sets
over lattices
In this section, we investigate some operations and
some properties of soft rough fuzzy sets over lattices
are discussed. In order to illustrate the roughness in
lattice Lwith respect to soft approximation spaces,
we first introduce two special kinds of soft sets over L.
Definition 3.1. Let S=(F, A) be a soft set over L
and δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then Sis called a C-soft set over L
if δ(a)=δ(b) and δ(c)=δ(d) imply δ(a∨c)=δ(b∨
d) and δ(a∧c)=δ(b∧d) for all a, b, c, d ∈L.
Definition 3.2. Let S=(F, A)beaC-soft set over
Land δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then Sis called a CC-soft set over
Lif for all c∈L,
(i) δ(c)=δ(x∨y) for x, y ∈L, there exist a, b ∈
Lsuch that δ(x)=δ(a) and δ(y)=δ(b) satis-
fying c=a∨b;
(ii) δ(c)=δ(x∧y) for x, y ∈L, there exist a, b ∈
Lsuch that δ(x)=δ(a) and δ(y)=δ(b) satis-
fying c=a∧b.
The following definition is from Zadeh’s expansion
principle.
Definition 3.3. Let μand νbe two fuzzy sets
over L. Define μ∨νand μ∧νover Las follows:
(μ∨ν)(x)=
x=a∨b
(μ(a)∧ν(b)), and (μ∧ν)(x)=
x=a∧b
(μ(a)∧ν(b)) for all x∈L, respectively.
Proposition 3.4. Let S=(F, A)be a C-soft set over
L.Ifμand νare any two fuzzy sets over L, then
μδ∨νδ⊆(μ∨ν)δ.
Proof. Let x=a∨b, a, b ∈L. Then
(μδ∨νδ)(x)=
x=a∨b
(μδ(a)∧νδ(b))
=
x=a∨b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)}).
Since S=(F, A)isaC-soft set over L,wehave
x=a∨b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)})
=
x=a∨b{μ(c)∧ν(d)|δ(c)=δ(a),δ(d)=δ(b)}
≤
x=a∨b{μ(c)∧ν(d)|δ(c∨d)
=δ(a∨b)=δ(x)}
={μ(c)∧ν(d)|δ(c∨d)=δ(x)}
≤{(μ∨ν)(y)|δ(y)=δ(x)}
=(μ∨ν)δ(x).
It follows that (μδ∨νδ)(x)≤(μ∨ν)δ(x), i.e.,
μδ∨νδ⊆(μ∨ν)δ.
Proposition 3.5. Let S=(F, A)be a C-soft set over
L.Ifμand νare any two fuzzy sets over L, then
μδ∧νδ⊆(μ∧ν)δ.
Proof. Let x=a∧b, a, b ∈L. Then
(μδ∧νδ)(x)=
x=a∧b
(μδ(a)∧νδ(b))
=
x=a∧b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)}).
Since S=(F, A)isaC-soft set over L,wehave
x=a∧b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)})
=
x=a∧b{μ(c)∧ν(d)|δ(c)=δ(a),δ(d)=δ(b)}
≤
x=a∧b{μ(c)∧ν(d)|δ(c∧d)
=δ(a∧b)=δ(x)}
={μ(c)∧ν(d)|δ(c∧d)=δ(x)}
≤{(μ∧ν)(y)|δ(y)=δ(x)}
=(μ∧ν)δ(x).
It follows that (μδ∧νδ)(x)≤(μ∧ν)δ(x), i.e.,
μδ∧νδ⊆(μ∧ν)δ.
The following example shows that the containment
in Propositions 3.4 and 3.5 is proper.
Example 3.6. Let L={0,a,b,c,d,1}. We define
the binary relation ≤in the Fig. 1. S=(F, A)isa
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Table 1
Soft set S
0abcd1
e1111011
e2000111
e3110111
soft set over Lwhich is given by Table 1. Then the
mapping δ:L→P(A) in soft approximation space
(L, δ)isgivenbyδ(0) =δ(a)={e1,e
3},δ(b)={e1},
δ(c)={e2,e
3},δ(d)=δ(1) ={e1,e
2,e
3}. Then we
can check that SisaC-soft set over L.If
we take μ=0.3
band ν=0.4
c, then μδ=
0.3
band νδ=0.4
b.Soμδ∨νδ=0.4
d,μδ∧
νδ=0.3
aand (μ∨ν)δ=0.3
d,0.3
1,(μ∧ν)δ=
0.3
0,0.3
a. Thus, μδ∨νδ$(μ∨ν)δ,μδ∧νδ$
(μ∧ν)δ.
If we strength the condition, we can obtain the
following result.
Proposition 3.7. Let S=(F, A)beaCC-soft set
over L.Ifμand νare any two fuzzy sets over L, then
μδ∨νδ=(μ∨ν)δ.
Proof. It follows from Proposition 3.4 that we only
need to show (μ∨ν)δ⊆μδ∨νδ.
(μ∨ν)δ(x)={(μ∨ν)(y)|δ(y)=δ(x)}
={
y=a∨b
μ(a)∧ν(b)|δ(a∨b)
=δ(x)}.
Since S=(F, A)isaCC-soft set over L, there
exist c, d ∈Lsuch that δ(a)=δ(c),δ(b)=δ(d) sat-
isfying x=c∨d.Sowehave
{
y=a∨b
μ(a)∧ν(b)|δ(a∨b)=δ(x)}
=
y=a∨b
{μ(a)∧ν(b)|δ(a)=δ(c),δ(b)=δ(d)}
=
y=a∨b
({μ(a)|δ(a)=δ(c)})∧({ν(b)|δ(b)
=δ(d)})
=({μ(a)|δ(a)=δ(c)})∧({ν(b)|δ(b)
=δ(d)})
=μδ(c)∧νδ(d)
≤
x=a∨b
μδ(a)∧νδ(b)
=(μδ∨νδ)(x).
Thus, (μ∨ν)δ⊆μδ∨νδ.
Proposition 3.8. Let S=(F, A)beaCC-soft set
over L.Ifμand νare any two fuzzy sets over L, then
μδ∧νδ=(μ∧ν)δ.
Proof. It follows from Proposition 3.5 that we only
need to show (μ∧ν)δ⊆μδ∧νδ.
(μ∧ν)δ(x)={(μ∧ν)(y)|δ(y)=δ(x)}
={
y=a∧b
μ(a)∧ν(b)|δ(a∧b)
=δ(x)}.
Since S=(F, A)isaCC-soft set over L, there
exist c, d ∈Lsuch that δ(a)=δ(c),δ(b)=δ(d) sat-
isfying x=c∧d.Sowehave
{
y=a∗b
μ(a)∧ν(b)|δ(a∧b)=δ(x)}
=
y=a∧b
{μ(a)∧ν(b)|δ(a)=δ(c),δ(b)
=δ(d)}
=(
y=a∧b{μ(a)|δ(a)=δ(c)})∧({ν(b)|δ(b)
=δ(d)})
=({μ(a)|δ(a)=δ(c)})∧({ν(b)|δ(b)
=δ(d)})
=μδ(c)∧νδ(d)
≤
x=a∧b
μδ(a)∧νδ(b)
=(μδ∧νδ)(x).
Thus, (μ∧ν)δ⊆μδ∧νδ.
Next, we consider lower soft rough fuzzy approx-
imations over L.
Proposition 3.9. Let S=(F, A)beaCC-soft set
over L.Ifμand νare any two fuzzy sets over L, then
μδ∨νδ⊆(μ∨ν)δ.
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Proof. Let x=a∨b, a, b ∈L. Then
(μδ∨νδ)(x)=
x=a∨b
(μδ(a)) ∧(νδ(b))
=
x=a∨b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)}).
Since S=(F, A)isaCC-soft set over L,wehave
x=a∨b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)})
=
x=a∨b{μ(c)∧ν(d)|δ(c)=δ(a),δ(d)=δ(b)}
≤
x=a∨b{μ(c)∧ν(d)|δ(c∨d)=δ(a∨b)
=δ(x)}
={(μ∨ν)(y)|δ(y)=δ(x)}
=(μ∨ν)δ(x).
Thus, (μδ∨νδ)(x)≤(μ∨ν)δ(x), i.e., μδ∨νδ⊆
(μ∨ν)δ.
Proposition 3.10. Let S=(F, A)be a CC-soft set
over L.Ifμand νare any two fuzzy sets over L, then
μδ∧νδ⊆(μ∧ν)δ.
Proof. Let x=a∧b, a, b ∈L. Then
(μδ∧νδ)(x)=
x=a∧b
(μδ(a)∧νδ(b))
=
x=a∧b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)}).
Since S=(F, A)isaCC-soft set over L,wehave
x=a∧b
({μ(c)|c∈L, δ(c)=δ(a)})
∧({ν(d)|d∈L, δ(d)=δ(b)})
=
x=a∧b{μ(c)∧ν(d)|δ(c)=δ(a),δ(d)=δ(b)}
≤
x=a∧b{μ(c)∧ν(d)|δ(c∧d)=δ(a∧b)
=δ(x)}
={(μ∧ν)(y)|δ(y)=δ(x)}
=(μ∧ν)δ(x).
Thus, (μδ∧νδ)(x)≤(μ∧ν)δ(x), i.e.,
μδ∧νδ⊆(μ∧ν)δ.
The following example shows that the con-
tainments in Propositions 3.9 and 3.10 are
proper.
Example 3.11. Let L={0,a,b,c,1}. We define the
binary relation ≤in the Fig. 2. S=(F, A) is a soft set
over Lwhich is given by Table 2. Then the mapping
δ:L→P(A) in soft approximation space (L, δ)is
given by δ(0) =δ(b)={e1},δ(a)=δ(c)={e1,e
2},
δ(1) ={e1,e
2,e
3}. Then we can check that Sis a
CC-soft set over L.Ifwetakeμ=0.4
band ν=
0.2
a,0.5
c, then μδ=∅,νδ=0.2
a,0.5
cand μδ∨
νδ=∅,μδ∧νδ=∅. On the other hand, (μ∨ν)δ=
0.2
a,0.3
c,(μ∧ν)δ=0.2
0,0.2
b. This means that
μδ∨νδ$(μ∨ν)δand μδ∧νδ$(μ∧ν)δ
Fig. 1. A lattice L.
Fig. 2. A lattice L.
Table 2
Soft set S
0abc1
e111111
e201011
e300001
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4. Characterizations of soft rough fuzzy
lattices (ideals, filters) over lattices
In this section, we characterize soft rough fuzzy
lattices (ideals, filters) over lattices. First, we give the
concept of soft rough fuzzy lattices (ideals, filters)
over L.
Definition 4.1. Let (F, A) be a soft set over Land
δ:L→P(A) be a mapping defined as δ(x)={a:
x∈F(a)}. Then the pair (L, δ) is called a soft approx-
imation space. For any fuzzy set μ∈F(L), the lower
and upper soft rough approximations of μare denoted
by μδand μδ, respectively, which are fuzzy sets in L
given by
μδ(x)={μ(y):y∈L, δ(y)=δ(x)},
μδ(x)={μ(y):y∈L, δ(y)=δ(x)}
for every x∈L.ifμδ/=μδ,
(i) μis called a lower (upper) soft rough fuzzy
lattice (ideal, filter) w.r.t. Sover L,ifμδ(μδ)
is a sublattice (ideal, filter) over L;
(ii) μis called a soft rough fuzzy lattice (ideal, fil-
ter) w.r.t. Sover L,ifμδand μδare fuzzy
sublattices (ideals, filters) over L.
Example 4.2. Consider the lattice Land the soft set
S=(F, A) in Example 3.11. It follows from Defini-
tion 4.1 that for a fuzzy set μ=0.3
0,0.4
b,0.2
c,
μδ=0.2
0,0.3
band
μδ=0.4
0,0.2
a,0.4
b,0.2
c.
It is easy to check that μδand μδare fuzzy sublat-
tices (ideals) over L. In other words, μis a soft rough
fuzzy lattice (ideal) over X.
Proposition 4.3. Let S=(F, A)be a C-soft set over
L.Ifμis a fuzzy sublattice over L, then μis an upper
soft rough fuzzy lattice over L.
Proof. For any x, y ∈L, since S=(F, A)isaC-soft
set over Land μis a fuzzy sublattice of X,
μδ(x)∧μδ(y)
=({μ(a)|δ(a)=δ(x)})
∧({μ(b)|δ(b)=δ(y)})
≤{μ(a)∧μ(b)|δ(a∨b)=δ(x∨y)}
={μ(a∨b)|δ(a∨b)=δ(x∨y)}
=μδ(x∨y).
μδ(x)∧μδ(y)
=({μ(a)|δ(a)=δ(x)})
∧({μ(b)|δ(b)=δ(y)})
≤{μ(a)∧μ(b)|δ(a∧b)=δ(x∧y)}
={μ(a∧b)|δ(a∧b)=δ(x∧y)}
=μδ(x∧y).
This implies that μδ(x∨y)∧μδ(x∧y)≥
μδ(x)∧μδ(y). So μδis a fuzzy sublattice over L.It
follows from Definition 4.1 that μis an upper soft
rough fuzzy lattice over L.
Proposition 4.4. Let S=(F, A)beaCC-soft set
over L.Ifμis a fuzzy sublattice over L, then μis a
lower soft rough fuzzy lattice over L.
Proof. For any x, y ∈L, since S=(F, A)isa
CC-soft set over Land μis a fuzzy sublattice
of L,
μδ(x)∧μδ(y)
=({μ(a)|δ(a)=δ(x)})
∧({μ(b)|δ(b)=δ(y)})
={μ(a)∧μ(b)|δ(a∨b)=δ(x∨y)}
≤{μ(a∨b)|δ(a∨b)=δ(x∨y)}
=μδ(x∨y).
μδ(x)∧μδ(y)
=({μ(a)|δ(a)=δ(x)})
∧({μ(b)|δ(b)=δ(y)})
={μ(a)∧μ(b)|δ(a∧b)=δ(x∧y)}
≤{μ(a∧b)|δ(a∧b)=δ(x∧y)}
=μδ(x∧y).
This implies that μδ(x∨y)∧μδ(x∧y)≥
μδ(x)∧μδ(y). So μδis a fuzzy sublattice over L.
AUTHOR COPY
2398 K.Y. Zhu and B.Q. Hu/Anewstudy on soft rough fuzzy lattices (ideals, filters) over lattices
It follows from Definition 4.1 that μis a lower soft
rough fuzzy lattice over L.
Definition 4.5. Let S=(F, A)beaC-soft set over
Land δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then Sis called a BC-soft set over L
if Iis an ideal over L,∀a, b ∈L,δ(a)=δ(b)ifand
only if there exist i1,i
2∈Isuch that a∨i1=b∨i2.
Theorem 4.6. Let S=(F, A)be a BC-soft set over
L.Ifμis a fuzzy ideal over L, then μis an upper soft
rough fuzzy ideal over L.
Proof. Let μbe a fuzzy ideal over L. Then μis a
fuzzy sublattice over L. It follows from Proposition
4.3 that μδis a fuzzy sublattice over L. Now let x≤
y, x, y ∈Lbut μδ(x)< μδ(y). Then μδ(x)< μδ(x∨
y)∧μδ(y). Choose t∈[0,1] such that μδ(x)<t<
μδ(x∨y)∧μδ(y). Since μ⊆μδ,wehaveμ(x)≤
μδ(x)<t. Then
μδ(x∨y)∧μδ(y)
=({μ(a)|δ(a)=δ(x∨y)})
∧({μ(b)|δ(b)=δ(y)})
={μ(a)∧μ(b)|δ(a)
=δ(x∨y),δ(b)=δ(y)}.
Since μis a fuzzy ideal of L, it follows from
Remark 2.10 that μtis an ideal over L. Because
δ(a)=δ(x∨y),δ(b)=δ(y) and S=(F, A)isaBC-
soft set over L, there exist i1,i
2,i
3,i
4∈μtsuch that
a∨i1=(x∨y)∨i2,b∨i3=y∨i4. Since [(x∨
y)∨i2]∨i4=[(x∨y)∨i2]∨i4,wehave[(x∨
y)∨i4]∨i2=[(x∨y)∨i2]∨i4, that is
[x∨(y∨i4)] ∨i2=[(x∨y)∨i2]∨i4,
[x∨(b∨i3)] ∨i2=(a∨i1)∨i4,
[(x∨b)∨i3]∨i2=(a∨i1)∨i4,
(x∨b)∨(i3∨i2)=a∨(i1∨i4).
It follows from Definition 4.5 that δ(x∨b)=δ(a),
where i3∨i2∈μt,i
1∨i4∈μt. Thus
μδ(x∧y)∧μδ(y)
={μ(a)∧μ(b)|δ(a)=δ(x∨y),δ(b)=δ(y)}
={μ(a)∧μ(b)|(x∨b)∨(i3∨i2)
=a∨(i1∨i4)}
={μ(a)∧μ(b)|δ(x∨b)=δ(a)}
>t.
This means that there exist a0,b
0∈Lsuch that
μ(a0)∧μ(b0)>t satisfying δ(x∨b0)=δ(a0). So
there exist i5,i
6∈μtsuch that (x∗b0)∨i5=a0∨
i6and μ(a0)∧μ(b0)>t. That is x∨(b0∨i5)=
a0∨i6and μ(a0)∧μ(b0)>t.Sox≤a0∨i6. Since
μis a fuzzy ideal over L,μ(x)≥μ(a0∨i6). Thus
μ(x)≥μ(x)∧μ(b0∨i5)
≥μ(a0∨i6)∧μ(b0∨i5)
=μ(a0)∧μ(i6)∧μ(b0)∧μ(i5)
≥t.
This is a contradicts with μ(x)<t.
Hence μδ(x)≥μδ(y) for all x≤y, x, y ∈L. This
implies that μδis a fuzzy ideal over L, that is, μis
an upper soft rough fuzzy ideal over L.
Remark 4.7. Let S=(F, A)beaCC-soft set over
Land δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. If for all c∈L, then the following
hold.
(i) δ(c)=δ(x∨y) for x, y ∈Lif and only if for
δ(x)=δ(a) and δ(y)=δ(b)wehavec=a∨b,
a, b ∈L.
(ii) δ(c)=δ(x∧y) for x, y ∈Lif and only if for
δ(x)=δ(a) and δ(y)=δ(b)wehavec=a∧b,
a, b ∈L.
Proof. It is straightforward.
Theorem 4.8. Let S=(F, A)beaCC-soft set over
L.Ifμis a fuzzy ideal over L, then μis a lower soft
rough fuzzy ideal over L.
Proof. Let μbe a fuzzy ideal over L. Then μis a
fuzzy sublattice over L. It follows from Proposition
4.4 that μδis a fuzzy sublattice over L. Now let x≤
y, x, y ∈L.
μδ(y)=μδ(x∨y)∧μδ(y)
=({μ(a)|δ(a)=δ(x∨y)})
∧({μ(b)|δ(b)=δ(y)})
={μ(a)∧μ(b)|δ(a)
=δ(x∨y),δ(b)=δ(y)}.
AUTHOR COPY
K.Y. Zhu and B.Q. Hu/Anewstudy on soft rough fuzzy lattices (ideals, filters) over lattices 2399
Since S=(F, A)isaCC-soft set over L, it fol-
lows from Remark 4.7 that a=c∨dfor δ(x)=
δ(c),δ(y)=δ(d), where c, d ∈L. Thus
μδ(x∨y)∧μδ(y)
={μ(a)∧μ(b)|δ(x)=δ(c),
δ(d)=δ(b),a =c∨d}
={μ(c∨d)∧μ(b)|δ(x)=δ(c),
δ(b)=δ(d),a =c∨d}
={μ(c∨d)|δ(x)=δ(c)}
∧{μ(b)|δ(b)=δ(d)}
=({μ(c∨d)|δ(x)=δ(c)})∧μδ(d).
Since μδ⊆μand μis a fuzzy ideal over L,c∨
d≥c,wehaveμ(c)≥μ(c∨d). Thus,
({μ(c∨d)|δ(x)=δ(c)})∧μδ(d)
≤{μ(c∨d)∧μ(d)|δ(x)=δ(c)}
≤{μ(c)∧μ(d)|δ(c)=δ(x)}
≤{μ(c)|δ(c)=δ(x)}
=μδ(x).
Thus, μδ(x)≥μδ(y) for all x≤y, x, y ∈L. This
implies that μδis a fuzzy ideal over L, that is, μis a
lower soft rough fuzzy ideal over L.
Definition 4.9. Let S=(F, A)beaC-soft set over
Land δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then Sis called a BC-soft set over
Lif Dis a filter over L,∀a, b ∈Lδ(a)=δ(b)ifand
only if there exist i1,i
2∈Dsuch that a∧i1=b∧i2.
Theorem 4.10. Let S=(F, A)be a BC-soft set over
L.Ifμis a fuzzy filter over L, then μis an upper soft
rough fuzzy filter over L.
Proof. Let μbe a fuzzy ideal over L. Then μis a fuzzy
sublattice over L. It follows from Proposition 4.3 that
μδis a fuzzy sublattice over L. Let x≤y, x, y ∈L
but μδ(y)< μδ(x). Then μδ(y)< μδ(x∧y)∧μδ(x).
Choose t∈[0,1] such that μδ(y)<t<μδ(x∧y)∧
μδ(x). Since μ⊆μδ,wehaveμ(y)≤μδ(y)<t.
Then
μδ(x∧y)∧μδ(x)
=({μ(a)|δ(a)=δ(x∧y)})
∧({μ(b)|δ(b)=δ(x)})
={μ(a)∧μ(b)|δ(a)
=δ(x∧y),δ(b)=δ(x)}.
Since μis a fuzzy filter over L, it follows from
Remark 2.10 that μtis a filter over L. Because δ(a)=
δ(x∧y),δ(b)=δ(y) and S=(F, A)isaBC-soft
set over L, there exist i1,i
2,i
3,i
4∈μtsuch that a∧
i1=(x∨y)∧i2,b∧i3=y∧i4. Since [(x∧y)∧
i2]∨i4=[(x∧y)∧i2]∨i4,wehave[(x∧y)∧
i4]∧i2=[(x∧y)∧i2]∧i4, that is
[y∨(x∧i4)] ∧i2=[(x∧y)∧i2]∧i4,
[y∧(b∨i3)] ∧i2=(a∧i1)∧i4,
[(y∨b)∧i3]∧i2=(a∧i1)∧i4,
(y∧b)∧(i3∧i2)=a∧(i1∧i4).
It follows from Definition 4.9 that δ(y∧b)=δ(a),
where i3∧i2∈μt,i
1∧i4∈μt. Thus
μδ(x∧y)∧μδ(x)
={μ(a)∧μ(b)|δ(a)
=δ(x∧y),δ(b)=δ(x)}
={μ(a)∧μ(b)|(x∧b)∨(i3∧i2)
=a∨(i1∧i4)}
={μ(a)∧μ(b)|δ(x∧b)=δ(a)}
>t.
This means that there exist a0,b
0∈Lsuch that
μ(a0)∧μ(b0)>t satisfying δ(x∨b0)=δ(a0). So
there exist i5,i
6∈μtsuch that (x∧b0)∧i5=a0∨
i6and μ(a0)∧μ(b0)>t. That is y∧(b0∨i5)=
a0∧i6and μ(a0)∧μ(b0)>t.Soy≥a0∨i6. Since
μis a fuzzy filter over L,μ(y)≥μ(a0∧i6).
Thus
μ(y)≥μ(y)∧μ(b0∧i5)
≥μ(a0∧i6)∧μ(b0∧i5)
=μ(a0)∧μ(i6)∧μ(b0)∧μ(i5)
≥t.
This is a contradicts with μ(y)<t.
AUTHOR COPY
2400 K.Y. Zhu and B.Q. Hu/Anewstudy on soft rough fuzzy lattices (ideals, filters) over lattices
Hence μδ(x)≤μδ(y) for all x≤y, x, y ∈L. This
implies that μδis a fuzzy filter over L, that is, μis an
upper soft rough fuzzy filter over L.
Theorem 4.11. Let S=(F, A)beaCC-soft set over
L.Ifμis a fuzzy filter over L, then μis a lower soft
rough fuzzy filter over L.
Proof. Let μbe a fuzzy ideal over L. Then μis a
fuzzy sublattice over L. It follows from Proposition
4.4 that μδis a fuzzy sublattice over L. Let x≤y, x,
y∈L.
μδ(x)
=μδ(x∧y)∧μδ(x)
=({μ(a)|δ(a)=δ(x∧y)})∧({μ(b)|δ(b)
=δ(x)})
={μ(a)∧μ(b)|δ(a)=δ(x∧y),δ(b)=δ(x)}.
Since S=(F, A)isaCC-soft set over L, it fol-
lows from Remark 4.7 that a=c∧dfor δ(x)=
δ(d),δ(y)=δ(c), where c, d ∈L. Thus
μδ(x∧y)∧μδ(y)
={μ(a)∧μ(b)|δ(y)=δ(c),
δ(d)=δ(b),a =c∧d}
={μ(c∧d)∧μ(b)|δ(y)=δ(c),
δ(b)=δ(d),a =c∧d}
=({μ(c∧d)|δ(y)=δ(c)})
∧({μ(b)|δ(b)=δ(d)})
=({μ(c∧d)|δ(y)=δ(c)})∧μδ(d).
Since μδ⊆μand μis a fuzzy filterl over L,c∧
d≤c,wehaveμ(c)≥μ(c∧d). Thus,
{μ(c∧d)|δ(x)
=δ(c)}∧μδ(d)
≤{μ(c∧d)∧μ(d)|δ(y)=δ(c)}
≤{μ(c)∧μ(d)|δ(c)=δ(y)}
≤{μ(c)|δ(c)=δ(y)}
=μδ(y).
Thus, μδ(x)≤μδ(y) for all x≤y, x, y ∈L. This
implies that μδis a fuzzy filter over L, that is, μis a
lower soft rough fuzzy filter over L.
Remark 4.12. The above theorems show that any soft
rough fuzzy lattice (ideal, filter) is a generalization of
a fuzzy lattice (ideal, filter) over lattices.
5. Soft rough fuzzy lattices (ideals, filters)
over lattices with respect to another fuzzy
soft set
In this section, we introduce the concept of approx-
imations on an information system with respect to
another information system, and lower and upper soft
rough fuzzy lattices (ideals, filters) with respect to
another fuzzy soft set.
Definition 5.1. Let S=(F, A) be a soft set over L
and δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Let T=(G, B) be a fuzzy soft set
defined over L. The lower and upper soft rough
approximations of Twith respect to Sare denoted by
(G, B)δ=(Gδ,B) and (G, B)δ=(Gδ,B), respec-
tively, which are two operators defined as
G(e)δ(x)={G(e)(y)|y∈L, δ(y)=δ(x)},
G(e)δ(x)={G(e)(y)|y∈L, δ(y)=δ(x)}
for all e∈B, x ∈L.
(i) If (G, B)δ=(G, B)δ, then Tis called soft
definable.
(ii) If (G, B)δ/=(G, B)δand G(e)δ(G(e)δ)isa
fuzzy sublattice (ideal, filter) over Lfor all e∈
B, then Tis called a lower (upper) soft rough
fuzzy sublattice (ideal, filter) with respect to
Sover L. Moreover, Tis called a lower
(upper) soft rough fuzzy lattice (ideal, filter)
with respect to Sover Lif G(e)δand G(e)δare
fuzzy sublattices (ideals, filter) with respect to
Sover Lfor all e∈B.
Example 5.2. We consider the lattice Land soft
set S=(F, A) in Example 3.11. Define a soft
set T=(G, B) as the following Table 3. By cal-
Table 3
Soft set S
0abc1
e10.80000.2
e20000.30.1
e30.70.50.30 0
AUTHOR COPY
K.Y. Zhu and B.Q. Hu/Anewstudy on soft rough fuzzy lattices (ideals, filters) over lattices 2401
culating, G(e1)δ=0.2
1,G(e1)δ=0.8
0,0.8
b,0.8
1,
G(e2)δ={
0.1
1},G(e1)δ={
0.3
0,0.3
b,0.1
1},G(e3)δ=
{0.3
0,0.3
b},G(e3)δ={
0.7
0,0.5
a,0.7
b,0.5
0}. It is easy to
check that (G, B)δand (G, B)δare sublattices (ideas)
over Lfor all e∈B. In other words, Tis a soft rough
fuzzy lattice (ideal) with respect to Sover L.
Definition 5.3. Let T=(G, B) and I=(H, C )
be two fuzzy soft sets over Lwith D=B∩
C/=∅. The ∨-operation and ∧-operation of T∨
Iand T∧Iare defined as T∨I=(G, B)∨
(H, C)=(K, D) and T∧I=(G, B)∨(H, C )=
(L, D), respectively, where K(a)=G(a)∨H(a) and
L(a)=G(a)∧H(a) for all a∈D.
Proposition 5.4. Let S=(F, A)be a C-soft set over
L. Let T1=(G1,B)and T2=(G2,C)be two fuzzy
soft sets over Lwith D=B∩C/=∅. Then
(1) (G1,B)δ∨(G2,C)δ⊆(G1∨G2,D)δ.
(2) (G1,B)δ∧(G2,C)δ⊆(G1∧G2,D)δ.
Proof. The proof is similar to that of Propositions 3.4
and 3.5.
Proposition 5.5. Let S=(F, A)beaCC-soft set
over L. Let T1=(G1,B)and T2=(G2,C)be two
fuzzy soft sets over Lwith D=B∩C/=∅. Then
(1) (G1,B)δ∨(G2,C)δ=(G1∨G2,D)δ.
(2) (G1,B)δ∧(G2,C)δ=(G1∧G2,D)δ.
Proof. The proof is similar to that of Propositions 3.7
and 3.8.
Proposition 5.6. Let S=(F, A)beaCC-soft set
over L. Let T1=(G1,B)and T2=(G2,C)be two
fuzzy soft sets over Xwith D=B∩C/=∅. Then
(1) (G1,B)δ∨(G2,C)δ⊆(G1∨G2,D)δ.
(2) (G1,B)δ∧(G2,C)δ⊆(G1∧G2,D)δ.
Proof. The proof is similar to that of Propositions 3.9
and 3.10.
Finally, we investigate the lower and upper soft
rough fuzzy sublattice (ideal, filter) with respect to a
fuzzy soft set.
Proposition 5.7. Let S=(F, A)be a C-soft set over
L.IfT=(G, B)is a fuzzy soft sublattice over L,
then T=(G, B)is a upper soft rough fuzzy lattice
with respect to Sover L.
Proof. The proof is similar to that of Proposi-
tion 4.3.
Proposition 5.8. Let S=(F, A)beaCC-soft set
over L.IfT=(G, B)is a fuzzy soft sublattice over
L, then T=(G, B)is a lower soft rough fuzzy lattice
with respect to Sover L.
Proof. The proof is similar to that of Proposi-
tion 4.4.
Definition 5.9. Let S=(F, A)beaC-soft set over
Land δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then Sis called a BC-soft set over L
if (G, B) is a fuzzy soft ideal over Land ∀a, b ∈L,
δ(a)=δ(b) if and only if there exist i1,i
2∈G(e)t
such that a∨i1=b∨i2, where G(e)t={x∈L:
G(e)(x)≥t}, for all e∈B,t∈[0,1].
Theorem 5.10. Let S=(F, A)be a BC-soft set over
L.IfT=(G, B)is a fuzzy ideal over L, then Tis a
upper soft rough fuzzy ideal over L.
Proof. The proof is similar to that of Proposition 4.6.
Theorem 5.11. Let S=(F, A)beaCC-soft set over
L.IfT=(G, B)is a fuzzy soft ideal over L, then T
is a lower soft rough fuzzy ideal over L.
Proof. The proof is similar to that of Proposition 4.8.
Definition 5.12. Let S=(F, A)beaC-soft set over
Land δ:L→P(A) be a mapping defined as δ(x)=
{a:x∈F(a)}. Then Sis called a BC-soft set over
Lif (G, B) is a fuzzy soft filter over Land ∀a, b ∈L,
δ(a)=δ(b) if and only if there exist i1,i
2∈G(e)t
such that a∧i1=b∧i2, where G(e)t={x∈L:
G(e)(x)≥t}, for all e∈B,t∈[0,1].
Theorem 5.13. Let S=(F, A)be a BC-soft set over
L.IfT=(G, B)is a fuzzy soft filter over L, then T
is a upper soft rough fuzzy filter over L.
Proof. The proof is similar to that of Proposition 4.10.
Theorem 5.14. Let S=(F, A)beaCC-soft set over
L.IfT=(G, B)is a fuzzy soft filter over L, then T
is a lower soft rough fuzzy filter over L.
Proof. The proof is similar to that of Proposition 4.11.
6. Conclusion
This paper aims at providing a framework to com-
bine soft sets, rough sets, fuzzy sets with lattices all
together, which propose the concept of soft rough
AUTHOR COPY
2402 K.Y. Zhu and B.Q. Hu/Anewstudy on soft rough fuzzy lattices (ideals, filters) over lattices
fuzzy lattices (ideals, filters) over lattices. The main
conclusions in this paper and the further work to do
are listed as follows.
(1) Some operations and fundamental properties
of soft rough fuzzy sets over lattices are investi-
gated, which can provide a theoretical research
method for other algebraic structures.
(2) By characterizations of lattices, we studied
roughness in lattices with respect to soft
approximation spaces. In particular, lower and
upper soft rough fuzzy lattices (ideals, filters)
over lattices have been investigated.
(3) We introduce the concept of approxima-
tions on an information system with respect
to another information system, lower and
upper soft rough fuzzy lattices (ideals, fil-
ters) with respect to another fuzzy soft set are
investigated.
As an extension of this work, the following topics
maybe considered:
(i) Constructing soft rough fuzzy sets to other
algebras, such as hyperrings, BL-algebras and
so on.
(ii) Studying soft fuzzy rough lattices.
(iii) Investigating decision making methods based
on soft rough fuzzy lattices.
Acknowledgments
This research is partially supported by a grant of
National Natural Science Foundation of China (Grant
nos. 11571010 and 61179038).
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