Decomposition of fuzzy soft sets with finite value spaces

Decomposition theorems of fuzzy soft sets

with finite value spaces

Feng Feng a,∗, Hamido Fujita b, Young Bae Jun c, Madad Khan d

aDepartment of Applied Mathematics, School of Science,Xi’an University of Posts and Telecommunications, Xi’an 710121, China

bFaculty of Software and Information Science,Iwate Prefectural University, 020-0193 Iwate, JapancDepartment of Mathematics Education (and RINS),

Gyeongsang National University, Chinju 660-701, Republic of KoreadDepartment of Mathematics, COMSATS Institute of Information Technology,

Abbottabad 22060, Pakistan

Abstract

The notion of fuzzy soft sets is a hybrid soft computing model that integrates bothgradualness and parametrization methods in harmony to deal with uncertainty.The decomposition of fuzzy soft sets is of great importance in both theory andpractical applications with regard to decision making under uncertainty. This studyaims to explore decomposition of fuzzy soft sets with finite value spaces. Scalaruni-product and int-product operations of fuzzy soft sets are introduced and somerelated properties are investigated. Using t-level soft sets, we define level equivalentrelations and show that the quotient structure of the unit interval induced by levelequivalent relations is isomorphic to the lattice consisting of all t-level soft sets of agiven fuzzy soft set. We also introduce the concepts of crucial threshold values andcomplete threshold sets. Finally, some decomposition theorems for fuzzy soft setswith finite finite value spaces are established, illustrated by an example concerningthe classification and rating of multimedia cell phones. The obtained results extendsome classical decomposition theorems of fuzzy sets since every fuzzy set can beviewed as a fuzzy soft set with a single parameter.

Key words: Fuzzy soft sets; Level soft sets; Lattice; Value space; Decompositiontheorems

∗ Corresponding author. Tel.: +86 29 88166086.Email addresses: [email protected] (Feng Feng), [email protected]

(Hamido Fujita), [email protected] (Young Bae Jun), [email protected](Madad Khan).

Preprint submitted to SCIENTIFIC WORLD JOURNAL 5 December 2013

1 Introduction

With the development of modern science and technology, modelling variousuncertainties has become an important task for a wide range of applicationsincluding data mining, pattern recognition, decision analysis, machine learningand intelligent systems. The concept of uncertainty is too complicate to be cap-tured within a single mathematical framework. In response to this situation, anumber of approaches including probability theory, fuzzy sets [28] and roughsets [18] have been developed. Generally speaking, these theories deal with un-certainty from distinct angle of views, namely randomness, gradualness, andgranulation, respectively. Molodtsov’s soft set theory [17] is a relatively newmathematical model for coping with uncertainty from a parametrization pointof view. Zhu and Wen [31] redefined and improved some set-theoretic opera-tions of soft sets that inherit all basic properties of operations on classical sets.Maji et al. [14] initiated the notion of fuzzy soft sets, which is a hybrid softcomputing model in which the viewpoints of gradualness and parametrizationfor dealing with uncertainty are combined effectively. Majumdar and Samanta[16] further considered generalized fuzzy soft sets and applied them to deci-sion making and medical diagnosis problems. Yang et al. [25] generalized fuzzysoft sets to interval-valued fuzzy soft sets. Up to now, fuzzy soft sets haveproven to be useful in various fields such as flood alarm prediction [12], med-ical diagnosis [6], combined forecasting [21], supply chains risk management[22], topology [20], decision making under uncertainty [8,19,23] and algebraicstructures [1,2,5,11,24,26,27,29,30].

The notion of level soft sets plays a crucial role in solving uncertain decisionmaking problems based on fuzzy soft sets [8]. In particular, it is important tofigure out how many different t-level soft sets could be obtained from a givenfuzzy soft set by choosing distinct threshold values t ∈ [0, 1]. On the otherhand, it is well-known that decomposition theorems are of great theoreticalimportance in exploring various types of fuzzy structures. Thus decompositionof fuzzy soft sets is a topic of both theoretical and practical value. Motivatedby this consideration, Feng et al. investigated some basic properties of levelsoft sets based on variable thresholds and obtained some decomposition theo-rems of fuzzy soft sets by considering variable thresholds [10]. Moreover, Fengand Pedrycz [7] carried out a detailed research on scalar products and decom-position of fuzzy soft sets. Particularly, They have shown that scalar productoperations can be regarded as semimodule actions and algebraic structureslike ordered idempotent semimodules of fuzzy soft sets over ordered semiringscan be constructed [7]. In most real applications, especially those involving theuse of computers and programs, we only need to consider a finite universe ofdiscourse associated with finite number of parameters. Consequently, in thisstudy we shall follow the research line above and concentrate on decompositionof fuzzy soft sets with finite value spaces.

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The remainder of this paper is organized as follows. Section 2 first recallssome basic notions concerning fuzzy sets, soft sets, and fuzzy soft sets. Section3 mainly introduces t-level soft sets and scalar uni-product operations of fuzzysoft sets. Then we explore some lattice structures associated with level softsets in detail. In Section 5, we consider the decomposition of fuzzy soft setswith finite value spaces and establish some useful decomposition theorems,supported by illustrative examples. Finally, the last section summarizes thestudy and suggests possible directions for future work.

2 Preliminaries

In this section, we briefly review some basic concepts concerning fuzzy sets,soft sets and fuzzy soft sets, respectively.

2.1 Fuzzy sets

The theory of fuzzy sets [28] provides an appropriate framework for rep-resenting and processing vague concepts by admitting a notion of a partialmembership. Recall that a fuzzy set µ in a universe U is defined by (and usu-ally identified with) its membership function µ : U → [0, 1]. For x ∈ U , themembership value µ(x) essentially specifies the degree to which x ∈ U belongsto the fuzzy set µ. By µ ⊆ ν, we mean that µ(x) ≤ ν(x) for all x ∈ U . Clearlyµ = ν if µ ⊆ ν and ν ⊆ µ. That is, µ(x) = ν(x) for all x ∈ U . Let t denote thefuzzy set in U with a constant membership value t ∈ [0, 1]; that is t(x) = t forall x ∈ U . In what follows, we denote by F (U) the set of all fuzzy sets in U .

Let t ∈ [0, 1] and µ ∈ F (U). Recall that the scalar product of t and µis a fuzzy set tµ ∈ F (U) defined by tµ(x) = t ∧ µ(x) for all x ∈ U . Thereare different definitions for fuzzy set operations. With the min-max systemproposed by Zadeh, fuzzy set intersection, union, and complement are definedas follows:

• (µ ∩ ν)(x) = min{µ(x), ν(x)},• (µ ∪ ν)(x) = max{µ(x), ν(x)},• µc(x) = 1− µ(x),

where µ, ν ∈ F (U) and x ∈ U .

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2.2 Soft sets

Soft set theory was proposed by Molodtsov [17] in 1999, which provides ageneral mechanism for uncertainty modelling in a wide variety of applications.Let U be the universe of discourse and E be the universe of all possible pa-rameters related to the objects in U . In most cases parameters are consideredto be attributes, characteristics or properties of objects in U . The pair (U,E)is also known as a soft universe. The power set of U is denoted by P(U).

Definition 2.1 [17] A pair S = (F,A) is called a soft set over U , whereA ⊆ E and F : A → P(U) is a set-valued mapping, called the approximatefunction of the soft set S.

By means of parametrization, a soft set gives a series of approximatedescriptions of a complicated object being perceived from distinct aspects.For each parameter ε ∈ A, the subset F (ε) ⊆ U is known as the set of ε-approximate elements [17]. It is worth noting that F (ε) may be arbitrary: someof them may be empty, and some may have nonempty intersections. In whatfollows, the collection of all soft sets over U with parameter sets contained inE is denoted by S E(U). Moreover, we denote by SA(U) the collection of allsoft sets over U with a fixed parameter set A ⊆ E.

Maji et al. [15] introduced the concept of soft M-subsets and soft M-equalrelations in the following manner:

Definition 2.2 [15] Let (F,A) and (G,B) be two soft sets over U. Then(F,A) is called a soft M-subset of (G,B), denoted (F,A)⊆M(G,B), if A ⊆ Band F (a) = G (a) (i.e., F (a) and G(a) are identical approximations) for alla ∈ A. Two soft sets (F,A) and (G,B) are said to be soft M-equal, denoted(F,A) =M (G,B), if (F,A) ⊆M (G,B) and (G,B) ⊆M (F,A) .

Another different type of soft subsets and soft equal relations can be de-fined as follows.

Definition 2.3 [9] Let (F,A) and (G,B) be two soft sets over U. Then (F,A)is called a soft F-subset of (G,B), denoted (F,A)⊆F (G,B), if A ⊆ B andF (a) ⊆ G (a) for all a ∈ A. Two soft sets (F,A) and (G,B) are said to be softF-equal, denoted (F,A) =F (G,B), if (F,A) ⊆F (G,B) and (G,B) ⊆F (F,A) .

It is easy to see that for two soft sets S = (F,A) and T = (G,B), if S is asoft M-subset of T then S is also a soft F-subset of T. However, the conversemay not be true (see Example 2.6 in [13]). As shown in [13], the soft equalrelations =M and =F coincide with each other. Hence in what follows, we justcall them soft equal relations and simply write = instead of =M or =F unlessstated otherwise.

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Definition 2.4 [3] Let S = (F,A) be a soft set over U . Then

(a) S is called a relative null soft set (with respect to the parameter set A),denoted by ΦA, if F (a) = ∅ for all a ∈ A.

(b) S is called a relative whole soft set (with respect to the parameter setA), denoted by UA, if F (a) = U for all a ∈ A.

2.3 Fuzzy soft sets

By combining fuzzy sets with soft sets, Maji et al. [14] initiated a hybridsoft computing model called fuzzy soft sets as follows.

Definition 2.5 [14] A pair S = (F , A) is called a fuzzy soft set over U , whereA ⊆ E and F is a mapping given by F : A → F (U).

Conventionally, the mapping F : A → F (U) is referred to as the approxi-mate function of the fuzzy soft set (F , A). It is easy to see that fuzzy soft setsextend Molodtsov’s soft sets by substituting fuzzy subsets for crisp subsets.Note also that a fuzzy set could be viewed as a fuzzy soft set whose param-eter set reduces to a singleton. This means that fuzzy soft sets can be seenas a parameterized extension of fuzzy sets and it can be used to model thosecomplicate fuzzy concepts which cannot be described using a single fuzzy setor simply by the intersection of some fuzzy sets.

As in the case of soft sets, different types of fuzzy soft subsets can beintroduced. Given two fuzzy soft sets (F , A) and (G, B) over U , we say that(F , A) is a fuzzy soft F-subset of (G, B), denoted by (F , A) ⊆F (G, B), if A ⊆ Band F (a) is a fuzzy subset of G(a) for all a ∈ A. On the other hand, (F , A) iscalled a fuzzy soft M-subset of (G, B), denoted by (F , A) ⊆M (G, B), if A ⊆ Band F (a) = G(a) for all a ∈ A. Two fuzzy soft sets (F , A) and (G, B) overU are said to be fuzzy soft equal if they are identical; that is, A = B andF (a) = G(a) for all a ∈ A. This is denoted by (F , A) = (G, B).

Definition 2.6 Let S = (F , A) be a fuzzy soft set over U and t ∈ [0, 1]. ThenS is called a t-constant fuzzy soft set, denoted by Ct

A, if F (a) = t for all a ∈ A.

In particular, C0A and C1

A are called the relative null fuzzy soft set andrelative whole fuzzy soft set (with respect to the parameter set A), respectively.

Definition 2.7 [10] Let(F , A

)and

(G, B

)be two fuzzy soft sets over U .

(1) The extended union of(F , A

)and

(G, B

)is defined as the the fuzzy soft

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set(H, C

)=

(F , A

)∪E

(G, B

)where C = A ∪B and ∀c ∈ C,

H (c) =

F (c) , if c ∈ A\B,

G (c) , if c ∈ B\A,

F (c) ∪ G (c) , if c ∈ A ∩B.

(2) The extended intersection of(F , A

)and

(G, B

)is defined as the the

fuzzy soft set(H, C

)=

(F , A

)∩E

(G, B

)where C = A∪B and ∀c ∈ C,

H (c) =

F (c) , if c ∈ A\B,

G (c) , if c ∈ B\A,

F (c) ∩ G (c) , if c ∈ A ∩B.

(3) The restricted intersection of(F , A

)and

(G, B

)is defined as the fuzzy

soft set(H, C

)=

(F , A

)∩R

(G, B

), where C = A ∩B 6= ∅ and H (c) =

F (c) ∩ G (c) for all c ∈ C.

(4) The restricted union of(F , A

)and

(G, B

)is defined as the fuzzy soft

set(H, C

)=

(F , A

)∪R

(G, B

), where C = A ∩ B 6= ∅ and H (c) =

F (c) ∪ G (c) for all c ∈ C.

In what follows, the collection of all fuzzy soft sets over U with parametersets contained in E is denoted by FS E(U). Taking any parameter set A ⊆ E,one can consider the collection consisting of all fuzzy soft sets over U with thefixed parameter set A, which is denote by FS A(U). The following result caneasily be obtained using the above definitions.

Proposition 2.8 Let(F , A

)and

(G, A

)be two fuzzy soft sets over U . Then

(1)(F , A

)∪E

(G, A

)=

(F , A

)∪R

(G, A

).

(2)(F , A

)∩E

(G, A

)=

(F , A

)∩R

(G, A

).

The first part of the above assertion states that for fuzzy soft sets inFS A(U), the extended union ∪E coincides with the restricted union ∪R.That is, the two soft union operations ∪R and ∪E will always lead to the sameresults when considering fuzzy soft sets with the same set of parameters. Thusin this case we shall use a uniform notation ∪ representing both ∪R and ∪E .Analogously ∩R and ∩E will be simply denoted by ∩ if the two operationcoincide with each other.

Now we illustrate the notion of fuzzy soft sets by an example as follows.

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Table 1Tabular representation of the fuzzy soft set S = (F , A).

U e2 e4 e5 e6 e8

p1 0.2 0.9 0.6 0.2 0.2

p2 0.6 0.6 0.2 0.2 0.9

p3 0.9 0.7 0.9 0.9 0.7

p4 0.6 0.2 0.2 0.7 0.6

p5 0.2 0.6 0.2 0.7 0.2

p6 0.9 0.2 0.7 0.6 0.9

Example 2.9 Suppose that there are six cell phones under consideration

U = {p1, p2, p3, p4, p5, p6}.

The set of parameters is given by E = {e1, e2, e3, e4, e5, e6, e7, e8}, where ei

respectively stand for “high quality of voice call”, “stylish design”, “friendlyuser interface”, “wonderful MP3/MP4 playback”, “low price”, “high resolutioncamera”, “popular brand” and “large screen”.

Now, let A = {e2, e4, e5, e6, e8} ⊆ E consist of some crucial features fordescribing “attractive multimedia cell phones”. We can arrange an expertgroup to evaluate these cell phones and the available information on thesemobile phones can be formulated as a fuzzy soft set S = (F , A). It providesa mathematical representation of the complicate fuzzy concept, called “at-tractive multimedia cell phones” in daily languages. Table 1 gives the tabularrepresentation of the fuzzy soft set S = (F , A).

Using this illustrative example, we can observe that some fuzzy concepts inthe real world are so complicate that they can hardly be described using a sin-gle fuzzy set or simple the intersection of some fuzzy sets. Alternatively thesecomplicate fuzzy concepts can jointly be represented as a family of fuzzy setsorganized by some useful parameters like those we list above for describing “at-tractive multimedia cell phones”. Based on the viewpoint of parametrization,each fuzzy set in a fuzzy soft set only produces an approximate (or partial)description of a complicated fuzzy concept, while the fuzzy soft set as a wholegives a complete representation.

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3 Level soft sets and scalar uni-products

To solve decision making problems based on fuzzy soft sets, Feng et al. [8]introduced the following notion called t-level soft sets of fuzzy soft sets.

Definition 3.1 [8] Let t ∈ [0, 1] and S = (F , A) be a fuzzy soft set over U .The t-level soft set of the fuzzy soft set S is a crisp soft set L(S; t) = (Ft, A)over U , where

Ft(a) = {x ∈ U : F (a)(x) ≥ t},for all a ∈ A.

In the above definition, t ∈ [0, 1] serves as a fixed threshold value onmembership grades. In practical applications, these thresholds might be chosenby decision makers and represent the strength of their general requirements[8]. Some basic properties of t-level soft sets have been investigated by Fengand Pedrycz [7]. Below we list two results, which show that the structure oft-level soft sets is compatible with some basic algebraic operations of fuzzysoft sets.

Proposition 3.2 [7] Let S = (F , A) and R = (G, B) be two fuzzy soft setsin FS E(U). Then for all t ∈ [0, 1], we have

(a) L(S ∪E R; t) = L(S; t) ∪E L(R; t).

(b) L(S ∩E R; t) = L(S; t) ∩E L(R; t).

Proposition 3.3 [7] Let S = (F , A) and R = (G, B) be two fuzzy soft setsin FS E(U) with A ∩B 6= ∅. Then for all t ∈ [0, 1], we have

(a) L(S ∪R R; t) = L(S; t) ∪R L(R; t).

(b) L(S ∩R R; t) = L(S; t) ∩R L(R; t).

We also have considered the following scalar product of a scalar value tand a fuzzy soft set in [7].

Definition 3.4 Let t ∈ [0, 1] and S = (F , A) ∈ FS E(U). Then the scalarproduct of t and S is defined to be a fuzzy soft set t¯S = (G, A) over U , suchthat

G(a)(x) = (tF (a))(x) = t ∧ F (a)(x),

where a ∈ A, x ∈ U .

By replacing a single value t with a set J ⊆ [0, 1] of thresholds, we imme-diately obtain an extension of the above concept.

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Definition 3.5 Let J ⊆ [0, 1] and S = (F , A) ∈ FS E(U). Then the scalaruni-product of J and S is defined as

J ¯∪ S =⋃

t∈Jt¯ S.

Clearly, if J = {t} is a singleton, then J ¯∪ S = t¯ S. In a dual way, wecan also define the following operation.

Definition 3.6 Let J ⊆ [0, 1] and S = (F , A) ∈ FS E(U). Then the scalarint-product of J and S is defined as

J ¯∩ S =⋂

t∈Jt¯ S.

For J = ∅, we define J ¯∪ S = J ¯∩ S = C0A. The following results give

some basic properties of scalar uni-product operations. Dually we can alsoinvestigate related properties of scalar int-product operations.

Proposition 3.7 Let J1, J2 ⊆ [0, 1] and S ∈ FS E(U). If J1 ⊆ J2, then wehave J1 ¯∪ S ⊆F J2 ¯∪ S.

Proof. The proof is straightforward and thus omitted. 2

Proposition 3.8 Let J ⊆ [0, 1] and S1, S2 ∈ FS E(U). If S1 ⊆F S2, then wehave J ¯∪ S1 ⊆F J ¯∪ S2.


Proposition 3.9 Let J ⊆ [0, 1] and S1, S2 ∈ FS E(U). If S1 ⊆M S2, then wehave J ¯∪ S1 ⊆M J ¯∪ S2.


4 Lattice structures associated with t-level soft sets

In this section, we begin with some basic notions in lattice theory andthen concentrate on exploring some lattice structures associated with t-levelsoft sets of a given fuzzy soft set. From an algebraic point of view, a lattice(L,∨,∧) is a nonempty set with two binary operations ∨ and ∧ such that

(1) (L,∨) and (L,∧) are commutative semigroups.

(2) a ∧ (a ∨ b) = a and a ∨ (a ∧ b) = a for all a, b ∈ L.

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Let (L,∨,∧) be a lattice. For every a ∈ L, it is easy to see that

a ∨ a = a ∨ (a ∧ (a ∨ a)) = a.

Similarly, we can deduce a ∧ a = a ∧ (a ∨ (a ∧ a)) = a. Thus in a lattice L,two operations ∨ and ∧ are both idempotent. That is, (L,∨) and (L,∧) aretwo semilattices.

If a lattice L has identity elements with respect to both ∨ and ∧, then Lis said to be bounded. Usually identity element of L with respect to operation∨ is denoted by ⊥, whereas the identity with respect to ∧ is denoted by >.In other words, (L,∨,⊥) and (L,∧,>) are two monoids if (L,∨,∧,⊥,>) is abounded lattice.

Definition 4.1 A lattice (L,∨,∧) is called a distributive lattice if

(1) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).

(2) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c).

for all a, b, c ∈ L.

Example 4.2 Let us consider the unit interval [0, 1]. For all a, b ∈ L, wedenote max{a, b} and min{a, b} by a∨b and a∧b, respectively. Then it is easyto verify that ([0, 1],∨,∧) is a distributive lattice. Moreover, ([0, 1],∨,∧) is abounded lattice with ⊥ = 0 and > = 1.

Proposition 4.3 Let J ⊆ [0, 1] and S ∈ FS E(U). Then J¯∪S = (∨J)¯S.

Proof. Let S = (F , A) be a fuzzy soft set over U . Note first that t∗ = ∨Jindeed exists since ([0, 1],∨,∧) is a complete lattice. For any t ∈ J , we writet¯ S = (Gt, A). Let a ∈ A and x ∈ U . By Definition 3.4, we have

Gt(a)(x) = t ∧ F (a)(x).

If we write J ¯∪ S = (H, A), then by the complete distributive laws in thecomplete lattice ([0, 1],∨,∧), we have

H(a)(x) =∨

t∈J

Gt(a)(x) =∨

t∈J

(t ∧ F (a)(x))

= (∨

t∈J

t) ∧ F (a)(x) = t∗ ∧ F (a)(x).

This implies that J ¯∪ S = t∗ ¯ S = (∨J)¯ S. 2

Now, we consider the collection of all fuzzy soft sets over U with a fixedparameter set A. This collection forms a lattice structure with respect to soft

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union and intersection operations as shown in [7].

Theorem 4.4 (FS A(U), ∪, ∩, C0A, C1

A) is a bounded distributive lattice.

Proof. Let Sk = (Fk, A) ∈ FS A(U) (k = 1, 2, 3). Write S1 ∪S2 = (H, A)and S2 ∪S1 = (R, A). Then for all t ∈ A, we have

H(t) = F1(t) ∪ F2(t) = F2(t) ∪ F1(t) = R(t).

This shows that the soft union operation of fuzzy soft sets is commutative.Moreover, we can prove that this operation is associative. By definition ofthe relative null fuzzy soft set C0

A, it is easy to see that S1 ∪ C0A = S1.

Hence (FS A(U), ∪, C0A) is a commutative monoid. Dually, we deduce that

(FS A(U), ∩, C1A) is a commutative monoid. Next, let S1 ∩ (H, A) = (K, A).

Then for all t ∈ A, we have

K(t) = F1(t) ∩ H(t) = F1(t) ∩ (F1(t) ∪ F2(t)) = F1(t),

and so S1 ∩ (S1 ∪S2) = S1. In a similar fashion, we can deduce S1 ∪ (S1 ∩S2) =S1. Thus (FS A(U), ∪, ∩, C0

A, C1A) is a bounded lattice. In addition, we write

S1 ∪S3 = (G, A) and S2 ∩S3 = (J , A). Then we have

F1(t) ∪ J(t) = (F1(t) ∪ F2(t)) ∩ (F1(t) ∪ F3(t)) = H(t) ∩ G(t),

which implies that S1 ∪ (S2 ∩S3) = (S1 ∪S2) ∩ (S1 ∪S3). Dually, we haveS1 ∩ (S2 ∪S3) = (S1 ∩S2) ∪ (S1 ∩S3). Therefore, (FS A(U), ∪, ∩, C0

A, C1A) is

a bounded distributive lattice. 2

As an immediate consequence of the above theorem, we get the followingresult in [4].

Corollary 4.5 (SA(U), ∪, ∩, ΦA, UA) is a bounded distributive lattice.

Let (L,∨,∧) be a lattice. A nonempty set M ⊆ L is called a sublattice ofL if a ∨ b ∈ M and a ∧ b ∈ M for all a, b ∈ M .

Proposition 4.6 Let S = (F , A) ∈ FS E(U) and s, t ∈ [0, 1]. Then

(a) L(S; s ∨ t) = L(S; s) ∩L(S; t).

(b) L(S; s ∧ t) = L(S; s) ∪L(S; t).

Proof. We only show that (a) is valid and (b) can be proved in a similar way.Let L(S; s) = (Fs, A), L(S; t) = (Ft, A) and L(S; s ∨ t) = (Fs∨t, A). Also,write L(S; s) ∩L(S; t) = (H, A). For all e ∈ A and u ∈ U , we have

u ∈ H(e) ⇔ u ∈ Fs(e) ∩ Ft(e) ⇔ F (e)(u) ≥ s ∨ t ⇔ u ∈ Fs∨t(e),

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and so H(e) = Fs∨t(e) for all e ∈ A. Thus L(S; s ∨ t) = L(S; s) ∩L(S; t) asrequired. 2

Let S = (F , A) be a fuzzy soft set over U . We denote the collection of all t-level soft sets of the fuzzy soft set S by L (S) = {L(S; t) : t ∈ [0, 1]}. Clearly,L (S) is a nonempty subset of SA(U) since UA = L(S; 0). In addition, byProposition 4.6 we can immediately deduce the following result.

Theorem 4.7 L (S) is a sublattice of the lattice (SA(U), ∪, ∩).

Definition 4.8 Let S ∈ FS E(U) and s, t ∈ [0, 1]. Then s and t are said tobe level equivalent, denoted by s ∼S t, if L(S; s) = L(S; t).

Definition 4.9 Let L1, L2 be two lattices. A mapping ψ : L1 → L2 is ahomomorphism of lattices if ψ(a∨ b) = ψ(a)∨ψ(b) and ψ(a∧ b) = ψ(a)∧ψ(b)for all a, b ∈ L1.

A homomorphism ψ : L1 → L2 of lattices is called a monomorphism(resp. epimorphism, isomorphism) if ψ is injective (resp. surjective, bijective).If ψ : L1 → L2 is an isomorphism of lattices, then L1, L2 are said to beisomorphic, written as L1

∼= L2.

Definition 4.10 Let ψ : L1 → L2 be a homomorphism of lattices. Then thekernel of ψ is a binary relation on L1 defined by

ker(ψ) = {(a, b) ∈ L1 × L1 : ψ(a) = ψ(b)}.

It is easy to check that ker(ψ) is an equivalence relation on L1. As a directconsequence, we deduce that the level equivalent relation ∼S defined above isan equivalence relation on the unit interval [0, 1].

Proposition 4.11 Let S be a fuzzy soft set over U and [0, 1]† = ([0, 1],∧,∨).Then the mapping ψS : [0, 1] → L (S) given by ψS(t) = L(S; t) is an epi-morphism of lattices with ker(ψS) =∼S.

Proof. Note first that from Example 4.2, ([0, 1],∨,∧) is a distributive lattice.Dually we deduce that [0, 1]† = ([0, 1],∧,∨) is also a distributive lattice. Usingthe notations given above, Proposition 4.6 implies ψS(s ∧ t) = ψS(s) ∪ψS(t)and ψS(s∨ t) = ψS(s) ∩ψS(t). Also it is clear that the mapping ψS : [0, 1] →L (S) is surjective. Thus it is an epimorphism of lattices. Moreover, it followsfrom Definition 4.8 and Definition 4.10 that ker(ψS) =∼S. 2

Now, let us consider the quotient set of the unit interval [0, 1] with re-spect to the level equivalent relation ∼S. We can define two operations on thequotient set [0, 1]/ ∼S as follows:

12

• [s]∼Su [t]∼S

= [s ∧ t]∼S,

• [s]∼St [t]∼S

= [s ∨ t]∼S,

where [s]∼Sdenotes the equivalence class of s ∈ [0, 1] under ∼S.

It can be shown that [0, 1]†/ ∼S= ([0, 1]/ ∼S,u,t) is a lattice. In addition,we have the following result.

Theorem 4.12 Let S be a fuzzy soft set over U and [0, 1]† = ([0, 1],∧,∨).Then [0, 1]†/ ∼S= ([0, 1]/ ∼S,u,t) ∼= (L (S), ∪, ∩).

Proof. Define a mapping ϕ : [0, 1]/ ∼S→ L (S) by

ϕ([t]∼S) = ψS(t) = L(S; t).

By Proposition 4.11, the mapping ψS : [0, 1] → L (S) given by ψS(t) =L(S; t) is an epimorphism of lattices. Thus ϕ is surjective. Also we have

ϕ([s]∼Su [t]∼S

) = ϕ([s ∧ t]∼S) = ψS(s ∧ t)

= ψS(s) ∪ψS(t) = ϕ([s]∼S) ∪ϕ([t]∼S

).

Similarly, we can get ϕ([s]∼St [t]∼S

) = ϕ([s]∼S) ∩ϕ([t]∼S

). This shows thatϕ : [0, 1]/ ∼S→ L (S) is an epimorphism of lattices. Now, assume thatϕ([s]∼S

) = ϕ([t]∼S). Then we deduce that

L(S; s) = ψS(s) = ϕ([s]∼S) = ϕ([t]∼S

) = ψS(t) = L(S; t),

which implies that ϕ is injective. Hence ϕ is an isomorphism of lattices andso [0, 1]†/ ∼S is isomorphic to (L (S), ∪, ∩). 2

Corollary 4.13 The lattice [0, 1]†/ ∼S= ([0, 1]/ ∼S,u,t) is a distributivelattice which can be embedded into the lattice (SA(U), ∪, ∩).

Proof. By Theorem 4.5, (SA(U), ∪, ∩) is a distributive lattice. Also weknow that L (S) is a sublattice of the lattice (SA(U), ∪, ∩). Finally, ac-cording to Theorem 4.12, [0, 1]†/ ∼S is isomorphic to the sublattice L (S).Therefore, [0, 1]†/ ∼S is distributive and can be embedded into the lattice(SA(U), ∪, ∩). 2

5 Decomposition theorems of fuzzy soft sets

As mentioned above, the notion of t-level soft sets acts as a key factor insolving adjustable fuzzy soft decision making problems. So it is meaningful toascertain how many different t-level soft sets could be derived from a given

13

fuzzy soft set by choosing distinct threshold value t ∈ [0, 1]. Motivated bythis point, we shall explore in this section the problem with regard to thedecomposition of a given fuzzy soft set in terms of its t-level soft sets.

Definition 5.1 [7] Let S = (F , A) ∈ FS E(U). The value space of the fuzzysoft set S is a set V (S) =

⋃a∈A{F (a)(x) : x ∈ U}. A scalar t ∈ [0, 1] is called

a crucial threshold value of the fuzzy soft set S if t ∈ V (S).

Using the above notions, the authors have established the following de-composition theorems of fuzzy soft sets in [7].

Theorem 5.2 Let S ∈ FS E(U). Then we have S =⋃

t∈[0,1]t¯ L(S; t).

Theorem 5.3 Let S ∈ FS E(U). Then we have S =⋃

t∈V (S)t¯ L(S; t).

The second decomposition theorem is especially useful when the valuespace of the fuzzy soft set S is finite since in this case we can obtain a finitedecomposition of S. In order to give a deeper insight into this issue, we proposethe following notions.

Definition 5.4 Let S ∈ FS E(U) and ∼S be the level equivalent relationinduced by S. Then J ⊆ [0, 1] is called a complete threshold set of S if itconsists of precisely one element from each equivalence class of [0, 1]/ ∼S.

Definition 5.5 Let S ∈ FS E(U) and ∼S be the level equivalent relationinduced by S. Then a mapping f : [0, 1]/ ∼S→ [0, 1] is called a thresholdchoice function of S if f([t]∼S

) ∈ [t]∼S. The image of the threshold choice

function f is denoted by Im(f).

In view of the above definitions, we immediately deduce that the image ofthe threshold choice function f is a complete threshold set. It is also evidentthat in general cases, both the threshold choice function and the completethreshold set of a given fuzzy soft set S might not be unique.

Proposition 5.6 Let S be a fuzzy soft set over U with a finite value spaceV (S) = {v1, · · · , vn} such that v1 < v2 < · · · < vn. We have

(1) If 1 ∈ V (S), then [0, 1]/ ∼S= {[0, v1], (v1, v2], · · · , (vn−1, vn]}.(2) If 1 /∈ V (S), then [0, 1]/ ∼S= {[0, v1], (v1, v2], · · · , (vn−1, vn], (vn, 1]}.

Proof. First, we assume that vn < 1. For t ∈ [0, 1], we have the followingcases:

• Case 1: L(S; t) = L(S; v1) = UA ⇔ t ∼S v1 ⇔ t ∈ [0, v1].• Case 2: L(S; t) = L(S; vi) ⇔ t ∼S vi ⇔ t ∈ (vi−1, vi], (i = 2, · · · , n).• Case 3: L(S; t) = L(S; 1) = ΦA ⇔ t ∼S 1 ⇔ t ∈ (vn, 1].

14

Thus we obtain that

[0, 1]/ ∼S= {[0, v1], (v1, v2], · · · , (vn−1, vn], (vn, 1]} .

Note also that if 1 ∈ V (S), then vn = 1 and (vn, 1] = ∅ should be removedin the above equality (Case 3 simply does not arise). Therefore, it follows that

[0, 1]/ ∼S= {[0, v1], (v1, v2], · · · , (vn−1, vn]}

holds when vn = 1. This completes the proof. 2

Remark 5.7 The above statement gives an explicit description of the struc-ture of the lattice [0, 1]†/ ∼S= ([0, 1]/ ∼S,u,t) discussed in Theorem 4.12and Corollary 4.13. It says that the level equivalent relation ∼S divides theunit interval [0, 1] into some subintervals and these subintervals actually forma distributive lattice under certain operations.

Corollary 5.8 Let S be a fuzzy soft set over U with a finite value space V (S).Then V (S) ∪ {1} is a complete threshold set of S.

Proof. Let S be a fuzzy soft set over U with a finite value space V (S) ={v1, · · · , vn} such that v1 < v2 < · · · < vn. First, we assume that vn < 1. ByProposition 5.6, we have

[0, 1]/ ∼S= {[0, v1], (v1, v2], · · · , (vn−1, vn], (vn, 1]} .

Let f : [0, 1]/ ∼S → [0, 1] be a mapping given by f([t]∼S) = ∨[t]∼S

. Then f isa threshold choice function of the fuzzy soft set S since every subinterval in[0, 1]/ ∼S contains its least upper bound. It follows that Im(f) = V (S)∪ {1}is a complete threshold set of S.

On the other hand, if vn = 1 ∈ V (S), then by Proposition 5.6, we have

[0, 1]/ ∼S= {[0, v1], (v1, v2], · · · , (vn−1, vn]} .

Using similar augments as above, we can show that Im(f) = V (S) is a com-plete threshold set of S. But it is clear that V (S) = V (S) ∪ {1} since vn = 1.Hence V (S) ∪ {1} is a complete threshold set of S. 2

The following statement explicitly characterizes the structure of the latticeL (S), consisting of all level soft sets of a fuzzy soft set S. By Theorem 4.12,the structure of L (S) is closely related to that of the lattice ([0, 1]/ ∼S,u,t)as described in Proposition 5.6.

Theorem 5.9 Let S = (F , A) be a fuzzy soft set over U with a finite valuespace V (S) = {v1, · · · , vn} such that v1 < v2 < · · · < vn. We have

15

(1) If 1 ∈ V (S), then the lattice L (S) is a finite ascending chain:

L(S; 1) = L(S; vn) ⊆F · · · ⊆F L(S; v2) ⊆F L(S; v1) = UA

(2) If 1 /∈ V (S), then the lattice L (S) is a finite ascending chain:

ΦA = L(S; 1) ⊆F L(S; vn) ⊆F · · · ⊆F L(S; v2) ⊆F L(S; v1) = UA

Proof. Define a mapping ϕ : [0, 1]/ ∼S→ L (S) by

ϕ([t]∼S) = L(S;∨[t]∼S

).

By Theorem 4.12, the mapping ϕ is an isomorphism of lattices. Then theabove result follows from Proposition 5.6 and its proof. 2

Theorem 5.10 Let S = (F , A) be a fuzzy soft set over U with a finite valuespace V (S) and J ⊆ [0, 1] be a finite set. Then we have

S =⋃

t∈Jt¯ L(S; t) ⇔ V (S) ⊆ J.

Proof. Let⋃

t∈Jt¯ L(S; t) = (G, A). First, we assume that V (S) ⊆ J . Thenby Theorem 5.2 and Theorem 5.3, we have

S =⋃

t∈V (S)t¯ L(S; t) ⊆F (G, A)

and

(G, A) ⊆F

⋃t∈[0,1]

t¯ L(S; t) = S.

It follows that S = (G, A) =⋃

t∈Jt¯ L(S; t).

Conversely, suppose that

(G, A) =⋃

t∈Jt¯ L(S; t) = S = (F , A).

If V (S) * J , then there exists a crucial threshold value F (a∗)(u∗) = t∗ ∈V (S) \ J for some a∗ ∈ A and u∗ ∈ U . Now, we write the finite set J asa disjoint union, namely J = J− ] J+, where J− = {r ∈ J : r < t∗} andJ+ = {r ∈ J : r ≥ t∗}. Using similar techniques and notations as in the proofof Theorem 5.3, we deduce

16

G(a∗)(u∗) =∨

t∈J

Gt(a)(x) =∨

t∈J

tFt(a)(x)

=

∨

t∈J−t ∧ Ft(a)(x)

∨

∨

t∈J+

t ∧ Ft(a)(x)

=

∨

t∈J−t ∧ 1

∨

∨

t∈J+

t ∧ 0

=

∨

t∈J−t < t∗ = F (a∗)(u∗).

This contradicts to the hypothesis (G, A) = (F , A). Hence V (S) ⊆ J . 2

Remark 5.11 The above result shows that the value space V (S) is the leastthreshold set for decomposing a fuzzy soft set S. In this case, we can findthat V (S) is of crucial importance since we cannot decompose a fuzzy softset correctly if any of its crucial threshold values is missing. This justifies theterm “crucial threshold values” (see Definition 5.1) for these scalars.

By means of scalar uni-products proposed in previous section, we can fur-ther obtain the following decomposition theorem.

Theorem 5.12 Let S be a fuzzy soft set over U with a finite value spaceV (S) = {v1, · · · , vn} such that v1 < v2 < · · · < vn. Then we have

S =⋃

2≤i≤n(vi−1, vi]¯∪ L(S; vi)

⋃[0, v1]¯∪ L(S; v1).

Proof. Using the definition of scalar uni-products, we have

[0, v1]¯∪ L(S; v1) =⋃

t∈[0,v1]t¯ L(S; v1) = v1 ¯ L(S; v1).

For 2 ≤ i ≤ n, it can be seen that

(vi−1, vi]¯∪ L(S; vi) =⋃

t∈(vi−1,vi]t¯ L(S; vi) = vi ¯ L(S; vi).

By Theorem 5.3, we know that S can be decomposed by using its crucialthreshold values. That is, S =

⋃1≤i≤nvi ¯ L(S; vi). Hence, it follows that

S =⋃

2≤i≤n(vi−1, vi]¯∪ L(S; vi)

⋃[0, v1]¯∪ L(S; v1).

This completes the proof. 2

Corollary 5.13 Let S be a fuzzy soft set over U with a finite value spaceV (S) = {v1, · · · , vn} such that v1 < v2 < · · · < vn. Then we have

S =⋃

2≤i≤n(vi−1, vi]¯∪ L(S; vi)

⋃[0, v1]¯∪ L(S; v1)

⋃(vn, 1]¯∪ L(S; 1).

17

Proof. If 1 ∈ V (S), then vn = 1 and (vn, 1] = ∅. Thus

(vn, 1]¯∪ L(S; 1) = ∅ ¯∪ L(S; 1) = C0A.

Otherwise, vn < 1 and so we have

⋃(vn, 1]¯∪ L(S; 1) = (vn, 1]¯∪ ΦA = C0

A.

Also it is clear that S = S ∪ C0A. Therefore, we deduce that

S =⋃

2≤i≤n(vi−1, vi]¯∪ L(S; vi)

⋃[0, v1]¯∪ L(S; v1)

⋃(vn, 1]¯∪ L(S; 1).

This completes the proof. 2

The above results reveal that a given fuzzy soft set with a finite value spacecould be linked to finite number of all its different t-level soft sets, which arederived from a partition of the unit interval [0, 1] determined by all crucialthreshold values of the given fuzzy soft set.

Example 5.14 Let us reconsider the fuzzy soft set S = (F , A) describing“attractive multimedia cell phones” in Example 2.9. We hope to give a properclassification and rating of these multimedia cell phones. Clearly, the valuespace of S is a finite set

V (S) = {0.2, 0.6, 0.7, 0.9}.

By Proposition 5.6, the level equivalent relation ∼S divides the unit interval[0, 1] into some subintervals and these subintervals actually form a distributivelattice. Specifically, we have

[0, 1]/ ∼S= {[0, 0.2], (0.2, 0.6], (0.6, 0.7], (0.7, 0.9], (0.9, 1]} .

With regard to the t-level soft sets of the fuzzy soft set S, we have:

• For v1 = 0.2, L(S; v1) = UA is the relative whole soft set with respect tothe parameter set A.

• For v2 = 0.6, L(S; v2) = T2 is a soft set with its tabular representationgiven by Table 2.



• For t ∈ (0.9, 1], L(S; t) = ΦA is the relative null soft set with respect to theparameter set A.

In addition, by Theorem 5.9 we also know that the lattice L (S), consisting

18

Table 2Tabular representation of the soft set T2 = L(S; 0.6).

U e2 e4 e5 e6 e8

p1 0 1 1 0 0

p2 1 1 0 0 1

p3 1 1 1 1 1

p4 1 0 0 1 1

p5 0 1 0 1 0

p6 1 0 1 1 1


U e2 e4 e5 e6 e8

p1 0 1 0 0 0

p2 0 0 0 0 1

p3 1 1 1 1 1

p4 0 0 0 1 0

p5 0 0 0 1 0

p6 1 0 1 0 1


U e2 e4 e5 e6 e8

p1 0 1 0 0 0

p2 0 0 0 0 1

p3 1 0 1 1 0

p4 0 0 0 0 0

p5 0 0 0 0 0

p6 1 0 0 0 1

of all level soft sets of a fuzzy soft set S, is a finite ascending chain:

ΦA = L(S; 1) ⊆F L(S; 0.9) ⊆F L(S; 0.7) ⊆F L(S; 0.6)) ⊆F L(S; 0.2) = UA.

19

Finally, by Corollary 5.13 we can obtain the following decomposition:

S = [0, 0.2]¯∪UA ∪ (0.2, 0.6]¯∪T2 ∪ (0.6, 0.7]¯∪T3 ∪ (0.7, 0.9]¯∪T4 ∪ (0.9, 1]¯∪ΦA.

It reveals that in total there are five distinct level soft sets which are corre-sponding to the given fuzzy soft set S on different segmentations of the unitinterval [0, 1] determined by all crucial threshold values. For instance, T2 isthe level soft set corresponding to S on the subinterval (0.2, 0.6]. Using thislevel soft set, we can get the following classification and rating of all the cellphones under our consideration:

{p3} Â {p6} Â {p2, p4} Â {p1, p5}.

This means that if we choose any threshold value 0.2 < t ≤ 0.6, then all thecell phones can be graded into four classes. Moreover, p3 turns out to be themost attractive multimedia cell phones, while p1 and p5 forms the class of leastattractive multimedia cell phones.

6 Conclusions

We have investigated the decomposition of fuzzy soft sets with finite valuespaces. We proposed scalar uni-product and int-product operations of fuzzysoft sets. We also defined level equivalent relations and investigated somelattice structures associated with level soft sets. It has been shown that thecollection L (S) of all t-level soft sets of a given fuzzy soft set S forms asublattice of the lattice (SA(U), ∪, ∩). In addition, we proved that the quotientstructure [0, 1]†/ ∼S= ([0, 1]/ ∼S,u,t) of the unit interval [0, 1] induced bylevel equivalent relations is isomorphic to the lattice L (S) of t-level soft setsand thus could be embedded into the lattice (SA(U), ∪, ∩). We also introducedcrucial threshold values and complete threshold sets. Moreover, we establishedsome decomposition theorems for fuzzy soft sets with finite value spaces andillustrated its possible practical applications with an example concerning theclassification and rating of multimedia cell phones. Note also that our resultsgeneralize those classical decomposition theorems of fuzzy sets since everyfuzzy set can simply be viewed as a fuzzy soft set with only one parameter. Toextend this work, one might consider decomposition of fuzzy soft sets basedon variable thresholds or other related applications of decomposition theoremsof fuzzy soft sets.

20

Acknowledgements

The authors are highly grateful to the anonymous referees for their in-sightful comments and valuable suggestions which greatly improve the qualityof this paper. This work was partially supported by National Natural ScienceFoundation of China (Program No. 11301415), Natural Science Basic ResearchPlan in Shaanxi Province of China (Program Nos. 2013JQ1020, 2012JM8022)and Scientific Research Program Funded by Shaanxi Provincial Education De-partment of China (Program Nos. 2013JK1098, 2013JK1182, 2013JK1130).

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Table 1Tabular representation of the fuzzy soft set S = (F , A).

U e2 e4 e5 e6 e8

p1 0.2 0.9 0.6 0.2 0.2

p2 0.6 0.6 0.2 0.2 0.9

p3 0.9 0.7 0.9 0.9 0.7

p4 0.6 0.2 0.2 0.7 0.6

p5 0.2 0.6 0.2 0.7 0.2

p6 0.9 0.2 0.7 0.6 0.9


U e2 e4 e5 e6 e8

p1 0 1 1 0 0

p2 1 1 0 0 1

p3 1 1 1 1 1

p4 1 0 0 1 1

p5 0 1 0 1 0

p6 1 0 1 1 1


U e2 e4 e5 e6 e8

p1 0 1 0 0 0

p2 0 0 0 0 1

p3 1 1 1 1 1

p4 0 0 0 1 0

p5 0 0 0 1 0

p6 1 0 1 0 1

24


U e2 e4 e5 e6 e8

p1 0 1 0 0 0

p2 0 0 0 0 1

p3 1 0 1 1 0

p4 0 0 0 0 0

p5 0 0 0 0 0

p6 1 0 0 0 1

25

Decomposition of fuzzy soft sets with finite value spaces

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