Decomposition of Fuzzy Soft Sets with Finite Value Spaces

The notion of fuzzy soft sets is a hybrid soft computing model that integrates both gradualness and parameterization methods in harmony to deal with uncertainty. The decomposition of fuzzy soft sets is of great importance in both theory and practical applications with regard to decision making under uncertainty. This study aims to explore decomposition of fuzzy soft sets with finite value spaces. Scalar uni-product and int-product operations of fuzzy soft sets are introduced and some related properties are investigated. Using t-level soft sets, we define level equivalent relations and show that the quotient structure of the unit interval induced by level equivalent relations is isomorphic to the lattice consisting of all t-level soft sets of a given fuzzy soft set. We also introduce the concepts of crucial threshold values and complete threshold sets. Finally, some decomposition theorems for fuzzy soft sets with finite value spaces are established, illustrated by an example concerning the classification and rating of multimedia cell phones. The obtained results extend some classical decomposition theorems of fuzzy sets, since every fuzzy set can be viewed as a fuzzy soft set with a single parameter.


Introduction
With the development of modern science and technology, modelling various uncertainties has become an important task for a wide range of applications including data mining, pattern recognition, decision analysis, machine learning, and intelligent systems. The concept of uncertainty is too complicated to be captured within a single mathematical framework. In response to this situation, a number of approaches including probability theory, fuzzy sets [1], and rough sets [2] have been developed. Generally speaking, these theories deal with uncertainty from distinct angle of views, namely, randomness, gradualness, and granulation, respectively. Molodtsov's soft set theory [3] is a relatively new mathematical model for coping with uncertainty from a parametrization point of view. Zhu and Wen [4] redefined and improved some set-theoretic operations of soft sets that inherit all basic properties of operations on classical sets. Maji et al. [5] initiated the notion of fuzzy soft sets, which is a hybrid soft computing model in which the viewpoints of gradualness and parametrization for dealing with uncertainty are combined effectively. Majumdar and Samanta [6] further considered generalized fuzzy soft sets and applied them to decision making and medical diagnosis problems. Yang et al. [7] generalized fuzzy soft sets to interval-valued fuzzy soft sets. Up to now, fuzzy soft sets have proven to be useful in various fields such as flood alarm prediction [8], medical diagnosis [9], combined forecasting [10], supply chains risk management [11], topology [12], decision making under uncertainty [13][14][15], and algebraic structures [16][17][18][19][20][21][22][23][24].
The notion of level soft sets plays a crucial role in solving uncertain decision-making problems based on fuzzy soft sets [13]. In particular, it is important to figure out how many different -level soft sets could be obtained from a given fuzzy soft set by choosing distinct threshold values ∈ [0, 1]. On the other hand, it is well known that decomposition theorems are of great theoretical importance in exploring various types of fuzzy structures. Thus decomposition of fuzzy soft sets is a topic of both theoretical and practical value. Motivated by this consideration, Feng et al. investigated some basic 2 The Scientific World Journal properties of level soft sets based on variable thresholds and obtained some decomposition theorems of fuzzy soft sets by considering variable thresholds [25]. Moreover, Feng and Pedrycz [26] carried out a detailed research on scalar products and decomposition of fuzzy soft sets. Particularly, They have shown that scalar product operations can be regarded as semimodule actions and algebraic structures like ordered idempotent semimodules of fuzzy soft sets over ordered semirings can be constructed [26]. In most real applications, especially those involving the use of computers and programs, we only need to consider a finite universe of discourse associated with finite number of parameters. Consequently, in this study, we shall follow the research line above and concentrate on decomposition of fuzzy soft sets with finite value spaces.
The remainder of this paper is organized as follows. Section 2 first recalls some basic notions concerning fuzzy sets, soft sets, and fuzzy soft sets. Section 3 mainly introduces -level soft sets and scalar uniproduct operations of fuzzy soft sets. Then we explore some lattice structures associated with level soft sets in detail. In Section 5, we consider the decomposition of fuzzy soft sets with finite value spaces and establish some useful decomposition theorems, supported by illustrative examples. Finally, the last section summarizes the study and suggests possible directions for future work.

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

Fuzzy Sets.
The theory of fuzzy sets [1] provides an appropriate framework for representing and processing vague concepts by admitting a notion of a partial membership. Recall that a fuzzy set in a universe is defined by (and usually identified with) its membership function : → [0, 1]. For ∈ , the membership value ( ) essentially specifies the degree to which ∈ belongs to the fuzzy set . By ⊆ ], we mean that ( ) ≤ ]( ) for all ∈ . Clearly = ] if ⊆ ] and ] ⊆ . That is, ( ) = ]( ) for all ∈ . Let̂denote the fuzzy set in with a constant membership value ∈ [0, 1]; that is,̂( ) = for all ∈ . In what follows, we denote by F( ) the set of all fuzzy sets in .
Let ∈ [0, 1] and ∈ F( ). Recall that the scalar product of and is a fuzzy set ∈ F( ) defined by ( ) = ∧ ( ) for all ∈ . There are different definitions for fuzzy set operations. With the min-max system proposed by Zadeh, fuzzy set intersection, union, and complement are defined as follows: where , ] ∈ F( ) and ∈ .

Soft Sets.
Soft set theory was proposed by Molodtsov [3] in 1999, which provides a general mechanism for uncertainty modelling in a wide variety of applications. Let be the universe of discourse and let be the universe of all possible parameters related to the objects in . In most cases, parameters are considered to be attributes, characteristics, or properties of objects in . The pair ( , ) is also known as a soft universe. The power set of is denoted by P( ).
Definition 1 (see [3]). A pair S = ( , ) is called a soft set over , where ⊆ and : → P( ) is a set-valued mapping, called the approximate function of the soft set S.
By means of parametrization, a soft set gives a series of approximate descriptions of a complicated object being perceived from distinct aspects. For each parameter ∈ , the subset ( ) ⊆ is known as the set of -approximate elements [3]. It is worth noting that ( ) may be arbitrary: some of them may be empty, and some may have nonempty intersections. In what follows, the collection of all soft sets over with parameter sets contained in is denoted by S ( ). Moreover, we denote by S ( ) the collection of all soft sets over with a fixed parameter set ⊆ .
Maji et al. [27] introduced the concept of soft M-subsets and soft M-equal relations in the following manner.
Definition 2 (see [27]). Let Another different type of soft subsets and soft equal relations can be defined as follows.
Definition 3 (see [28]). Let It is easy to see that, for two soft sets S = ( , ) and T = ( , ), if S is a soft M-subset of T, then S is also a soft F-subset of T. However, the converse may not be true (see Example 2.6 in [29]). As shown in [29], the soft equal relations = and = coincide with each other. Hence in what follows, we just call them soft equal relations and simply write = instead of = or = unless stated otherwise.
Definition 4 (see [30]). Let S = ( , ) be a soft set over . Then (a) S is called a relative null soft set (with respect to the parameter set ), denoted by Φ , if ( ) = 0 for all ∈ ; (b) S is called a relative whole soft set (with respect to the parameter set ), denoted by U , if ( ) = for all ∈ . Definition 5 (see [5]). A pair S = (̃, ) is called a fuzzy soft set over , where ⊆ and̃is a mapping given bỹ: → F( ).
Conventionally, the mapping̃: → F( ) is referred to as the approximate function of the fuzzy soft set (̃, ). It is easy to see that fuzzy soft sets extend 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 parameter set reduces to a singleton. This means that fuzzy soft sets can be seen as a parameterized extension of fuzzy sets and it can be used to model those complicate fuzzy concepts which cannot be described using a single fuzzy set or simply by the intersection of some fuzzy sets.
(1) The extended union of (̃, ) and (̃, ) is defined as the fuzzy soft set (̃, ) = (̃, )∪ E (̃, ), where = ∪ and for all ∈ , (2) The extended intersection of (̃, ) and (̃, ) is defined as the fuzzy soft set (̃, ) = (̃, )∩ E (̃, ), where = ∪ and for all ∈ , In what follows, the collection of all fuzzy soft sets over with parameter sets contained in is denoted by FS ( ). Taking any parameter set ⊆ , one can consider the collection consisting of all fuzzy soft sets over with the fixed parameter set , which is denoted by FS ( ). The following result can easily be obtained using the above definitions.
The first part of the above assertion states that, for fuzzy soft sets in FS ( ), 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 same results when considering fuzzy soft sets with the same set of parameters. Thus in 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 operations coincide with each other. Now we illustrate the notion of fuzzy soft sets by an example as follows.
Example 9. Suppose that there are six cell phones under consideration The set of parameters is given by where , respectively, stand for "high quality of voice call, " "stylish design, " "friendly user interface, " "wonderful MP3/MP4 playback, " "low price, " "high resolution camera, " "popular brand" and "large screen". Now, let = { 2 , 4 , 5 , 6 , 8 } ⊆ consist of some crucial features for describing "attractive multimedia cell phones. " We can arrange an expert group to evaluate these cell phones and the available information on these mobile phones can be formulated as a fuzzy soft set S = (̃, ). It provides a mathematical representation of the complicate fuzzy concept, called "attractive multimedia cell phones" in daily languages. Table 1 gives the tabular representation of the fuzzy soft set S = (̃, ).
Using this illustrative example, we can observe that some fuzzy concepts in the real world are so complicated that they can hardly be described using a single fuzzy set or simply the intersection of some fuzzy sets. Alternatively, these complicate fuzzy concepts can jointly be represented as a family of fuzzy sets organized by some useful parameters like those we list above for describing "attractive 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 whole gives a complete representation.

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Level Soft Sets and Scalar Uni-Products
To solve decision-making problems based on fuzzy soft sets, Feng et al. [13] introduced the following notion called -level soft sets of fuzzy soft sets.
Definition 10 (see [13]). Let ∈ [0, 1] and S = (̃, ) be a fuzzy soft set over . The -level soft set of the fuzzy soft set S is a crisp soft set (S; ) = ( , ) over , where for all ∈ .
In the above definition, ∈ [0, 1] serves as a fixed threshold value on membership grades. In practical applications, these thresholds might be chosen by decision makers and represent the strength of their general requirements [13]. Some basic properties of -level soft sets have been investigated by Feng and Pedrycz [26]. Below, we list two results, which show that the structure of -level soft sets is compatible with some basic algebraic operations of fuzzy soft sets.
By replacing a single value with a set ⊆ [0,1] of thresholds, we immediately obtain an extension of the above concept.
For = 0, we define ⊙ ∪ = ⊙ ∩ =C 0 . The following results give some basic properties of scalar uniproduct operations. Dually, we can also investigate related properties of scalar int-product operations.
Proof. The proof is straightforward and thus omitted. Proof. The proof is straightforward and thus omitted. Proof. The proof is straightforward and thus omitted.

Lattice Structures Associated with -Level Soft Sets
In this section, we begin with some basic notions in lattice theory and then concentrate on exploring some lattice structures associated with -level soft sets of a given fuzzy soft set. From an algebraic point of view, a lattice ( , ∨, ∧) is a nonempty set with two binary operations ∨ and ∧ such that (1) ( , ∨) and ( , ∧) are commutative semigroups; (2) ∧ ( ∨ ) = and ∨ ( ∧ ) = for all , ∈ .
As an immediate consequence of the above theorem, we get the following result in [31].

Definition 28. Let
: 1 → 2 be a homomorphism of lattices. Then the kernel of is a binary relation on 1 defined by It is easy to check that ker( ) is an equivalence relation on 1 . As a direct consequence, we deduce that the level equivalent relation ∼ S defined above is an equivalence relation on the unit interval which implies that is injective. Hence is an isomorphism of lattices and so [0, 1] † /∼ S is isomorphic to (L(S),∪,∩).

Decomposition Theorems of Fuzzy Soft Sets
As mentioned above, the notion of -level soft sets acts as a key factor in solving adjustable fuzzy soft decision-making problems. So it is meaningful to ascertain how many different -level soft sets could be derived from a given fuzzy soft set by choosing distinct threshold value ∈ [0, 1]. Motivated by this point, we will explore in this section the problem with regard to the decomposition of a given fuzzy soft set in terms of its -level soft sets.
Using the above notions, the authors have established the following decomposition theorems of fuzzy soft sets in [26]. The second decomposition theorem is especially useful when the value space of the fuzzy soft set is finite since in this case we can obtain a finite decomposition of . In order to give a deeper insight into this issue, we propose the following notions. In view of the above definitions, we immediately deduce that the image of the threshold choice function is a complete threshold set. It is also evident that, in general cases, both the threshold choice function and the complete threshold set of a given fuzzy soft set S might not be unique.

Proposition 37. Let be a fuzzy soft set over with a finite value space
One has the following.
(i) Case 1: Thus we obtain that Note also that if 1 ∈ ( ), then V = 1 and (V , 1] = 0 should be removed in the above equality (Case 3 simply does not arise). Therefore, it follows that holds when V = 1. This completes the proof. Proof. Let be a fuzzy soft set over with a finite value space ( ) = {V 1 , . . . , V } such that V 1 < V 2 < ⋅ ⋅ ⋅ < V . First, we assume that V < 1. By Proposition 37, we have Using similar augments as above, we can show that Im( ) = ( ) is a complete threshold set of . But it is clear that ( ) = ( )∪{1}, since V = 1. Hence, ( )∪{1} is a complete threshold set of .
The following statement explicitly characterizes the structure of the lattice L( ), consisting of all level soft sets of a fuzzy soft set . By Theorem 30, the structure of L( ) is closely related to that of the lattice ([0, 1]/∼ , ⊓, ⊔) as described in Proposition 37.
Remark 42. The above result shows that the value space ( ) is the least threshold set for decomposing a fuzzy soft set .
In this case, we can find that ( ) is of crucial importance since we cannot decompose a fuzzy soft set correctly if any of its crucial threshold values is missing. This justifies the term "crucial threshold values" (see Definition 32) for these scalars.
By means of scalar uniproducts proposed in the previous section, we can further obtain the following decomposition theorem.
Theorem 43. Let be a fuzzy soft set over with a finite value space Proof. Using the definition of scalar uniproducts, we have For 2 ≤ ≤ , it can be seen that By Theorem 34, we know that can be decomposed by using its crucial threshold values. That is, =⋃ 1≤ ≤ V ⊙ ( ; V ). Hence, it follows that This completes the proof.
Corollary 44. Let be a fuzzy soft set over with a finite value space Proof. If 1 ∈ ( ), then V = 1 and (V , 1] = 0. Thus Otherwise, V < 1 and so we havẽ Also it is clear that =∪C 0 . Therefore, we deduce that This completes the proof.
The above results reveal that a given fuzzy soft set with a finite value space could be linked to finite number of all its different -level soft sets, which are derived from a partition of the unit interval [0, 1] determined by all crucial threshold values of the given fuzzy soft set.
Example 45. Let us reconsider the fuzzy soft set S = (̃, ) describing "attractive multimedia cell phones" in Example 9. We hope to give a proper classification and rating of these multimedia cell phones. Clearly, the value space of S is a finite set (S) = {0.2, 0.6, 0.7, 0.9} .
With regard to the -level soft sets of the fuzzy soft set S, we have the following.
The Scientific World Journal 9 Table 2: Tabular representation of the soft set T 2 = (S; 0.6).  Table 3: Tabular representation of the soft set T 3 = (S; 0.7).    Table 2. (iii) For V 3 = 0.7, (S; V 3 ) = T 3 is a soft set with its tabular representation given by Table 3. (iv) For V 2 = 0.9, (S; V 4 ) = T 4 is a soft set with its tabular representation given by Table 4. (v) For ∈ (0.9, 1], (S; ) = Φ is the relative null soft set with respect to the parameter set .
In addition, by Theorem 40, we also know that the lattice L(S), consisting of all level soft sets of a fuzzy soft set S, is a finite ascending chain as follows: It reveals that in total there are five distinct level soft sets which are corresponding to the given fuzzy soft set S on different segmentations of the unit interval [0, 1] determined by all crucial threshold values. For instance, T 2 is the level soft set corresponding to S on the subinterval (0.2, 0.6]. Using this level soft set, we can get the following classification and rating of all the cell phones under our consideration: This means that if we choose any threshold value 0.2 < ≤ 0.6, then all the cell phones can be graded into four classes. Moreover, 3 turns out to be the most attractive multimedia cell phones, while 1 and 5 form the class of least attractive multimedia cell phones.

Conclusions
We have investigated the decomposition of fuzzy soft sets with finite value spaces. We proposed scalar uniproduct and int-product operations of fuzzy soft sets. We also defined level equivalent relations and investigated some lattice structures associated with level soft sets. It has been shown that the collection L(S) of all -level soft sets of a given fuzzy soft set S forms a sublattice of the lattice (S ( ),∪,∩). In addition, we proved that the quotient structure [0, 1] † /∼ S = ([0, 1]/∼ S , ⊓, ⊔) of the unit interval [0, 1] induced by level equivalent relations is isomorphic to the lattice L(S) of -level soft sets and thus could be embedded into the lattice (S ( ),∪,∩). We also introduced crucial threshold values and complete threshold sets. Moreover, we established some decomposition theorems for fuzzy soft sets with finite value spaces and illustrated its possible practical applications with an example concerning the classification and rating of multimedia cell phones. Note also that our results generalize those classical decomposition theorems of fuzzy sets since every fuzzy set can simply be viewed as a fuzzy soft set with only one parameter. To extend this work, one might consider decomposition of fuzzy soft sets based on variable thresholds or other related applications of decomposition theorems of fuzzy soft sets.