On Some Nonclassical Algebraic Properties of Interval-Valued Fuzzy Soft Sets

Interval-valued fuzzy soft sets realize a hybrid soft computing model in a general framework. Both Molodtsov's soft sets and interval-valued fuzzy sets can be seen as special cases of interval-valued fuzzy soft sets. In this study, we first compare four different types of interval-valued fuzzy soft subsets and reveal the relations among them. Then we concentrate on investigating some nonclassical algebraic properties of interval-valued fuzzy soft sets under the soft product operations. We show that some fundamental algebraic properties including the commutative and associative laws do not hold in the conventional sense, but hold in weaker forms characterized in terms of the relation =L. We obtain a number of algebraic inequalities of interval-valued fuzzy soft sets characterized by interval-valued fuzzy soft inclusions. We also establish the weak idempotent law and the weak absorptive law of interval-valued fuzzy soft sets using interval-valued fuzzy soft J-equal relations. It is revealed that the soft product operations ∧ and ∨ of interval-valued fuzzy soft sets do not always have similar algebraic properties. Moreover, we find that only distributive inequalities described by the interval-valued fuzzy soft L-inclusions hold for interval-valued fuzzy soft sets.


Introduction
In recent years, uncertainty modelling has become an important research topic in computational intelligence and related fields. It is worth noting that uncertainty is multifaceted and as such cannot be captured within a single mathematical framework. In response to this fact, a number of different approaches such as probability theory, fuzzy sets [1], and rough sets [2] have been developed, which are capable of addressing various types of uncertainties from differing viewpoints including randomness, gradualness, and granulation, respectively. Molodtsov [3] proposed soft set theory as a generic mathematical model for dealing with uncertainty from a parameterization point of view in about a decade ago. Since then there has been a rapid development in this theory and its applications to algebraic structures [4][5][6][7][8][9][10], topology [11,12], knowledge acquisition [13], decision making [14], and others [15,16]. Soft sets are also closely related to many other soft computing models [17][18][19]. Algebraic operations of soft sets were initiated by Molodtsov [3] and systematically investigated by Maji et al. [20]. Ali et al. [21] defined some new operations in soft set theory and further studied algebraic structures of soft sets associated with these new operations in [22].
On the other hand, some researchers considered extending soft sets with fuzzy set theory [23][24][25][26][27]. Yang et al. [25] introduce interval-valued fuzzy soft sets which realize a common extension of both Molodtsov's soft sets and intervalvalued fuzzy sets. There is no doubt that soft subsets play a fundamental role in soft set theory. Maji et al. [20] initiated the notion of soft subsets in a very strict manner. Other types of soft subsets were also proposed by relaxing the conditions for defining Maji's soft subsets in [18,28,29]. Feng and Li [30] investigated these different types of soft subsets and the related soft equal relations in a systematic way. They also considered the free soft algebras associated with soft product operations.
The present study aims to investigate the above line of exploration in the more general setting of interval-valued fuzzy soft sets. We will investigate some nonclassical algebraic 2 The Scientific World Journal properties of interval-valued fuzzy soft sets with respect to soft product operations, which are distinct from those of interval-valued fuzzy sets. Note that some of the results obtained here are natural extensions to those corresponding results concerning Molodtsov's soft sets or Maji's fuzzy soft sets.
The remainder of the present paper is organized as follows. Section 2 first recalls some basic notions related to interval-valued fuzzy soft sets. Section 3 mainly discuss four types of interval-valued fuzzy soft subsets and the corresponding interval-valued fuzzy soft equal relations. Generalized commutative laws and generalized associative laws of interval-valued fuzzy soft sets are obtained as well. In Section 4, we propose many algebraic inequalities of IVF soft sets and use them to explore some nonclassical algebraic properties of interval-valued fuzzy soft sets with respect to soft product operations. Finally, the last section gives conclusions to complete this study and suggests potential directions for future work.

Preliminaries
Let be an initial universe of objects and let (or simply ) be the set of all parameters associated with objects in , called a parameter space. In most cases parameters are considered to be attributes, characteristics, or properties of objects in . The pair ( , ) is also known as a soft universe. We denote the power sets of by P( ). The concept of soft sets is defined as follows.
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 produces a series of approximate descriptions of a complicated object being perceived from various points of view. In other words, a soft set S = ( , ) over the universe can be regarded as a parameterized family of subsets of the universe . For any parameter ∈ , the subset ( ) ⊆ may be interpreted as the set of -approximate elements [3]. Note that ( ) may be arbitrary: some of them may be empty, and some may have nonempty intersections [3].
Maji et al. [23] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced the notion called fuzzy soft sets, which can be seen as a fuzzy generalization of Molodtsov's soft sets.
Definition 2 (see [23]). Let F( ) be the set of all fuzzy subsets in a universe . Let be a set of parameters and ⊆ . A pair (̃, ) is called a fuzzy soft set over , wherẽis a mapping given bỹ: → F( ).
In the above definition, fuzzy subsets in the universe are used as substitutes for the crisp subsets of . Hence, it is clear that every (classical) soft set may be considered as a fuzzy soft set. Using the standard notations, the fuzzy set̃( ) can be written as̃( ) = {( ,̃( )( )) : ∈ }. Now we consider the set = {[ , ] : 0 ≤ ≤ ≤ 1} and the order relation ≤ given by (1) Then L = ( , ≤ ) is a complete lattice. An interval-valued fuzzy set on a universe is a mapping : → . The union, intersection, and complement of interval-valued fuzzy sets can be obtained by canonically extending fuzzy settheoretic operations to intervals. The set of all interval-valued fuzzy sets on is denoted by I( ).
It is easy to see that fuzzy soft sets are a special case of interval-valued fuzzy soft sets since interval-valued fuzzy sets are extensions of fuzzy sets. This ensures that the results obtained in this paper can easily be applied to fuzzy soft sets or Molodtsov's soft sets.
The above two operations ∧ and ∨ will be referred to as soft product operations of interval-valued fuzzy soft sets in general. We denote by S ( , ) the collection of all intervalvalued fuzzy soft sets over with parameter space . For more details on interval-valued fuzzy soft sets and related terminologies used below, we refer to the papers [25,28,31].

Four Types of Interval-Valued Fuzzy Soft Subsets
For the sake of simplicity, the term "interval-valued fuzzy" will be abbreviated to IVF in what follows. The following type of soft subsets was first considered by Maji et al. [20] for Molodtsov's soft sets. Here we extend it to the more general seating of IVF soft sets.    Another kind of IVF soft subsets was initiated by Yang et al. [25] in the following way.
Definition 7 (see [25]). Let (̃, ) and (̃, ) be two IVF soft sets over . Then (̃, ) is called a IVF soft F-subset of (̃, ),  ( However, the reverse implication may not be true as illustrated by the following example.
Proof. It is straightforward and thus omitted.
By Proposition 9, we know that the IVF soft equal relations = and = coincide with each other. Hence, in what follows we write ≡ (called IVF soft identical relation) instead of = or = unless stated otherwise.
It is clear that⊆ and⊆ can be viewed as binary relations on the collection S ( , ) of all interval-valued fuzzy soft sets over with parameter space . From this point of view,⊆ and⊆ are also referred to as the IVF soft M-inclusion and IVF soft F-inclusion on S ( , ), respectively.
In a similar fashion, we can also verify that the IVF soft -inclusion⊆ is a partial order on S ( , ).

Proposition 11. The IVF soft -inclusion⊆ is a partial order on S ( , ).
Proof. The proof is similar to that of Proposition 10 and thus omitted.
Yang and Jun [28] proposed the following type of IVF soft subsets, which extends IVF soft -subsets in a substantial way.
Given two IVF soft sets A = (̃, ) and B = (̃, ) over , one easily observes that A⊆ B implies A⊆ B. However, the reverse may not be true as illustrated by the following example.
Example 13. Consider the two IVF soft sets A = (̃, ) and B 2 = (2, 2 ) shown in Tables 1 and 3. Then we can see that A⊆ B 2 sincẽ( 1 ) ⊆2( 3 ) and̃( 2 ) =2( 2 ), but A⊆ B 2 does not hold since ̸ ⊆ 2 . Moreover, it is worth noting that the IVF soft sets A and B 2 are not IVF soft -equal since we see that B 2 is not an IVF soft -subset of A.

4
The Scientific World Journal Motivated by Jun and Yang's IVF soft -subsets, we further introduced a new kind of IVF soft subsets as follows.
Two IVF soft sets (̃, ) and (̃, ) are said to be IVF soft Obviously,⊆ and⊆ are binary relations on S ( , ), which are called the IVF soft J-inclusion and IVF soft Linclusion, respectively.

Proposition 16. The IVF soft L-inclusion⊆ is a quasiorder on S ( , ).
Proof. The proof is similar to that of Proposition 15 and thus omitted.
Suppose that A = (̃, ) and B = (̃, ) are two IVF soft sets over . By definition, it can be verified that However, it is worth noting that the reverse implications do not hold in general. Note also that the IVF soft -inclusion⊆ and the IVF soft -inclusion⊆ are only quasiorders instead of partial orders on S ( , ). In general, these relations are not antisymmetric. As illustration, we consider an example as follows.
Example 17. Consider the two IVF soft sets A = (̃, ) and Tables 1 and 4. Then we have A⊆ B 3 . In addition, we can deduce that A and B 3 are IVF soft -equal since we also have B 3⊆ A. However, it is clear that A ≡ B 3 does not hold. This shows that the IVF soft -inclusion⊆ is not antisymmetric. Similarly, one can verify that A = B 3 , which implies that the IVF soft -inclusion⊆ is not antisymmetric. Thus,⊆ and⊆ are only quasiorders instead of partial orders.
Remark 18. In view of the above discussion, we have investigated four different types of soft subsets, including IVF soft -subsets, IVF soft -subsets, IVF soft -subsets, and IVF soft L-subsets. In conclusion, Figure 1 illustrates the relations among different types of IVF soft subsets, in which a single arrow → denotes that is a generalization of . These results extend the corresponding results for Molodtsov's soft sets in [30].
As an immediate consequence of the above results, we also obtain the following relations among three types of IVF soft equal relations.

Corollary 19.
Suppose that A = (̃, ) and B = (̃, ) are two IVF soft sets over . Then we have The Scientific World Journal 5 It is worth noting that the IVF soft equal relations ≡, = , and = are essentially distinct notions. For more details, one can refer to the discussion regarding these relations for Molodtsov's soft sets in [29].
At the end of this section, we characterize some basic algebraic properties of soft product operations ∧ and ∨ in virtue of IVF soft -equal relations. First, we consider properties regarding commutativity of soft product operations.
Hence, by definition of IVF soft -subsets, we have (̃, × )⊆ (̃, × ). The reverse IVF soft -inclusion follows analogously. Therefore, we have The second assertion can be proved in a similar fashion and thus omitted.
Remark 21. The above results are referred to as the generalized commutative laws of IVF soft sets due to the following reason. On one hand, it is obvious that the two IVF soft sets obtained by calculation from the left and right sides are distinct since they have different parameter sets. Thus, commutative laws do not hold in the conventional sense, which are characterized in virtue of the IVF soft identical relation ≡.
On the other hand, we have shown that commutative laws are valid in a weaker sense, characterized by using IVF soft -equal relations = . Note that Theorem 20 extends Theorem 5.13 (called the generalized soft commutative laws) in [30].
Next, we investigate properties of soft product operations by considering associative laws.
The second assertion can be proved in a similar fashion and thus omitted.

Basic Inequalities of IVF Soft Sets.
In this subsection, we first investigate some basic inequalities of IVF soft sets characterized by IVF soft -inclusions, which are useful in subsequent discussions.
By considering ∨-product of IVF soft sets, we can get the following results which are analogous to Propositions 28, 29, and 30, respectively.  The above result (called the weak idempotent law of IVF soft sets) indicates that the ∧-product operation of IVF soft sets is idempotent with respect to IVF soft -equal relations. On the other hand, it should be noted that we cannot form the parallel result for the ∨-product operation of IVF soft sets as illustrated by the following example.
Remark 39. The above results indicate that IVF soft sets have some nonclassical algebraic properties, compared with interval-valued fuzzy sets. By Definitions 4 and 5, the operations ∧ and ∨ of IVF soft sets are, respectively, defined by means of the intersection ∩ and union ∪ of intervalvalued fuzzy sets. In the theory of interval-valued fuzzy sets, operations ∩ and ∪ are both idempotent since (I( ), ∩, ∪) is a lattice. Furthermore, we know that ∩ and ∪ are dual to each other, which always satisfy similar or parallel algebraic properties. Nevertheless, from Theorem 37 and Example 38, one can observe that the operations ∧ and ∨ of IVF soft sets do not always have similar algebraic properties. In fact, as shown above we have while (̃, ) = (̃, ) ∧ (̃, ) .
It is also worth noting that which means that the ∧-product operation of IVF soft sets is idempotent with respect to the IVF soft -equal relation = , but not in the stronger sense of = . Thus, by using the IVF soft -inclusion⊆ , we only have some idempotent inequalities shown in Propositions 34 and 35.

Distributive Inequality of IVF Soft Sets.
In the above discussion, we have shown some nonclassical algebraic properties concerning soft product operations of IVF soft sets by considering idempotent laws. Next, we will investigate other interesting properties with regard to operations ∧ and ∨ of IVF soft sets by considering distributive laws.
In a similar fashion, one can verify the following distributive inequalities of IVF soft sets given by Liu et al. [29] without proofs. Remark 42. It is interesting to note that the operations ∧ and ∨ of IVF soft sets show some nonclassical algebraic properties when considering distributive laws. In the theory of intervalvalued fuzzy sets, the inclusion relation ⊆ is a partial order on the set I( ) of all interval-valued fuzzy sets on a given universe and we also know that (I( ), ∩, ∪) forms a distributive lattice. Thus, the operations ∩ and ∪ of intervalvalued fuzzy sets satisfy both the left and the right distributive laws.   However, it can be found that only distributive inequalities (described by the IVF soft -inclusion⊆ , which is a quasiorder on S ( , )) hold for IVF soft sets. In fact, neither the left nor the right distributive laws hold even in the weakest sense of IVF soft -equal relations. To illustrate this, we consider an example as follows.
4.4. Absorptive Inequality of IVF Soft Sets. As mentioned above, the set (I( ), ∩, ∪) of all interval-valued fuzzy sets on forms a distributive lattice. Thus, it is well known that the following types of absorptive laws are valid in the theory of interval-valued fuzzy sets. (2) ∪ ( ∩ ]) = .
Considering IVF soft sets, we have the following result which shows that the absorptive law similar to the first assertion in Proposition 44 holds in a much weaker form characterized in terms of IVF soft -equal relations.
Note also that the reverse IVF soft -inclusion holds according to Proposition 25.
The above result is called the weak absorptive law of IVF soft sets. From its proof above, we can also obtain the following absorptive inequalities of IVF soft sets described by IVF soft -inclusions.
It can easily be verified that the reverse IVF softinclusions do not hold in general. Thus, we cannot replace = by = in Theorem 45.
More interesting, it is worth noting that the absorptive law of IVF soft sets similar to the second assertion in Proposition 44 might not hold even in the weakest sense of = . To illustrate this, we consider an example as follows.

Conclusions
This study has been devoted to exploring some nonclassical algebraic properties of IVF soft sets under the soft product operations. Some algebraic inequalities of IVF soft sets characterized by IVF soft inclusions have been obtained and the relations among four different types of interval-valued fuzzy soft subsets have been ascertained. It has been shown that the commutative, associative, idempotent, and absorptive laws of IVF soft sets do not hold in the conventional sense described by IVF soft identical relations but hold in some weaker forms characterized in terms of the IVF soft equal relations = or = . We have found that the soft product operations ∧ and ∨ of IVF soft sets do not always have similar algebraic properties and only certain types of distributive inequalities instead of distributive laws hold for IVF soft sets. Some of the results obtained here extend the corresponding results for Molodtsov's soft sets in [29,30]. As future work, we will further investigate free IVF soft algebras. It will be also interesting to consider potential applications of different types of interval-valued fuzzy soft subsets in decision making under uncertainty.