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On modular inequalities of intervalvalued fuzzy soft sets characterized by soft Jinclusions
Journal of Inequalities and Applications volume 2014, Article number: 360 (2014)
Abstract
This study aims to explore modular inequalities of intervalvalued fuzzy soft sets characterized by Jun’s soft Jinclusions. Using soft product operations of intervalvalued fuzzy soft sets, we first investigate some basic properties of soft Jinclusions and soft Linclusions. Then a new concept called upward directed intervalvalued fuzzy soft sets is defined and some equivalent characterizations are presented. Furthermore, we consider modular laws in lattice theory and find that classical modular inequalities in lattice theory are not valid for intervalvalued fuzzy soft sets. Finally, we present some interesting inequalities of intervalvalued fuzzy soft sets by virtue of soft Jinclusions and related notions.
MSC:03E72.
1 Introduction
It is worth noting that uncertainty arise from various domains has different nature and cannot be captured within a single mathematical framework. In addition to probability theory and statistics, we currently have some advanced soft computing methods such as fuzzy sets [1], rough sets [2], and also soft sets [3]. Molodtsov’s soft set theory provides a relatively new mathematical approach to dealing with uncertainty from a parameterization point of view. In the past decades, a rapid development in this theory and various applications have been witnessed [4–24].
Some researchers endeavored to enrich soft sets by combining them with other soft computing models such as rough sets and fuzzy sets. Using soft sets as the granulation structures, Feng et al. [21] initiated soft approximation spaces and soft rough sets, which generalize Pawlak’s rough sets based on soft sets. On the other hand, Maji et al. [25] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of Molodtsov’s soft sets. Furthermore, Yang et al. [26] introduced intervalvalued fuzzy soft sets which realize a common extension of both Molodtsov’s soft sets and intervalvalued fuzzy sets. It should be noted that there are several different kinds of soft inclusions (also known as soft subsets) in the literature [22, 23, 27, 28]. Feng and Li [29] investigated different types of soft subsets and the related soft equal relations in a systematic way. They also considered some interesting algebraic properties of soft product operations for Molodtsov’s soft sets. Liu et al. [30] investigated these issues in the more general setting of intervalvalued fuzzy soft sets. They revealed some nonclassical algebraic properties of intervalvalued fuzzy soft sets with respect to soft product operations, which are distinct from those of intervalvalued fuzzy sets. As a continuation to this line of research, in the present paper we will focus on modular inequalities of intervalvalued fuzzy soft sets characterized by Jun’s soft Jinclusions.
The remainder of this study is organized as follows. Section 2 introduces some fundamental concepts and useful results regarding intervalvalued fuzzy soft sets. Section 3 investigates some basic properties of Jun’s soft Jinclusions and Liu’s soft Linclusions. In Section 4, we define upward directed intervalvalued fuzzy soft sets and concentrate on discussing algebraic properties of soft product operations concerning idempotency. Section 5 is devoted to the exploration of modular inequalities of intervalvalued fuzzy soft sets. Finally, the last section summarizes the study and suggests possible future works.
2 Preliminaries
Let U be a universe and ${E}_{U}$ (or simply E) be the set of all parameters associated with objects in U, which is called a parameter space. We denote the power sets of U by $\mathcal{P}(U)$. Then soft sets are defined as follows.
Definition 2.1 [3]
A pair $\mathfrak{S}=(F,A)$ is called a soft set over U, where $A\subseteq E$ and $F:A\to \mathcal{P}(U)$ is a setvalued mapping, called the approximate function of the soft set $\mathfrak{S}$.
By virtue of parametrization, a soft set could provide a series of approximate descriptions of a complicated object being perceived from various points of view. For any parameter $\u03f5\in A$, the subset $F(\u03f5)\subseteq U$ is called an ϵapproximate set, consisting of all ϵapproximate elements [3].
Next, let us consider the set ${L}^{I}=\{[x,y]:0\le x\le y\le 1\}$ and the order relation ${\le}_{{L}^{I}}$ given by
Then ${\mathcal{L}}^{I}=({L}^{I},{\le}_{{L}^{I}})$ is a complete lattice. An intervalvalued fuzzy set on a universe U is a mapping $\mu :U\to {L}^{I}$. The union, intersection and complement of intervalvalued fuzzy sets can be obtained by canonically extending fuzzy settheoretic operations to intervals. The set of all intervalvalued fuzzy sets on U is denoted by $\mathcal{I}(U)$.
Definition 2.2 [27]
Let $(U,E)$ be a soft universe and $A\subseteq E$. A pair $\mathfrak{I}=(\tilde{F},A)$ is called an intervalvalued fuzzy soft set over U, where $\tilde{F}$ is a mapping given by $\tilde{F}:A\to \mathcal{I}(U)$.
The mapping $\tilde{F}:A\to \mathcal{I}(U)$ is also called the approximate function of the intervalvalued fuzzy soft set $\mathfrak{I}=(\tilde{F},A)$. The following two operations ∧ and ∨ will be referred to as soft product operations of intervalvalued fuzzy soft sets in general.
Definition 2.3 [26]
Let $\mathfrak{A}=(\tilde{F},A)$ and $\mathfrak{B}=(\tilde{G},B)$ be two intervalvalued fuzzy soft sets over U. The ∧product (also called AND operation) of intervalvalued fuzzy soft sets $\mathfrak{A}$ and $\mathfrak{B}$ is an intervalvalued fuzzy soft set defined by $\mathfrak{A}\wedge \mathfrak{B}=(H,A\times B)$, where $H(a,b)=F(a)\cap G(b)$ for all $(a,b)\in A\times B$.
Definition 2.4 [26]
Let $\mathfrak{A}=(\tilde{F},A)$ and $\mathfrak{B}=(\tilde{G},B)$ be two intervalvalued fuzzy soft sets over U. The ∨product (also called OR operation) of intervalvalued fuzzy soft sets $\mathfrak{A}$ and $\mathfrak{B}$ is an intervalvalued fuzzy soft set defined by $\mathfrak{A}\vee \mathfrak{B}=(H,A\times B)$, where $H(a,b)=F(a)\cup G(b)$ for all $(a,b)\in A\times B$.
We denote by ${\mathcal{S}}^{I}(U,E)$ the collection of all intervalvalued fuzzy soft sets over U with parameter space E. For more details on intervalvalued fuzzy soft sets and related terminologies used below, we refer to the papers [26, 27, 31]. For convenience, we abbreviate the term ‘intervalvalued fuzzy’ as IVF in what follows.
There are some different types of soft inclusion relations in the literature. Here we mainly introduce two of them, namely Jun’s inclusion ${\tilde{\subseteq}}_{J}$ in [27] and Liu’s inclusion ${\tilde{\subseteq}}_{L}$ in [28]. The readers who are interested in soft subsets and related topics are referred to papers [27–29].
Definition 2.5 [27]
Let $(\tilde{F},A)$ and $(\tilde{G},B)$ be two IVF soft sets over U. Then $(\tilde{F},A)$ is called a IVF soft Jsubset of $(\tilde{G},B)$, denoted $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$, if for every $a\in A$ there exists $b\in B$ such that $\tilde{F}(a)\subseteq \tilde{G}(b)$. Two IVF soft sets $(\tilde{F},A)$ and $(\tilde{G},B)$ are said to be IVF soft Jequal, denoted $(\tilde{F},A){=}_{J}(\tilde{G},B)$, if $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$ and $(\tilde{G},B)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)$.
Motivated by Jun and Yang’s IVF soft Jsubsets, Liu et al. [28] further introduced the following kinds of IVF soft subsets.
Definition 2.6 [28]
Let $(\tilde{F},A)$ and $(\tilde{G},B)$ be two IVF soft sets over U. Then $(\tilde{F},A)$ is called a IVF soft Lsubset of $(\tilde{G},B)$, denoted $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$, if for every $a\in A$ there exists $b\in B$ such that $\tilde{F}(a)=\tilde{G}(b)$. Two IVF soft sets $(\tilde{F},A)$ and $(\tilde{G},B)$ are said to be IVF soft Lequal, denoted $(\tilde{F},A){=}_{L}(\tilde{G},B)$, if $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$ and $(\tilde{G},B)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)$.
Note that ${\tilde{\subseteq}}_{J}$ and ${\tilde{\subseteq}}_{L}$ are binary relations on ${\mathcal{S}}^{I}(U,E)$, which are called IVF soft Jinclusion and IVF soft Linclusion, respectively. The following result is easily verified in virtue of the above definitions.
Proposition 2.7 If $\mathfrak{A},\mathfrak{B}\in {\mathcal{S}}^{I}(U,E)$, then $\mathfrak{A}\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}\mathfrak{B}\Rightarrow \mathfrak{A}\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}\mathfrak{B}$.
Two IVF soft sets $\mathfrak{A}=(\tilde{F},A)$ and $\mathfrak{B}=(\tilde{G},B)$ are said to be identical, denoted by $\mathfrak{A}\equiv \mathfrak{B}$, if they have the same parameter sets as well as approximate functions. That is, $A=B$ and $\tilde{F}(a)=\tilde{G}(a)$ for all $a\in A$. As an immediate consequence of Proposition 2.7 and the definition of IVF soft identical relations, we get a result as follows.
Corollary 2.8 Suppose that $\mathfrak{A}=(\tilde{F},A)$ and $\mathfrak{B}=(\tilde{G},B)$ are two IVF soft sets over U. Then we have
It is worth noting that all the reverse implications in Proposition 2.7 and Corollary 2.8 do not hold in general. For more details, one can refer to the discussion regarding Molodtsov’s soft sets in [28, 29].
The abovementioned concepts are useful in characterizing some fundamental algebraic properties of soft product operations ∧ and ∨. To show this, we consider the following result.
Theorem 2.9 [30] (Generalized commutative laws of IVF soft sets)
Let $(\tilde{F},A)$ and $(\tilde{G},B)$ be two IVF soft sets over U. Then we have

(1)
$(\tilde{F},A)\wedge (\tilde{G},B){=}_{L}(\tilde{G},B)\wedge (\tilde{F},A)$;

(2)
$(\tilde{F},A)\vee (\tilde{G},B){=}_{L}(\tilde{G},B)\vee (\tilde{F},A)$.
In view of results, we can see that the commutative laws do not hold in the conventional sense, which are characterized by IVF soft identical relation. Another important fact concerns algebraic properties regarding distributivity of soft product operations.
Theorem 2.10 [30]
Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. Then we have

(1)
$((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\wedge (\tilde{H},C))\vee ((\tilde{G},B)\wedge (\tilde{H},C))$;

(2)
$((\tilde{F},A)\wedge (\tilde{G},B))\vee (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\vee (\tilde{H},C))\wedge ((\tilde{G},B)\vee (\tilde{H},C))$.
In a similar fashion, Liu et al. [28] proposed the following distributive inequalities of IVF soft sets.
Theorem 2.11 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. Then we have

(1)
$(\tilde{F},A)\wedge ((\tilde{G},B)\vee (H,C))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\wedge (\tilde{G},B))\vee ((\tilde{F},A)\wedge (\tilde{H},C))$;

(2)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\vee (\tilde{G},B))\wedge ((\tilde{F},A)\vee (\tilde{H},C))$.
Remark 2.12 It is interesting to point out that the soft product operations of IVF soft sets possess some nonclassical algebraic properties, as shown by the above results. Comparing with intervalvalued fuzzy sets, we can find that $(\mathcal{I}(U),\cap ,\cup )$ forms a distributive lattice, while only distributive inequalities (described by the IVF soft Linclusion ${\tilde{\subseteq}}_{L}$) hold for IVF soft sets. Thus neither the left nor the right distributive laws hold even in the weakest sense of IVF soft Jequal relations.
3 Some basic properties of IVF soft inclusions
Here we propose several basic inequalities of IVF soft sets characterized by IVF soft inclusions, which are useful in subsequent discussions.
Proposition 3.1 [30]
Let $(\tilde{F},A)$ and $(\tilde{G},B)$ be IVF soft sets over U. Then we have

(1)
$(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee (\tilde{G},B)$;

(2)
$(\tilde{G},B)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee (\tilde{G},B)$;

(3)
$(\tilde{F},A)\wedge (\tilde{G},B)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)$;

(4)
$(\tilde{F},A)\wedge (\tilde{G},B)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$.
Proposition 3.2 [30]
The IVF soft Linclusion ${\tilde{\subseteq}}_{L}$ is a preorder on ${\mathcal{S}}^{I}(U,E)$.
Proposition 3.3 [30]
The IVF soft Jinclusion ${\tilde{\subseteq}}_{J}$ is a preorder on ${\mathcal{S}}^{I}(U,E)$.
By generic properties of preorders, we can deduce the following two results.
Corollary 3.4 The IVF soft Lequal relation ${=}_{L}$ is an equivalence relation on ${\mathcal{S}}^{I}(U,E)$.
Corollary 3.5 The IVF soft Jequal relation ${=}_{J}$ is an equivalence relation on ${\mathcal{S}}^{I}(U,E)$.
Proposition 3.6 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$, then we have

(1)
$(\tilde{H},C)\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\vee (\tilde{G},B)$;

(2)
$(\tilde{H},C)\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\vee (\tilde{H},C)$;

(3)
$(\tilde{F},A)\vee (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\vee (\tilde{H},C)$;

(4)
$(\tilde{F},A)\vee (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\vee (\tilde{G},B)$.
Proof To prove the first assertion, let $(\tilde{H},C)\vee (\tilde{F},A)=(\tilde{L},C\times A)$ and $(\tilde{H},C)\vee (\tilde{G},B)=({\tilde{R}}_{1},C\times B)$. By hypothesis, we have $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$ and so for every $a\in A$ there exists $b\in B$ such that $\tilde{F}(a)=\tilde{G}(b)$. For any $(c,a)\in C\times A$, we deduce that
for some $(c,b)\in B\times C$. Hence $(\tilde{H},C)\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\vee (\tilde{G},B)$.
Next, we show the second assertion. Let $(\tilde{G},B)\vee (\tilde{H},C)=({\tilde{R}}_{2},B\times C)$. By the generalized commutative laws of IVF soft sets in Theorem 2.9, we have
Note also that we have verified that $(\tilde{L},C\times A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}({\tilde{R}}_{1},C\times B)$ in the first assertion. It follows that
by transitivity of the preorder ${\tilde{\subseteq}}_{L}$ on ${\mathcal{S}}^{I}(U,E)$.
The proofs of the other two assertions can be obtained in a similar fashion and thus omitted. □
Moreover, one can easily verify a result as follows.
Proposition 3.7 Let $({\tilde{F}}_{i},{A}_{i})$, $({\tilde{G}}_{i},{B}_{i})$ be IVF soft sets over U and $i=1,2$. If $({\tilde{F}}_{1},{A}_{1})\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}({\tilde{G}}_{1},{B}_{1})$ and $({\tilde{F}}_{2},{A}_{2})\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}({\tilde{G}}_{2},{B}_{2})$, then we have
Considering ∧product of IVF soft sets, one can verify the following results which are analogous to Proposition 3.6 and 3.7, respectively.
Proposition 3.8 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$, then we have

(1)
$(\tilde{H},C)\wedge (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\wedge (\tilde{G},B)$;

(2)
$(\tilde{H},C)\wedge (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\wedge (\tilde{H},C)$;

(3)
$(\tilde{F},A)\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\wedge (\tilde{H},C)$;

(4)
$(\tilde{F},A)\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\wedge (\tilde{G},B)$.
Proposition 3.9 Let $({\tilde{F}}_{i},{A}_{i})$, $({\tilde{G}}_{i},{B}_{i})$ be IVF soft sets over U and $i=1,2$. If $({\tilde{F}}_{1},{A}_{1})\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}({\tilde{G}}_{1},{B}_{1})$ and $({\tilde{F}}_{2},{A}_{2})\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}({\tilde{G}}_{2},{B}_{2})$, then we have
Regarding Jun’s inclusion ${\tilde{\subseteq}}_{J}$ of IVF soft sets, we obtained the following similar results in [30].
Proposition 3.10 Let $(\tilde{F},A)$, $(\tilde{G},B)$ and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)$, then we have

(1)
$(\tilde{F},A)\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\wedge (\tilde{H},C)$;

(2)
$(\tilde{F},A)\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\wedge (\tilde{G},B)$;

(3)
$(\tilde{H},C)\wedge (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\wedge (\tilde{G},B)$;

(4)
$(\tilde{H},C)\wedge (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\wedge (\tilde{H},C)$;

(5)
$(\tilde{F},A)\vee (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\vee (\tilde{H},C)$;

(6)
$(\tilde{F},A)\vee (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\vee (\tilde{G},B)$;

(7)
$(\tilde{H},C)\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)\vee (\tilde{G},B)$;

(8)
$(\tilde{H},C)\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{G},B)\vee (\tilde{H},C)$.
Proposition 3.11 Let $({\tilde{F}}_{i},{A}_{i})$, $({\tilde{G}}_{i},{B}_{i})$ be IVF soft sets over U and $i=1,2$. If $({\tilde{F}}_{1},{A}_{1})\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}({\tilde{G}}_{1},{B}_{1})$ and $({\tilde{F}}_{2},{A}_{2})\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}({\tilde{G}}_{2},{B}_{2})$, then we have
and
4 Upward directed IVF soft sets and idempotency
Now, we investigate algebraic properties of soft product operations of IVF soft sets by considering idempotency. First, we recall some important results proposed by Liu et al. [30].
Proposition 4.1 [30]
Let $(\tilde{F},A)\in {\mathcal{S}}^{I}(U,E)$. Then $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee (\tilde{F},A)$.
Proposition 4.2 [30]
Let $(\tilde{F},A)\in {\mathcal{S}}^{I}(U,E)$. Then $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\wedge (\tilde{F},A)$.
Theorem 4.3 [30]
Let $(\tilde{F},A)\in {\mathcal{S}}^{I}(U,E)$. Then $(\tilde{F},A){=}_{J}(\tilde{F},A)\wedge (\tilde{F},A)$.
The last result indicates that the ∧product operation of IVF soft sets is idempotent with respect to IVF soft Jequal relations, and is referred to as the weak idempotent law of IVF soft sets. Nevertheless, the parallel result regarding ∨product operations of IVF soft sets does not hold.
We know that ∩ and ∪ are both idempotent since $(\mathcal{I}(U),\cap ,\cup )$ is a lattice in the theory of intervalvalued fuzzy sets. The two operations ∩ and ∪ are dual to each other, which always satisfy similar or parallel algebraic properties. According to Definition 2.3 and Definition 2.4, the operations ∧ and ∨ of IVF soft sets are defined in terms of the intersection ∩ and union ∪ of intervalvalued fuzzy sets, respectively. Nevertheless, it is interesting to see that the operations ∧ and ∨ of IVF soft sets do not always have similar algebraic properties. In fact, as shown by some illustrative examples in [30] we have
but
Note also that
which shows that the ∧product operation of IVF soft sets is idempotent with respect to the IVF soft Jequal relation ${=}_{J}$, but not in the stronger sense of ${=}_{L}$. Thus considering the IVF soft Linclusion ${\tilde{\subseteq}}_{L}$, we only have some idempotent inequalities as shown in Proposition 4.1 and Proposition 4.2. In view of these results, we conclude that IVF soft sets possess some nonclassical algebraic properties, compared with intervalvalued fuzzy sets. The interested readers could refer to [28, 30] for more details.
Recall that a nonempty set A together with a preorder ≤ is called an upward directed set if every pair of elements in A has an upper bound. That is, for every $a,b\in A$, there exists $c\in A$ such that $a\le c$ and $b\le c$.
Definition 4.4 Let $(\tilde{F},A)$ be an IVF soft set over U with $A\ne \mathrm{\varnothing}$. Then $(\tilde{F},A)$ is said to be upward directed if for every ${a}_{1},{a}_{2}\in A$, there exists ${a}_{3}\in A$ such that
Example 4.5 Let $U=\{a,b,c,d,e\}$ be the universe and the parameter space $E=\{{e}_{1},{e}_{2},{e}_{3},{e}_{4},{e}_{5}\}$. Assume that the parameter set $A=\{{e}_{1},{e}_{2},{e}_{4}\}$ and $\mathfrak{A}=(\tilde{F},A)$ is an IVF soft set over U with tabular representation given by Table 1. By definition, one can verify that $\mathfrak{A}=(\tilde{F},A)$ is an upward directed IVF soft set over U.
The following statements justify the term upward directed IVF soft sets and illustrate some intuitive ideas for introducing such a notion in the theory of IVF soft sets.
Proposition 4.6 Let $\mathfrak{S}=(\tilde{F},A)\in {\mathcal{S}}^{I}(U,E)$ and ${C}_{\mathfrak{S}}=\{\tilde{F}(a)a\in A\}$. Then $\mathbb{D}$ is an upward directed IVF soft set over U if and only if $({C}_{\mathfrak{S}},\subseteq )$ is an upward directed set.
Proof First, suppose that $\mathfrak{S}=(\tilde{F},A)$ is an upward directed IVF soft set over U. Then by the definition of upward directed IVF soft sets, we have $A\ne \mathrm{\varnothing}$ and so ${C}_{\mathfrak{S}}=\{\tilde{F}(a)a\in A\}$ is a nonempty subset of $\mathcal{I}(U)$. Moreover, for every ${a}_{1},{a}_{2}\in A$, there exists ${a}_{3}\in A$ such that $\tilde{F}({a}_{1})\cup \tilde{F}({a}_{2})\subseteq \tilde{F}({a}_{3})$. It follows that $\tilde{F}({a}_{1})\subseteq \tilde{F}({a}_{3})$ and $\tilde{F}({a}_{2})\subseteq \tilde{F}({a}_{3})$. This shows that ${C}_{\mathfrak{S}}$ is an upward directed set with respect to the partial order ⊆.
Conversely, let $({C}_{\mathfrak{S}},\subseteq )$ be an upward directed set. Then $A\ne \mathrm{\varnothing}$ since ${C}_{\mathfrak{S}}$ is nonempty. For every ${a}_{1},{a}_{2}\in A$, by the definition of upward directed sets, the pair of elements $\tilde{F}({a}_{1})$ and $\tilde{F}({a}_{2})$ has an upper bound in ${C}_{\mathfrak{S}}$. Hence there exists ${a}_{3}\in A$ such that $\tilde{F}({a}_{1})\subseteq \tilde{F}({a}_{3})$ and $\tilde{F}({a}_{2})\subseteq \tilde{F}({a}_{3})$. It follows that $\tilde{F}({a}_{1})\cup \tilde{F}({a}_{2})\subseteq \tilde{F}({a}_{3})$. Therefore, $\mathfrak{S}=(\tilde{F},A)$ is an upward directed IVF soft set over U. □
Proposition 4.7 Let $(\tilde{F},A)\in {\mathcal{S}}^{I}(U,E)$ and $A\ne \mathrm{\varnothing}$. Then $(\tilde{F},A)$ is an upward directed IVF soft set over U if and only if $(\tilde{F},A){=}_{J}(\tilde{F},A)\vee (\tilde{F},A)$.
Proof Let us denote $(\tilde{F},A)\vee (\tilde{F},A)$ by $(\tilde{R},A\times A)$. First, assume that $(\tilde{F},A)$ is an upward directed IVF soft set over U. For every $({a}_{1},{a}_{2})\in A\times A$, there exists ${a}_{3}\in A$ such that
since $(\tilde{F},A)$ is upward directed. This shows that
But by Proposition 4.3, we also have $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{L}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee (\tilde{F},A)$, and so
Hence we deduce that $(\tilde{F},A){=}_{J}(\tilde{F},A)\vee (\tilde{F},A)$.
Conversely, let $(\tilde{F},A){=}_{J}(\tilde{F},A)\vee (\tilde{F},A)$. Then in particular, we have
Now for every ${a}_{1},{a}_{2}\in A$, there exists ${a}_{3}\in A$ such that
Hence by definition, $(\tilde{F},A)$ is an upward directed IVF soft set over U. □
Corollary 4.8 Let $\mathfrak{S}=(\tilde{F},A)\in {\mathcal{S}}^{I}(U,E)$ and $A\ne \mathrm{\varnothing}$. Then the following are equivalent:

(1)
$\mathbb{D}$ is an upward directed IVF soft set over U.

(2)
${C}_{\mathfrak{S}}=\{\tilde{F}(a)a\in A\}$ is an upward directed set with respect to ⊆.

(3)
$(\tilde{F},A)\vee (\tilde{F},A){=}_{J}(\tilde{F},A)$.

(4)
$(\tilde{F},A)\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)$.
Proof According to Proposition 4.6 and Proposition 4.7, we have (1) ⇔ (2) and (1) ⇔ (3), respectively. Thus we only need to show that (1) ⇔ (4). In fact, let us denote $(\tilde{F},A)\vee (\tilde{F},A)$ by $(\tilde{G},A\times A)$. Then by Definition 4.4, $(\tilde{F},A)$ is an upward directed IVF soft set over U if and only if for every ${a}_{1},{a}_{2}\in A$, there exists ${a}_{3}\in A$ such that
Clearly, this is equivalent to $(\tilde{G},A\times A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)$, completing our proof. □
5 Modular inequalities of IVF soft sets
Let $(L,\wedge ,\vee ,\le )$ be a lattice and $a,b,c\in L$. Then one can verify that
In particular, for every intervalvalued fuzzy sets μ, ν, λ, one can deduce that
Actually, the modular law
holds for all $\mu ,\nu ,\lambda \in \mathcal{I}(U)$ since $(\mathcal{I}(U),\cap ,\cup )$ forms a distributive lattice. Considering IVF soft sets and soft product operations, we encounter a situation in contrast to the above. First, we have following types of modular inequalities of IVF soft sets.
Theorem 5.1 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. Then we have
Proof By the distributive inequalities in Theorem 2.10, we have
and it implies that
Note also that
by Proposition 3.1. Thus from Proposition 3.10, it follows that
Since ${\tilde{\subseteq}}_{J}$ is transitive on ${\mathcal{S}}^{I}(U,E)$, we deduce that
which completes the proof. □
It is interesting to see that the reverse soft Jinclusion does not hold in general. Actually we will illustrate this fact with an example in the following. Using the generalized commutative laws of IVF soft sets in Theorem 2.9, we have the following consequences of Theorem 5.1.
Corollary 5.2 Let $(\tilde{F},A)$, $(\tilde{G},B)$ and $(\tilde{H},C)$ be IVF soft sets over U. Then we have

(1)
$((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))$;

(2)
$((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))$;

(3)
$((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))$.
Proof We only prove the first soft Jinclusion; the proofs of other soft Jinclusions can be obtained using similar techniques. Note first that
by the generalized commutative laws of IVF soft sets. It follows that
Thus we deduce that
according to Proposition 3.10. But by Theorem 5.1, we also have
Hence we conclude that
since ${\tilde{\subseteq}}_{J}$ is transitive on ${\mathcal{S}}^{I}(U,E)$. □
In a similar fashion, one can verify the following statements.
Corollary 5.3 Let $(\tilde{F},A)$, $(\tilde{G},B)$ and $(\tilde{H},C)$ be IVF soft sets over U. Then we have

(1)
$(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))$;

(2)
$(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))$;

(3)
$(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))$;

(4)
$(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))$.
Corollary 5.4 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. Then we have

(1)
$(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)$;

(2)
$(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)$;

(3)
$(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)$;

(4)
$(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)$.
Corollary 5.5 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. Then we have

(1)
$((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)$;

(2)
$((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)$;

(3)
$((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)$;

(4)
$((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)$.
Theorem 5.6 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$, then we have
Proof By the distributive inequalities in Theorem 2.11, we have
and so
Since $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ by the hypothesis, we deduce that
by Proposition 3.10. Then it follows that
Consequently, we can obtain
by transitivity of the preorder ${\tilde{\subseteq}}_{J}$ on ${\mathcal{S}}^{I}(U,E)$. □
Example 5.7 Suppose that $U=\{{h}_{1},{h}_{2},{h}_{3},{h}_{4},{h}_{5},{h}_{6}\}$ is the universe and $E=\{{e}_{1},{e}_{2},{e}_{3},{e}_{4},{e}_{5}\}$ is the parameter space. For the parameter sets $A=\{{e}_{1},{e}_{2}\}$, $B=\{{e}_{3}\}$, and $C=\{{e}_{4},{e}_{5}\}$, let $\mathfrak{S}=(\tilde{F},A)$, $\mathfrak{T}=(\tilde{G},B)$, and $\mathfrak{K}=(\tilde{H},C)$ be three IVF soft sets over U, where
It is clear that $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ since $\tilde{F}({e}_{1})\subseteq \tilde{H}({e}_{4})$ and $\tilde{F}({e}_{2})\subseteq \tilde{H}({e}_{5})$. Let us write $(\tilde{T},B\times C)$ for $(\tilde{G},B)\wedge (\tilde{H},C)$ where $\tilde{T}(b,c)=\tilde{G}(b)\cap \tilde{H}(c)$ for all $(b,c)\in B\times C$. Then $B\times C=\{({e}_{3},{e}_{4}),({e}_{3},{e}_{5})\}$ and by calculation, one obtains
and
Then let us write $(\tilde{F},A)\vee (\tilde{T},B\times C)$ as $\mathfrak{L}=(\tilde{L},A\times (B\times C))$, where
for all $(a,(b,c))\in A\times (B\times C)$. It is easy to see that
Proceeding with detailed calculations, one can obtain the IVF soft set $\mathfrak{L}$ with its tabular representation shown in Table 2.
Next, we write $(\tilde{M},A\times B)$ for $(\tilde{F},A)\vee (\tilde{G},B)$ where $\tilde{M}(a,b)=\tilde{F}(a)\cup \tilde{G}(b)$ for all $(a,b)\in A\times B$. Let us denote $(\tilde{H},C)\vee (\tilde{H},C)$ by $\mathfrak{N}=(\tilde{N},C\times C)$, where $\tilde{N}({c}_{1},{c}_{2})=\tilde{H}({c}_{1})\cup \tilde{H}({c}_{2})$ for all $({c}_{1},{c}_{2})\in C\times C$. It is easy to see that $A\times B=\{({e}_{1},{e}_{3}),({e}_{2},{e}_{3})\}$ and $C\times C=\{({e}_{4},{e}_{4}),({e}_{4},{e}_{5}),({e}_{5},{e}_{4}),({e}_{5},{e}_{5})\}$. By calculation, we get
and
Also we can obtain the IVF soft set $\mathfrak{N}=(\tilde{N},C\times C)$ with its tabular representation given by Table 3.
Moreover, let $\mathfrak{R}=(\tilde{R},(A\times B)\times (C\times C))=(\tilde{M},A\times B)\wedge (\tilde{N},C\times C)$, where $\tilde{R}((a,b),({c}_{1},{c}_{2}))=\tilde{M}(a,b)\cap \tilde{N}({c}_{1},{c}_{2})=(\tilde{F}(a)\cup \tilde{G}(b))\cap (\tilde{H}({c}_{1})\cup \tilde{H}({c}_{2}))$ for all $((a,b),({c}_{1},{c}_{2}))\in (A\times B)\times (C\times C)$. It is not difficult to check that
Proceeding with detailed calculations, we get the IVF soft set ℜ with its tabular representation shown in Table 4.
Also, let ${\mathfrak{R}}_{1}=({\tilde{R}}_{1},(A\times B)\times C)=(\tilde{M},A\times B)\wedge (\tilde{H},C)$. Then we have
and tabular representation of the IVF soft set ${\mathfrak{R}}_{1}$ is shown in Table 5.
Now, in view of Table 2 and Table 4 one can verify that $\mathfrak{L}\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}\mathfrak{R}$. That is,
Nevertheless, it is easily seen from Table 2 and Table 5 that $\mathfrak{L}{\tilde{\u2288}}_{J}{\mathfrak{R}}_{1}$ since $\tilde{L}({e}_{1},({e}_{3},{e}_{5}))\u2288\tilde{R}(({e}_{1},{e}_{3}),{e}_{4})$, $\tilde{L}({e}_{1},({e}_{3},{e}_{5}))\u2288\tilde{R}(({e}_{1},{e}_{3}),{e}_{5})$, $\tilde{L}({e}_{1},({e}_{3},{e}_{5}))\u2288\tilde{R}(({e}_{2},{e}_{3}),{e}_{4})$, and $\tilde{L}({e}_{1},({e}_{3},{e}_{5}))\u2288\tilde{R}(({e}_{2},{e}_{3}),{e}_{5})$. That is,
does not hold in general.
Remark 5.8 In view of Theorem 5.1, Theorem 5.6 and Example 5.7, we can find that under soft product operations, IVF soft sets possess some interesting algebraic properties which differ from those of intervalvalued fuzzy sets. In particular, one can see that usual modular inequalities in lattice theory are not valid for IVF soft sets.
Using the generalized commutative laws of IVF soft sets in Theorem 2.9, we have the following consequences of Theorem 5.6.
Corollary 5.9 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$, then we have

(1)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\vee (\tilde{G},B))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$;

(2)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\vee (\tilde{F},A))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$;

(3)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\vee (\tilde{F},A))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$.
Proof We only verify the first assertion; the proofs of the others can be obtained in a similar fashion. First, by the generalized commutative laws of IVF soft sets, we also know that
and so
Now, from Proposition 3.6, it follows that
But according to Theorem 5.6,
Hence we conclude that
by transitivity of the preorder ${\tilde{\subseteq}}_{J}$ on ${\mathcal{S}}^{I}(U,E)$. □
Using similar techniques as above, we obtain the following results.
Corollary 5.10 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$, then we have

(1)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(2)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(3)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{F},A)\vee (\tilde{G},B))$;

(4)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B))\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{F},A)\vee (\tilde{G},B))$.
Corollary 5.11 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$, then we have

(1)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(2)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(3)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{F},A)\vee (\tilde{G},B))$;

(4)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{H},C)\vee (\tilde{H},C))\wedge ((\tilde{F},A)\vee (\tilde{G},B))$.
Corollary 5.12 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$, then we have

(1)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\vee (\tilde{F},A))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$;

(2)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{G},B)\vee (\tilde{F},A))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$;

(3)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\vee (\tilde{G},B))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$;

(4)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}((\tilde{F},A)\vee (\tilde{G},B))\wedge ((\tilde{H},C)\vee (\tilde{H},C))$.
Theorem 5.13 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ and $(\tilde{H},C)$ is upward directed, then we have
Proof First, according to Theorem 5.6,
But by the hypothesis, $(\tilde{H},C)$ is a upward directed IVF soft set, and so
by Proposition 4.7. It follows that
and so by Proposition 3.10, we deduce that
Thus by transitivity of ${\tilde{\subseteq}}_{J}$, we have
On the other hand, we also have
according to Theorem 5.1. Therefore, we finally conclude that
completing the proof. □
Using the generalized commutative laws of IVF soft sets in Theorem 2.9, we have the following consequences of Theorem 5.13.
Corollary 5.14 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ and $(\tilde{H},C)$ is upward directed, then we have

(1)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B)){=}_{J}((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)$;

(2)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B)){=}_{J}((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)$;

(3)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C)){=}_{J}((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)$.
Proof By the generalized commutative laws of IVF soft sets, we have
and so
That is,
and
Then according to Proposition 3.10, we deduce that
and
Hence it follows that
Now, according to Theorem 5.13,
Therefore, we conclude that
by transitivity of IVF soft Jequal relations. The proofs of other soft Jequalities can be obtained using similar techniques. □
The proofs of the following results are similar to that of Corollary 5.14 and thus omitted.
Corollary 5.15 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ and $(\tilde{H},C)$ is upward directed, then we have

(1)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A){=}_{J}((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)$;

(2)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A){=}_{J}((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)$;

(3)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A){=}_{J}((\tilde{G},B)\vee (\tilde{F},A))\wedge (\tilde{H},C)$;

(4)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A){=}_{J}((\tilde{F},A)\vee (\tilde{G},B))\wedge (\tilde{H},C)$.
Corollary 5.16 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ and $(\tilde{H},C)$ is upward directed, then we have

(1)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A){=}_{J}(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))$;

(2)
$((\tilde{H},C)\wedge (\tilde{G},B))\vee (\tilde{F},A){=}_{J}(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(3)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A){=}_{J}(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(4)
$((\tilde{G},B)\wedge (\tilde{H},C))\vee (\tilde{F},A){=}_{J}(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))$.
Corollary 5.17 Let $(\tilde{F},A)$, $(\tilde{G},B)$, and $(\tilde{H},C)$ be IVF soft sets over U. If $(\tilde{F},A)\phantom{\rule{0.2em}{0ex}}{\tilde{\subseteq}}_{J}\phantom{\rule{0.2em}{0ex}}(\tilde{H},C)$ and $(\tilde{H},C)$ is upward directed, then we have

(1)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B)){=}_{J}(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))$;

(2)
$(\tilde{F},A)\vee ((\tilde{H},C)\wedge (\tilde{G},B)){=}_{J}(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(3)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C)){=}_{J}(\tilde{H},C)\wedge ((\tilde{G},B)\vee (\tilde{F},A))$;

(4)
$(\tilde{F},A)\vee ((\tilde{G},B)\wedge (\tilde{H},C)){=}_{J}(\tilde{H},C)\wedge ((\tilde{F},A)\vee (\tilde{G},B))$.
6 Conclusions
This paper focused on exploring modular inequalities of IVF soft sets characterized by Jun’s soft Jinclusions. It has been shown that Jun’s soft Jinclusions and Liu’s soft Linclusions are preorders; hence the soft equal relations ${=}_{J}$ and ${=}_{L}$ derived from them are equivalence relations on the collection ${\mathcal{S}}^{I}(U,E)$ of all IVF soft sets over U with parameter space E. These soft inclusions proved to be useful in characterizing some fundamental algebraic properties of soft product operations. Moreover, upward directed IVF soft sets were introduced and some equivalent characterizations were presented. We finally considered modular laws in lattice theory and proposed modular inequalities of IVF soft sets using soft Jinclusions and some related notions mentioned above. As future work, we will further investigate other types of interesting inequalities of IVF soft sets.
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Acknowledgements
The authors are highly grateful to the anonymous referees for their insightful comments and valuable suggestions which greatly improve the quality of this paper. This work was partially supported by National Natural Science Foundation of China (Program No. 11301415), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2013JQ1020), Shaanxi Provincial Research Plan for Young Scientific and Technological New Stars (Program No. 2014KJXX73) and Scientific Research Program Funded by Shaanxi Provincial Education Department of China (Program No. 2013JK1098).
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Liu, X., Feng, F., Yager, R.R. et al. On modular inequalities of intervalvalued fuzzy soft sets characterized by soft Jinclusions. J Inequal Appl 2014, 360 (2014). https://doi.org/10.1186/1029242X2014360
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Keywords
 intervalvalued fuzzy soft sets
 intervalvalued fuzzy sets
 soft product operations
 soft inclusions
 modular inequalities