Inverse packet fuzzy sets and its characteristics

Abstract

Based on inverse packet sets and fuzzy set, the concept of inverse packet fuzzy sets is proposed in this paper by means of introducing dynamic characteristics into a fuzzy set $A$ and improving the fuzzy set $A$ . The inverse packet fuzzy sets is composed of a pair of fuzzy sets generated by the fuzzy set $A$ and has dynamic characteristic and fuzzy characteristic. Furthermore, on certain conditions, it can be restored to the fuzzy set $A$ or the inverse packet sets generated by the Cantor set $X=\ker(A)$ . As a result, an inverse packet fuzzy sets can be expressed as a series of inverse packet sets by defining its scalar multiplication and its cut sets. Meanwhile some attribute-dependent characteristics of inverse packet fuzzy sets are obtained. The new set model will both enrich the set theory and provide a new basic tool for further practical applications.

Keywords

Inverse packet sets fuzzy set inverse packet fuzzy sets attribute-dependent characteristics

1. Introduction

Fuzzy sets [1] were proposed by ZADEN L.A in 1965 and are an extension of Cantor sets. It has fuzzy properties represented by membership functions. Since then, the theory of fuzzy sets has been deeply and extensively studied, such as random fuzzy sets, fuzzy logic, fuzzy numbers, fuzzy reasoning and so on [1, 2, 3, 4]. With the improvement of the theory, its applications in various fields are also more and more extensive, for example, fuzzy control is a common one. On the other hand, the inverse packet sets [5, 6, 7] was proposed by Kaiquan SHI in 2012, and it was obtained by introducing dynamic properties into the Cantor set and improving it. Its dynamic properties is expressed by a pair of Cantor sets composed of an internal inverse packet set and the outer inverse packet set generated by the Cantor set $X$ . After that, the inverse packet sets has been continuously supplemented in theory and applied to solve practical problems, such as intelligent fusion mining-information discovery, iterative intelligent camouflage and reduction of information, multi-attribute venture capital decision-making, and intelligent retrieval of big data [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

With the further development of science and technology, the uncertainty and dynamic characteristics of search objects are becoming more and more prominent. For example, patients in the suspected COVID-19 patient collection in Shandong Province have both ambiguity and dynamic changes. Therefore, we consider how to introduce dynamic properties into a fuzzy set, or replace the basic set $X$ of an inverse packet sets with a fuzzy set $A$ . Based on this idea, we propose the concept of inverse packet fuzzy sets, which not only has fuzzy properties but also dynamic properties. And it can be shown that it is an extension of the fuzzy set $A$ , or an extension of an inverse packet sets generated by the Cantor set $\ker(A)$ . Using the properties of fuzzy sets and inverse packet sets, the relation theorems, generation theorems and attribute-dependent theorems of inverse packet fuzzy sets can be obtained.

2. Preliminaries

2.1 Fuzzy set

Let $U$ be a finite nonempty universe, call the map

$\displaystyle A:U\to[0,1]$ $\displaystyle x|\to A(x)$

a fuzzy set [1], written briefly as a FS, in which $A(x)$ is called the membership function of $A$ .

Obviously, if $A(x)=1$ or $A(x)=0$ for every element $x\in U$ , the fuzzy set $A$ is a Cantor set. So a FS is an extended form of the Cantor set.

Supposed that $A$ is a fuzzy set, $\lambda$ is a real number and $\lambda\in[0,1]$ , then $A^{\lambda},A^{\dot{\lambda}},\ker(A),\lambda A$ are respectively called $\lambda-$ cut set, $\lambda-$ strong cut set, the core and scalar multiplication of $A$ , and

$\displaystyle A^{\lambda}=\{x\in U|A(x)\geqslant\lambda\},$ $\displaystyle A^{\dot{\lambda}}=\{x\in U|A(x)>\lambda\},$ $\displaystyle\ker(A)=\{x\in U|A(x)=1\},$ $\displaystyle\lambda A(x)=\lambda\wedge A(x).$

2.2 Inverse packet sets [5]

Let $X\subset U$ be a nonempty Cantor set, $\alpha\subseteq V$ is its attribute set, and $\alpha\neq\emptyset$ , then $\bar{X}^{F},\bar{X}^{\bar{F}},(\bar{X}^{F},\bar{X}^{\bar{F}})$ are respectively called an internal inverse packet set (an internal IPS), an outer inverse packet set (an outer IPS), and an inverse packet sets (an IPS) generated by $X$ , and

$\displaystyle\bar{X}^{F}=X\cup\{y\in C_{U}(X)|f(y)\in X,f\in F\},$ $\displaystyle\bar{X}^{\bar{F}}=X-\{x\in X|f(x)\bar{\in}X,\bar{f}\in\bar{F}\},$

if the attribute sets $\alpha,\alpha^{F},\alpha^{\bar{F}}$ of $X,\bar{X}^{F},\bar{X}^{\bar{F}}$ respectively meet

$\displaystyle\alpha^{F}=\alpha\cup\{\beta\in C_{V}(\alpha)|f(\beta)\in\alpha,f% \in F\},$ (1) $\displaystyle\alpha^{\bar{F}}=\alpha-\{\alpha_{i}\in\alpha|f(\alpha)\bar{\in}% \alpha,\bar{f}\in\bar{F}\},$ (2)

in which $U$ is a finite element universe and $V$ is a finite attribute universe, $f$ , $\bar{f}$ are migration functions, $F$ , $\bar{F}$ are migration function families, and $f\in F,\bar{f}\in\bar{F},C_{U}(X)=\{y|y\in U,y\bar{\in}X\},C_{V}(\alpha)=\{% \beta|\beta\in V,\beta\bar{\in}\alpha\}$ .

By the above definitions, it is obvious that if $F=\bar{F}=\emptyset$ , then

$\displaystyle(\bar{X}^{F},\bar{X}^{\bar{F}})=X.$

Further analyzing the concept of IPS, we can know if some additional attributes are continually transferred into the attribute setof $X$ , meanwhile some attributes in $\alpha$ are continually removed out of $\alpha$ , $X$ can generate a family of IPS $\{(\bar{X}_{i}^{F},\bar{X}_{j}^{\bar{F}})|i\in I,j\in J\}$ in which $I$ , $J$ are index sets, which shows the dynamic change process of the finite Cantor set $X$ and characteristics depending on its attributes perfectly.

Making use of the above statement, we propose the concept of inverse packet fuzzy sets by introducing the dynamic characteristics of IPS into a fuzzy set $A$ to expand set theory in Section 3.

More detail contents of theory and applications about FS and IPS can be found in literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

3. Inverse packet fuzzy sets

In the paper, we only consider the case of nonempty finite universes and nonempty finite attribute sets for convenience.

Definition 1: Let $A$ be a fuzzy set on the finite universe $U$ , $\alpha\in V$ be the attribute set of $A$ and $\alpha\neq\emptyset$ , call $\bar{A}^{F}$ an internal inverse packet fuzzy set generated by $A$ , written shortly as an internal IPFS, and call $\bar{A}^{F}(x)$ the membership function of $\bar{A}^{F}$ , and

$\displaystyle\bar{A}^{F}:U\to[0,1]$

(3) $\displaystyle x\mapsto\bar{A}^{F}(x)$

if the attribute set $\alpha^{F}$ of $\bar{A}^{F}$ satisfies Eq. (1), in which $\bar{A}^{F}(x)$ , $A(x)$ are the membership functions of $\bar{A}^{F}$ and $A$ differently.

Definition 2: Let $A$ be a fuzzy set on the finite universe $U$ , call $\bar{A}^{\bar{F}}$ an outer inverse packet fuzzy set generated by $A$ , written briefly as an outer IPFS, and call $\bar{A}^{\bar{F}}(x)$ the membership function of $\bar{A}^{\bar{F}}$ , and

$\displaystyle\bar{A}^{\bar{F}}:U\to[0,1]$ (4) $\displaystyle x\mapsto\bar{A}^{\bar{F}}(x)$

if the attribute set $\alpha^{\bar{F}}$ of $A^{\bar{F}}$ satisfies Eq. (2), in which $\bar{A}^{\bar{F}}(x)$ , $A(x)$ are the membership functions of $\bar{A}^{\bar{F}}$ , $A$ respectively.

Definition 3: The couple of fuzzy sets composed of $\bar{A}^{F}$ and $\bar{A}^{\bar{F}}$ is called an inverse packet fuzzy sets generated by $A$ , written briefly as an IPFS, and

$\displaystyle(\bar{A}^{F},\bar{A}^{\bar{F}}).$ (5)

The set $\Phi(A)$ is called IPFS family generated by the fuzzy set $A$ , and

$\displaystyle\Phi(A)=\{(\bar{A}_{i}^{F},\bar{A}_{j}^{\bar{F}})|i\in I,j\in J\}.$ (6)

Combining the theories of FS and IPS with definitions 1–3, the following propositions 1–3 are easily obtained.

Proposition 1: If $(\bar{A}^{F},\bar{A}^{\bar{F}})\in\Phi(A)$ , and $\bar{A}^{F}(x),A(x),\bar{A}^{\bar{F}}(x)$ are the membership functions of $\bar{A}^{F},A,\bar{A}^{\bar{F}}$ respectively, then

$\displaystyle\bar{A}^{\bar{F}}(x)\leqslant A(x)\leqslant\bar{A}^{F}(x).$ (7)

Proposition 2: If $(\bar{A}^{F},\bar{A}^{\bar{F}})\in\Phi(A)$ , then

$\displaystyle\bar{A}^{\bar{F}}\subseteq A\subseteq\bar{A}^{F}.$ (8)

The relation among $\bar{A}^{F},A,\bar{A}^{\bar{F}}$ are showed in Fig. 1.

Figure 1.
The relation among $\bar{A}^{F},A,\bar{A}^{\bar{F}}$ .

Proposition 3: If $\bar{A}^{\bar{F}}(x)=A(x)=\bar{A}^{F}(x)$ for $\forall x\in U$ , then $(\bar{A}^{F},\bar{A}^{\bar{F}})$ is reverted to the fuzzy set $A$ . Especially if $\bar{A}^{\bar{F}}(x)=A(x)=\bar{A}^{F}(x)=1$ or $\bar{A}^{\bar{F}}(x)=A(x)=\bar{A}^{F}(x)=0$ for $\forall x\in U$ , then $(\bar{A}^{F},\bar{A}^{\bar{F}})$ is reverted to the Cantor set $X=\ker(A)$ .

Proposition 4: If $\bar{A}^{\bar{F}},A,\bar{A}^{F}$ are Cantor sets, then $(\bar{A}^{F},\bar{A}^{\bar{F}})$ is an IPS. It is also said that IPS is a special case of IPFS, on the other hand, IPFS is the extension of IPS.

Supposed that $\bar{A}_{i}^{F}$ is an internal IPFS generated by $A$ , and $\bar{A}_{i+1}^{F}$ is an internal IPFS generated by $\bar{A}_{i}^{F}(i=1,2,3,\ldots)$ , then an internal IPFS chain can be gotten as follows

$\displaystyle\bar{A}_{i}^{F}\subseteq\bar{A}_{i+1}^{F}.$ (9)

Similarly supposed that $\bar{A}_{j}^{\bar{F}}$ is an outer IPFS generated by $A$ , and $\bar{A}_{j+1}^{\bar{F}}$ is an outer IPFS generated by $\bar{A}_{j}^{\bar{F}}(j=1,2,3,\ldots)$ , then an internal IPFS of $A$ chain can be done as follows

$\displaystyle\bar{A}_{j+1}^{\bar{F}}\subseteq\bar{A}_{j}^{\bar{F}}.$ (10)

Supposed that $(\bar{A}_{i}^{F},\bar{A}_{j}^{\bar{F}}),(\bar{A}_{i+1}^{F},\bar{A}_{j+1}^{\bar% {F}})\in\Phi(A)$ , and Eqs (9) and (10) are true, then

$\displaystyle(\bar{A}_{i}^{F},\bar{A}_{j}^{\bar{F}})\begin{array}[]{l}% \subseteq\\ \supset\\ \end{array}(\bar{A}_{i+1}^{F},\bar{A}_{j+1}^{\bar{F}})$ (11)

Note that $\bar{A}_{i+1}^{F}$ , $\bar{A}_{j+1}^{\bar{F}}$ , $(\bar{A}_{i+1}^{F},\bar{A}_{j+1}^{\bar{F}})$ are also internal IPFS, outer IPFS, IPFS generated by $A$ in order in Eqs (9)–(11). Thus $A$ can generate some IPFS nested sets which forms $\Psi(A)$ , and

$\displaystyle\Psi(A)=\{(\bar{A}_{i}^{F},\bar{A}_{i}^{\bar{F}})_{k}\begin{array% }[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{i}^{F},\bar{A}_{i}^{\bar{F}})_{k}{|}k\in K\}.$ (12)

where $K$ is an index set.

It is obvious that nested sets Eqs (9)–(12) show the dynamic change process of $A$ .

Theorem 1: (The first relation theorem of IPFS and FS) If $\bar{F}=F=\emptyset$ , then IPFS is restored to $A$ , and

$\displaystyle(\bar{A}^{F},\bar{A}^{\bar{F}})_{F=F=\emptyset}=A.$ (13)

Proof: Since $F=\emptyset$ , then $\{\alpha^{\prime}|\beta\in V,\beta\bar{\in}\alpha,f(\beta)=\alpha^{\prime}\in% \alpha,f\in F\}=\emptyset$ . According to definiton 1, $\alpha^{F}=\alpha\cup\{\alpha^{\prime}|\beta\in V,\beta\bar{\in}\alpha,f(\beta% )=\alpha^{\prime}\in\alpha,f\in F\}=\alpha$ , it is also said that the internal IPFS $\bar{A}^{F}$ is restored to $A$ . In the same way, we can prove if $\bar{F}=\emptyset$ , then the internal IPFS $\bar{A}^{F}$ is restore to $A$ . Thus the Eq. (13) is satisfied. $(\bar{A}^{F},\bar{A}^{\bar{F}})$ is reverted to the fuzzy set $A$ .

Theorem 2: (The second relation theorem of IPFS and FS) If $\bar{F}=F=\emptyset$ , then IPFS family is restored to $A$ , and

$\displaystyle(\bar{A}_{i}^{F},\bar{A}_{j}^{\bar{F}})_{F=\bar{F}=\emptyset}=A.$ (14)

Theorem 3: (The relation theorem of FS and Cantor set) Supposed that is the attribute set of $A$ and $X$ , then

$\displaystyle X=\ker A,$ (15)

where $A$ is a fuzzy set and $X$ is a Cantor set.

The proof of theorem 2 is similar to theorem 1, and the proof of theorem 3 is easy to obtain by the concepts of fuzzy set and Cantor set [1], so omitted.

Theorem 3 shows the relationship between a fuzzy set and its corresponding Cantor set which has the same attribute set.

Theorem 4: (The generation theorem of IPFS) Supposed that is the attribute set of $A$ , then $(\bar{A}^{F},\bar{A}^{\bar{F}})$ is generated by $A$ , if there exist attribute set pair $(\Delta\alpha,\nabla\alpha)$ that satisfies

$\displaystyle(\Delta\alpha,\nabla\alpha)\neq\emptyset,$ (16)

where $\Delta\alpha=\{\beta\in V|\beta\bar{\in}\alpha,f(\beta)\in\alpha\},\nabla% \alpha=\{\alpha_{i}\in\alpha|\bar{f}(\alpha_{i})\bar{\in}\alpha\}$ .

Proof: I. Since $(\Delta\alpha,\nabla\alpha)\neq\emptyset$ , then $\Delta\alpha=\{\beta\in V|\beta\bar{\in}\alpha,f(\beta)\in\alpha\}\neq\emptyset$ , and $\nabla\alpha=\{\alpha\in A|\linebreak\bar{f}(\alpha)\bar{\in}\alpha\}\neq\emptyset$ , i.e. $\alpha^{F}=\alpha\cup\Delta\alpha\neq\alpha,\alpha^{\bar{F}}=\alpha-\nabla% \alpha\neq\alpha$ . Thus there exist the pair of fuzzy sets $(\bar{A}^{F},\bar{A}^{\bar{F}})$ which satisfy $\bar{A}^{F}-A\neq\emptyset$ according to Eqs (1)–(5), the theorem 4 is proved.

Corollary 1 follows easily from theorem 3 and theorem 4.

Corollary 1: If $(\alpha^{F},\alpha^{\bar{F}})$ is the attribute sets of the IPFS $(\bar{A}^{F},\bar{A}^{\bar{F}})$ and the IPS $(X^{F},X^{\bar{F}})$ , then $\bar{A}^{F},\linebreak\bar{A}^{\bar{F}},X^{F},X^{\bar{F}}$ satisfy

$\displaystyle(X^{F},X^{\bar{F}})=(\ker(\bar{A}^{F}),\ker(\bar{A}^{\bar{F}})).$ (17)
4. Scalar multiplication and $\lambda$ -cut sets of IPFS

By the concepts of $\lambda$ -cut set and $\lambda$ -strong cut set of fuzzy set $A^{[1]}$ , we present the following concepts about IPFS.

Definition 4: Let $\lambda\in[0,1]$ be a real number and $(\bar{A}^{F},\bar{A}^{\bar{F}})$ be an IPFS generated by $A$ , then $\lambda(\bar{A}^{F},\bar{A}^{\bar{F}})$ is called the scalar multiplication of $\lambda$ and $(\bar{A}^{F},\bar{A}^{\bar{F}})$ , and

$\displaystyle\lambda(\bar{A}^{F},\bar{A}^{\bar{F}})=(\lambda\bar{A}^{F},% \lambda\bar{A}^{\bar{F}}),$ (18)

where $\lambda\bar{A}^{F}(x)=\lambda\wedge\bar{A}^{F}(x),\lambda\bar{A}^{\bar{F}}(x)=% \lambda\wedge\bar{A}^{\bar{F}}(x)$ .

Definition 5: Let $\lambda\in[0,1]$ be a real number and $\bar{A}^{F}$ be an internal IPFS generated by $A$ , then $\bar{A}^{F,\lambda},\bar{A}^{F,\dot{\lambda}}$ are respectively called $\lambda-$ cut set and $\lambda-$ strong cut set of $\bar{A}^{F}$ , and

$\displaystyle\bar{A}^{F,\lambda}=\{x|A^{F}(x)\geqslant\lambda\},$ (19) $\displaystyle\bar{A}^{F,\dot{\lambda}}=\{x|A^{F}(x)>\lambda\};$ (20)

let $\bar{A}^{\bar{F}}$ be an outer IPFS generated by $A$ , $\bar{A}^{\bar{F},\lambda},\bar{A}^{\bar{F},\dot{\lambda}}$ are respectively called $\lambda-$ cut set and $\lambda-$ strong cut set of $\bar{A}^{\bar{F}}$ , and

$\displaystyle\bar{A}^{\bar{F},\lambda}=\{x|A^{\bar{F}}(x)\geqslant\lambda\},$ (21) $\displaystyle\bar{A}^{F,\dot{\lambda}}=\{x|A^{\bar{F}}(x)>\lambda\};$ (22)

let $(\bar{A}^{F},\bar{A}^{\bar{F}})$ be an IPFS generated by $A$ , then $(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda}),(\bar{A}^{F,\dot{\lambda}},% \bar{A}^{\bar{F},\dot{\lambda}})$ are respectivelcalled $\lambda-$ cut set and $\lambda-$ strong cut set of $(\bar{A}^{F},\bar{A}^{\bar{F}})$ , where $\lambda$ is a real number, and $\lambda\in[0,1]$ .

Definition 4 indicates that Eqs (19)–(22) are Cantor sets, furthermore, $\lambda-$ cut set or $\lambda-$ strong cut set of an internal IPFS can be thought as internal IPSs of $A^{\lambda}$ , $\lambda-$ cut set and $\lambda-$ strong cut set of an outer IPFS are outer IPSs of $A^{\lambda}$ for any real number $\lambda\in[0,1]$ . Thus $(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda}),(\bar{A}^{\bar{F},\dot{\lambda% }})$ are IPSs generated by $A^{\lambda}$ . And they satisfy

$\displaystyle(\bar{A}^{F,\dot{\lambda}},\bar{A}^{\bar{F},\dot{\lambda}})% \subseteq(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda}),$ (23)

where $A^{\lambda}$ is $\lambda-$ cut set of $A$ , Eq. (23) means $\bar{A}^{F,\dot{\lambda}}\subseteq\bar{A}^{F,\lambda}$ and $\bar{A}^{\bar{F},\dot{\lambda}}\subseteq\bar{A}^{\bar{F},\lambda}$ .

According to definition 4 and definition 5, we can obtain the following theorems.

Theorem 5: (The first relation theorem of scalar multiplications) Let $\lambda_{1}$ , $\lambda_{2}\in[0,1]$ be real numbers, if $\lambda_{1}<\lambda_{2}$ , then

$\displaystyle\lambda_{1}(\bar{A}^{F},\bar{A}^{\bar{F}})\subseteq\lambda_{2}(% \bar{A}^{F},\bar{A}^{\bar{F}}).$ (24)

where $(\bar{A}^{F},\bar{A}^{\bar{F}})\in\Phi(A)$ .

Theorem 6: (The second relation theorem of IPFS) Let $(\bar{A}_{1}^{F},\bar{A}_{1}^{\bar{F}}),(\bar{A}_{2}^{F},\bar{A}_{2}^{\bar{F}}% )\in\Phi(A)$ , and $\lambda\in[0,1]$ be a a real number, If $(\bar{A}_{1}^{F},\bar{A}_{1}^{\bar{F}})\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{2}^{F},\bar{A}_{2}^{\bar{F}})$ , then

$\displaystyle\lambda(\bar{A}_{1}^{F},\bar{A}_{1}^{\bar{F}})\begin{array}[]{l}% \subseteq\\ \supset\\ \end{array}\lambda(\bar{A}_{2}^{F},\bar{A}_{2}^{\bar{F}}).$ (25)

Theorem 7: (The first theorem of cut sets) Let $\lambda_{1},\lambda_{2}\in[0,1]$ be real numbers, if $\lambda_{1}<\lambda_{2}$ , then

$\displaystyle(\bar{A}^{F,\lambda_{2}},\bar{A}^{\bar{F},\lambda_{2}})\subseteq(% \bar{A}^{F,\lambda_{1}},\bar{A}^{\bar{F},\lambda_{1}}),(\bar{A}^{F,\dot{% \lambda}_{2}},\bar{A}^{\bar{F},\dot{\lambda}_{2}})\subseteq(\bar{A}^{F,\dot{% \lambda}_{1}},\bar{A}^{\bar{F},\dot{\lambda}_{1}}).$ (26)

where $(\bar{A}^{F,\lambda_{2}},\bar{A}^{\bar{F},\lambda_{2}}),(\bar{A}^{F,\lambda_{1% }},\bar{A}^{\bar{F},\lambda_{1}}),(\bar{A}^{F,\dot{\lambda}_{2}},\bar{A}^{\bar% {F},\dot{\lambda}_{2}}),(\bar{A}^{F,\dot{\lambda}_{1}},\bar{A}^{\bar{F},\dot{% \lambda}_{1}})$ are cut sets and strong cut sets of $(\bar{A}^{F},\bar{A}^{\bar{F}})\in\Phi(A)$ respectively, and Eq. (26) means

$\displaystyle\bar{A}^{F,\lambda_{2}}\subseteq\bar{A}^{F,\lambda_{1}},\bar{A}^{% \bar{F},\lambda_{2}}\subseteq\bar{A}^{\bar{F},\lambda_{1}},A^{F,\dot{\lambda}_% {2}}\subseteq\bar{A}^{F,\dot{\lambda}_{1}},\bar{A}^{\bar{F},\dot{\lambda}_{2}}% \subseteq\bar{A}^{\bar{F},\dot{\lambda}_{1}}.$

Based on the decomposition theorems of fuzzy set [1] and Eqs (18)–(26), we can easily obtain the decomposition theorems of IPFS.

Theorem 8: (The first decomposition theorem of IPFS) Let $\lambda\in[0,1]$ be a real number, $(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda}),(\bar{A}^{F,\dot{\lambda}},% \bar{A}^{\bar{F},\dot{\lambda}})$ be $\lambda-$ cut sets and $\lambda-$ strong cut sets of $(\bar{A}^{F},\bar{A}^{\bar{F}})$ , then

$\displaystyle(\bar{A}^{F},\bar{A}^{\bar{F}})=\mathop{\cup}\limits_{\lambda\in[% 0,1]}\lambda(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda}),(\bar{A}^{F},\bar{% A}^{\bar{F}})=\mathop{\cup}\limits_{\lambda\in[0,1]}\lambda(\bar{A}^{F,\dot{% \lambda}},\bar{A}^{\bar{F},\dot{\lambda}}).$ (27)

Corollary 2: Let $\lambda\in[0,1]$ be a real number, and $(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda})$ be $\lambda-$ cut sets of $(\bar{A}^{F},\bar{A}^{\bar{F}})$ , then

$\displaystyle(\bar{A}^{F}(x),\bar{A}^{\bar{F}}(x))=(\mathop{\vee}\limits_{x\in% \bar{A}^{F,\lambda}}\lambda,\mathop{\vee}\limits_{x\in\bar{A}^{\bar{F},\lambda% }}\lambda),$ (28)

for $x\in U$ , where Eq. (28) expresses $\bar{A}^{F}(x)=\mathop{\vee}\limits_{x\in\bar{A}^{F,\lambda}}\lambda,\bar{A}^{% \bar{F}}(x)=\mathop{\vee}\limits_{x\in\bar{A}^{\bar{F},\lambda}}\lambda$ .

Corollary 3: If $\lambda\in[0,1]$ is a real number, $(\bar{A}^{F,\dot{\lambda}},\bar{A}^{\bar{F},\dot{\lambda}})$ is $\lambda-$ strong cut sets of $(\bar{A}^{F},\bar{A}^{\bar{F}})$ , then

$\displaystyle(\bar{A}^{F}(x),\bar{A}^{\bar{F}}(x))=(\mathop{\vee}\limits_{x\in% \bar{A}^{F,\dot{\lambda}}}\lambda,\mathop{\vee}\limits_{x\in\bar{A}^{\bar{F},% \dot{\lambda}}}\lambda),$ (29)

for $x\in U$ , where Eq. (28) expresses $\bar{A}^{F}(x)=\mathop{\vee}\limits_{x\in\bar{A}^{F,\dot{\lambda}}}\lambda,% \bar{A}^{\bar{F}}(x)=\mathop{\vee}\limits_{x\in\bar{A}^{\bar{F},\dot{\lambda}}}\lambda$ .

Theorem 9: (The second decomposition theorem of IPFS) Let $(\bar{A}^{F},\bar{A}^{\bar{F}})\in\Phi(A)$ , and $\lambda$ be a real number, if there exists a map

$\displaystyle H:[0,1]\to\Phi(A)$ (30) $\displaystyle\lambda|\to H(\lambda),$

that meets $(\bar{A}^{F,\dot{\lambda}},\bar{A}^{\bar{F},\dot{\lambda}})\subseteq H(\lambda% )\subseteq(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda})$ for $\lambda\in[0,1]$ , then

$\displaystyle\text{I. }(\bar{A}^{F},\bar{A}^{\bar{F}})=\mathop{\cup}\limits_{% \lambda\in[0,1]}\lambda H(\lambda)$ (31) $\displaystyle\text{II. }\lambda_{1}<\lambda_{2}\Rightarrow H(\lambda_{2})% \subseteq H(\lambda_{1}).$

Proof: I. Due to $(\bar{A}^{F,\dot{\lambda}},\bar{A}^{\bar{F},\dot{\lambda}})\subseteq H(\lambda% )\subseteq(\bar{A}^{F,\lambda},\bar{A}^{\bar{F},\lambda})$ , we have

$\displaystyle\lambda(\bar{A}^{F,\dot{\lambda}},\bar{A}^{\bar{F},\dot{\lambda}}% )\subseteq\lambda H(\lambda)\subseteq\lambda(\bar{A}^{F,\lambda},\bar{A}^{\bar% {F},\lambda})$

according to Eqs (24) and (27), we can obtain

$\displaystyle(\bar{A}^{F},\bar{A}^{\bar{F}})=\mathop{\cup}\limits_{\lambda\in[% 0,1]}\lambda(\bar{A}^{F,\dot{\lambda}},\bar{A}^{\bar{F},\dot{\lambda}})% \subseteq\mathop{\cup}\limits_{\lambda\in[0,1]}\lambda H(\lambda)\subseteq% \mathop{\cup}\limits_{\lambda\in[0,1]}\lambda(\bar{A}^{F,\lambda},\bar{A}^{% \bar{F},\lambda})=(\bar{A}^{F},\bar{A}^{\bar{F}}),$

thus

$\displaystyle(\bar{A}^{F},\bar{A}^{\bar{F}})=\mathop{\cup}\limits_{\lambda\in[% 0,1]}\lambda H(\lambda).$

II. For $x\in U$ , if $x\in\bar{A}^{F,\lambda_{2}}$ then $\bar{A}^{F}(x)\geqslant\lambda_{2}>\lambda_{1}$ . So $x\in\bar{A}^{F,\dot{\lambda}_{1}}$ , namely $\bar{A}^{F,\dot{\lambda}_{1}}\subseteq\bar{A}^{F,\lambda_{2}}$ . In the same way, we have $\bar{A}^{\bar{F},\dot{\lambda}_{1}}\subseteq\bar{A}^{\bar{F},\lambda_{2}}$ , thus $(\bar{A}^{F,\lambda_{2}},\bar{A}^{\bar{F},\lambda_{2}})\subseteq(\bar{A}^{F,% \dot{\lambda}_{1}},\bar{A}^{\bar{F},\dot{\lambda}_{1}})$ , so

$\displaystyle H(\lambda_{2})\subseteq(\bar{A}^{F,\lambda_{2}},\bar{A}^{\bar{F}% ,\lambda_{2}})\subseteq(\bar{A}^{F,\dot{\lambda}_{1}},\bar{A}^{\bar{F},\dot{% \lambda}_{1}})\subseteq H(\lambda_{1}).$

Decomposition theorems of IPFS propose a theoretical foundation and a practical method that expresses an IPFS with the corresponding IPS.

5. The attribute-dependent characteristics of IPFS

In the part, we discuss deeply the above knowledge and obtain the attribute-dependent relationship of IPFS.

Theorem 9: (The attribute-dependent theorem of internal IPFSs) Let $\alpha_{1}^{F},\alpha_{2}^{F},\ldots,\alpha_{n}^{F}$ be the attribute sets of internal IPFS $\bar{A}_{1}^{F},\bar{A}_{2}^{F},\ldots,\bar{A}_{n}^{F}$ generated by fuzzy set $A$ in order, if $\alpha_{1}^{F}\subseteq\alpha_{2}^{F}\subseteq\ldots\subseteq\alpha_{n}^{F}$ , then

$\displaystyle\text{I. }\bar{A}_{1}^{F}\subseteq\bar{A}_{2}^{F}\subseteq\ldots% \subseteq\bar{A}_{n}^{F},$ $\displaystyle\text{II. }\lambda\bar{A}_{1}^{F}\subseteq\lambda\bar{A}_{2}^{F}% \subseteq\ldots\subseteq\lambda\bar{A}_{n}^{F},\text{for }\forall\lambda\in[0,% 1],$ $\displaystyle\text{III. }\bar{A}_{1}^{F,\lambda}\subseteq\bar{A}_{2}^{F,% \lambda}\subseteq\ldots\subseteq\bar{A}_{n}^{F,\lambda,},\bar{A}_{1}^{F,\dot{% \lambda}}\subseteq\bar{A}_{2}^{F,\dot{\lambda}}\subseteq\ldots\subseteq\bar{A}% _{n}^{F,\dot{\lambda},}\text{for }\forall\lambda\in[0,1].$

Proof: Since $\alpha_{i}^{F}\subseteq\alpha_{i+1}^{F},i=1,2,\ldots,n-1$ , we suppose that $\Delta\alpha_{i}=\alpha_{i+1}^{F}-\alpha_{i}^{F}$ , namely, $\alpha_{i+1}^{F}=\alpha_{i}^{F}\cup\Delta\alpha_{i}$ , then the attribute set $\alpha_{i+1}^{F}$ of $\bar{A}_{i+1}^{F}$ satisfy the condition of definition 1, so $\bar{A}_{i+1}^{F}$ can be regarded as an internal IPFS generated by $\bar{A}_{i}^{F}$ , so $\bar{A}_{i}^{F}(x)\leqslant\bar{A}_{i+1}^{F}(x)$ for $x\in U$ , thus $\bar{A}_{i}^{F}\subseteq\bar{A}_{i+1}^{F},i=1,2,\ldots,n-1.$

II and III can be easily obtained by definition 4, definition 5 and I.

Theorem 10: (The attribute-dependent theorem of outer IPFSs) Assume that $\alpha_{1}^{\bar{F}},\alpha_{2}^{\bar{F}},\ldots,\alpha_{n}^{\bar{F}}$ are the attribute sets of outer IPFS $\bar{A}_{1}^{\bar{F}},\bar{A}_{2}^{\bar{F}},\ldots,\bar{A}_{n}^{\bar{F}}$ generated by fuzzy $A$ in order, if $\alpha_{n}^{\bar{F}}\subseteq\alpha_{n-1}^{\bar{F}}\subseteq\ldots\subseteq% \alpha_{2}^{\bar{F}}\subseteq\alpha_{1}^{\bar{F}}$ , then

$\displaystyle\text{I. }\bar{A}_{n}^{\bar{F}}\subseteq\bar{A}_{n-1}^{\bar{F}}% \subseteq\ldots\subseteq\bar{A}_{2}^{\bar{F}}\subseteq\bar{A}_{1}^{\bar{F}},$ $\displaystyle\text{II. }\lambda\bar{A}_{n}^{\bar{F}}\subseteq\lambda\bar{A}_{n% -1}^{\bar{F}}\subseteq\ldots\subseteq\lambda\bar{A}_{2}^{\bar{F}}\subseteq% \lambda\bar{A}_{1}^{\bar{F}},\text{for }\forall\lambda\in[0,1]$ $\displaystyle\text{III. }\bar{A}_{n}^{\bar{F},\lambda}\subseteq\bar{A}_{n-1}^{% \bar{F},\lambda}\subseteq\ldots\subseteq\bar{A}_{2}^{\bar{F},\lambda}\subseteq% \bar{A}_{1}^{\bar{F},\lambda},\bar{A}_{n}^{\bar{F},\dot{\lambda}}\subseteq\bar% {A}_{n-1}^{\bar{F},\dot{\lambda}}\subseteq\ldots\subseteq\bar{A}_{2}^{\bar{F},% \dot{\lambda}}\subseteq\bar{A}_{1}^{\bar{F},\dot{\lambda}}\text{ for }\forall% \lambda\in[0,1]$

Theorem 11: (The attribute-dependent theorem of IPFSs) Assume that $(\alpha_{1}^{F},\alpha_{1}^{\bar{F}}),(\alpha_{2}^{F},\alpha_{2}^{\bar{F}}),% \ldots,(\alpha_{n}^{F},\linebreak\alpha_{n}^{\bar{F}})$ are the attribute sets of $(\bar{A}_{1}^{F},\bar{A}_{1}^{\bar{F}}),(\bar{A}_{2}^{F},\bar{A}_{2}^{\bar{F}}% ),\ldots,(\bar{A}_{n}^{F},\bar{A}_{n}^{\bar{F}})$ in order, if $(\alpha_{1}^{F},\alpha_{1}^{\bar{F}})\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}\alpha_{2}^{\bar{F}})\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}\ldots\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}\alpha_{n}^{\bar{F}})$ , then

$\displaystyle\text{I. }(\bar{A}_{1}^{F},\bar{A}_{1}^{\bar{F}})\begin{array}[]{% l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{2}^{F},\bar{A}_{2}^{\bar{F}})\begin{array}[]{l}\subseteq% \\ \supset\\ \end{array}\ldots\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{n}^{F},\bar{A}_{n}^{\bar{F}}),$ $\displaystyle\text{II. }\lambda(\bar{A}_{1}^{F},\bar{A}_{1}^{\bar{F}})\begin{% array}[]{l}\subseteq\\ \supset\\ \end{array}\lambda(\bar{A}_{2}^{F},\bar{A}_{2}^{\bar{F}})\begin{array}[]{l}% \subseteq\\ \supset\\ \end{array}\ldots\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}\lambda(\bar{A}_{n}^{F},\bar{A}_{n}^{\bar{F}}),\text{for }\forall% \lambda\in[0,1],$ $\displaystyle\text{III. }(\bar{A}_{1}^{F,\lambda},\bar{A}_{1}^{\bar{F},\lambda% })\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{2}^{F,\lambda},\bar{A}_{2}^{\bar{F},\lambda})\begin{array% }[]{l}\subseteq\\ \supset\\ \end{array}\ldots\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{n}^{F,\lambda},\bar{A}_{n}^{\bar{F},\lambda}),\text{for }% \forall\lambda\in[0,1].$ $\displaystyle\text{IV. }(\bar{A}_{1}^{F,\dot{\lambda}},\bar{A}_{1}^{\bar{F},% \dot{\lambda}})\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{2}^{F,\dot{\lambda}},\bar{A}_{2}^{\bar{F},\dot{\lambda}})% \begin{array}[]{l}\subseteq\\ \supset\\ \end{array}\ldots\begin{array}[]{l}\subseteq\\ \supset\\ \end{array}(\bar{A}_{n}^{F,\dot{\lambda}},\bar{A}_{n}^{\bar{F},\dot{\lambda}})% ,\text{for }\forall\lambda\in[0,1].$

The proofs of theorem 10 and theorem 11 are similar to theorem 9, so omitted.

Theorem 12: (The identification relation of internal IPFSs) Assumed that the attribute sets of internal IPFS $\bar{A}_{i}^{F},\bar{A}_{i+1}^{F},(i=1,2,\ldots,n)$ are $\alpha_{i}^{F},\alpha_{i+1}^{F}$ respectively, if $\text{IDE}(\bar{A}_{i}^{F},\bar{A}_{i+1}^{F})$ , then

$\displaystyle\text{IDE}(\alpha_{i}^{F},\alpha_{i+1}^{F}),$

where IDE $=$ identification.

Proof: Assumed that $\text{UNI}(\alpha_{i}^{F},\alpha_{i+1}^{F})$ , then $\bar{A}_{i}^{F},\bar{A}_{i+1}^{F},(i=1,2,\ldots,n)$ have the same attribute set, according to the relationship between set and its attribute set, we can obtain that $\bar{A}_{i}^{F},\bar{A}_{i+1}^{F},(i=1,2,\ldots,n)$ can not be identified, namely $\text{UNI}(\bar{A}_{i}^{F},\bar{A}_{i+1}^{F})$ that contradicts $\text{IDE}(\bar{A}_{i}^{F},\bar{A}_{i+1}^{F})$ . Thus theorem 12 is proved.

Theorem 13: (The identification relation between outer IPFSs) Assumed that the attribute sets of outer IPFS $\bar{A}_{i}^{\bar{F}},\bar{A}_{i+1}^{\bar{F}},(i=1,2,\ldots,n)$ are $\alpha_{i}^{\bar{F}},\alpha_{i+1}^{\bar{F}}$ respectively, if $\text{IDE}(\bar{A}_{i}^{\bar{F}},\bar{A}_{i+1}^{\bar{F}})$ , then

$\displaystyle\text{IDE}(\alpha_{i}^{\bar{F}},\alpha_{i+1}^{\bar{F}}).$

Theorem 14: (The identification relation of IPFSs) Assumed that the attribute sets of IPFS $(\bar{A}_{i}^{F},\linebreak\bar{A}_{i}^{\bar{F}}),(\bar{A}_{i+1}^{F},\bar{A}_{% i+1}^{\bar{F}})(i=1,2,\ldots,n)$ are $(\alpha_{i}^{F},\alpha_{i}^{\bar{F}}),(\alpha_{i+1}^{F}\alpha_{i+1}^{\bar{F}})$ respectively, if $\text{IDE}((\bar{A}_{i}^{F},\bar{A}_{i}^{\bar{F}}),(\bar{A}_{i+1}^{F},% \linebreak\bar{A}_{i+1}^{\bar{F}}))$ , then

$\displaystyle\text{IDE}((\alpha_{i}^{F},\alpha_{i}^{\bar{F}}),(\alpha_{i+1}^{F% }\alpha_{i+1}^{\bar{F}})).$

The proofs of theorem 13 and theorem 14 are similar to theorem 12, so omitted.

6. Discussion

Using Introducing dynamic characteristics into a fuzzy set $A$ and improving it, the model of inverse packet fuzzy sets is proposed. It has vague characteristic and dynamic characteristic. Due to all of its $\lambda-$ cut sets and $\lambda-$ strong cut sets can be regarded as inverse packet sets generated by $\ker(A)$ . And any inverse packet fuzzy sets can be expressed as the union of all its $\lambda-$ cut sets or $\lambda-$ strong cut sets for $\forall\lambda\in[0,1]$ . So, we can turn fuzzy problems into deterministic ones. It is also said that we can use inverse packet sets theory to discuss inverse packet fuzzy sets. On the other hand, inverse packet fuzzy sets has the attribute-dependent characteristics. Therefore, its attribute sets can be used to study the dynamic properties of inverse packet fuzzy sets. The presented set model will enrich set theory and, to a certain extent, facilitate practical applications.

Footnotes

Acknowledgments

The paper is supported by the Ministry of Education Humanities and Social Sciences Research Youth Foundation (Grant No: 19YJC910011), the Project of Shandong Province Higher Educational Science and Technology Pro-gram (Grant No: J18KB099).

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