Journal of Applied Mathematics

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Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 479783 | 18 pages | https://doi.org/10.1155/2012/479783

On Generalised Interval-Valued Fuzzy Soft Sets

Xiaoqiang Zhou,1,2 Qingguo Li ,1 and Lankun Guo3

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

2College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

3College of Computer and Communication, Hunan University, Changsha 410082, China

Academic Editor: Jong Hae Kim
Received10 Aug 2011
Revised18 Nov 2011
Accepted22 Nov 2011
Published06 Feb 2012

Abstract

Soft set theory, initiated by Molodtsov, can be used as a new mathematical tool for dealing with imprecise, vague, and uncertain problems. In this paper, the concepts of two types of generalised interval-valued fuzzy soft set are proposed and their basic properties are studied. The lattice structures of generalised interval-valued fuzzy soft set are also discussed. Furthermore, an application of the new approach in decision making based on generalised interval-valued fuzzy soft set is developed.

1. Introduction

Most of our real-life problems in social science, economics, medical science, engineering, environmental science, and many other fields have various uncertainties. To deal with these uncertainties, many kinds of theories have been proposed such as theory of probability [1], fuzzy set theory [2], rough set theory [3], intuitionistic fuzzy set theory [4], and interval mathematics [5–7]. Unfortunately, each of these theories has its inherent difficulties, which was pointed out by Molodtsov in [8]. To overcome these difficulties, Molodtsov [8] proposed the soft set theory, which has become a new completely generic mathematical tool for modeling uncertainties.

Recently, the soft set theory has been widely focused in theory and application after Molodtsov’s work. Maji and Biswas [9] first introduced the concepts of soft subset, soft superset, soft equality, null soft set, and absolute soft set. They also gave some operations on soft set and verified De Morgan’s laws. Ali et al. [10] corrected some errors of former studies and defined some new operations on soft sets. Afterwards, Ali et al. [11] further studied some important properties associated with the new operations and investigated some algebraic structures of soft sets. Sezgin and AtagΓΌn [12] extended the theoretical aspect of operations on soft sets. Soft mappings, soft equality, kernels and closures of soft set relations, and soft set relation mappings were presented in [13–15]. On the other hand, soft set theory has a rich potential for application in many fields. Especially, it has been successfully applied to soft decision making [16–18] and some algebra structures such as groups [19, 20], ordered semigroups [21], rings [22], semirings [23], BCK/BCI-algebras [24–26], d-algebras [27], and BL-algebras [28].

Clearly, all of these works mentioned above are based on the classical soft set theory. To improve the capability of soft set theory in dealing with more complex real-life problems, some fuzzy extensions of soft set theory have been studied by many scholars [29–36]. Particularly, Maji et al. [29] firstly proposed the concept of the fuzzy soft set. Roy and Maji [30] presented an application of fuzzy soft set in decision making. Yang et al. [31] defined the interval-valued fuzzy soft set which is based on a combination of the interval-valued fuzzy set and soft set. Majumdar and Samanta [32] generalized the concept of fuzzy soft sets; that is, a degree of which is attached with the parameterization of fuzzy sets while defining a fuzzy soft set.

However, in many practical applications, specially in fuzzy decision-making problems, the membership functions of objects and parameters are very individual, which are dependent on evaluation of experts in general and thus cannot be lightly confirmed. For example, concerning the fuzzy concept β€œcapability”, there are three experts who give their evaluations to that of someone as 0.6, 0.76, and 0.8, respectively. Clearly, it is more practical and reasonable to evaluate someone’s capability by an interval-valued data [0.6, 0.8] than a certain single value. In this case, therefore, we can make use of interval-valued fuzzy sets which assign to each object or parameter an interval that approximates the β€œreal’’ (but unknown) membership degree. This paper aims to further generalize the concept of generalised fuzzy soft sets by combining the generalised fuzzy soft sets [32] and interval-valued fuzzy sets [7] and obtain a new soft set model named generalised interval-valued fuzzy soft set. It can be viewed as an interval-valued fuzzy extension of the generalised fuzzy soft set theory [32] or a generalization of the interval-valued fuzzy soft set theory [31].

The rest of this paper is organized as follows. In Section 2, the notions of soft set, fuzzy soft set, generalised fuzzy soft set, and interval-valued fuzzy soft set are recalled. In Section 3, the concept and operations of generalised interval-valued fuzzy soft sets are proposed and some of their properties are investigated. Section 4 studies the lattice structures of generalised interval-valued fuzzy soft set. Section 5 introduces the concept of generalised comparison table, which is applied to decision making based on generalised interval-valued fuzzy soft set. Some illustrative examples are also employed to show that the method presented here is not only reasonable but also more efficient in practical applications. Finally, Section 6 presents the conclusion.

2. Preliminary

In this section, we briefly review the concepts of soft sets, fuzzy soft sets, generalised fuzzy soft sets, interval-valued fuzzy soft set, and so on. Further details could be found in [7, 8, 29, 31, 32, 37]. Throughout this paper, unless otherwise stated, π‘ˆ refers to an initial universe, 𝐸 is a set of parameters, 𝑃(π‘ˆ) is the power set of π‘ˆ, and 𝛼,𝛽,𝛾 are fuzzy subset of 𝐴,𝐡,πΆβŠ†πΈ, respectively.

Definition 2.1 (see [8]). A pair (𝐹,𝐴) is called a soft set over π‘ˆ where 𝐹 is a mapping given by πΉβˆΆπ΄β†’π‘ƒ(π‘ˆ).

In other words, a soft set over π‘ˆ is a parameterized family of subsets of the universe π‘ˆ. For πœ€βˆˆπ΄, 𝐹(πœ€) may be considered as the set of πœ€-elements of the soft set (𝐹,𝐴) or as the set of πœ€-approximate elements of the soft set.

Definition 2.2 (see [29]). Let 𝒫(π‘ˆ) denote the set of all fuzzy subsets of π‘ˆ. Then a pair (𝐹,𝐴) is called a fuzzy soft set over π‘ˆ, where 𝐹 is a mapping from 𝐴 to 𝒫(π‘ˆ).

From the definition, it is clear that 𝐹(𝑒) is a fuzzy set on π‘ˆ for any π‘’βˆˆπ΄. The modified definition of fuzzy soft set by Majumdar and Samanta is as follows.

Definition 2.3 (see [32]). Let π‘ˆ be an initial universal set, 𝐸 a set of parameters, and the pair (π‘ˆ,𝐸) a soft universe. Let πΉβˆΆπΈβ†’π’«(π‘ˆ) and πœ‡ be a fuzzy subset of 𝐸; that is, πœ‡βˆΆπΈβ†’[0,1]. Let πΉπœ‡βˆΆπΈβ†’π’«(π‘ˆ)Γ—[0,1] be a function defined as follows: πΉπœ‡(𝑒)=(𝐹(𝑒),πœ‡(𝑒)), where 𝐹(𝑒)βˆˆπ’«(π‘ˆ). Then πΉπœ‡ is called a generalised fuzzy soft set over (π‘ˆ,𝐸).

Definition 2.4 (see [7]). An interval-valued fuzzy set 𝑋 on a universe π‘ˆ is a mapping π‘‹βˆΆπ‘ˆβ†’Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1].

The set of all interval-valued fuzzy sets on π‘ˆ is denoted by β„±(π‘ˆ). Suppose that π‘‹βˆˆβ„±(π‘ˆ), for all β„Žβˆˆπ‘ˆ,πœ‡π‘‹(β„Ž)=[πœ‡βˆ’π‘‹(β„Ž),πœ‡+𝑋(β„Ž)] is called the degree of membership of an element β„Ž to 𝑋. And πœ‡βˆ’π‘‹(β„Ž) and πœ‡+𝑋(β„Ž) are referred to as the lower and upper degrees of membership of β„Ž to 𝑋, where 0β‰€πœ‡βˆ’π‘‹(β„Ž)β‰€πœ‡+𝑋(β„Ž)≀1.

Definition 2.5 (see [7]). Let 𝑋 and π‘Œ be two interval-valued fuzzy sets on universe π‘ˆ. Then the union, intersection, and complement of vague sets are defined as follows: ξƒ―β„Žπ‘‹βˆͺπ‘Œ=ξ€Ίπœ‡βˆ’π‘‹(β„Ž)βˆ¨πœ‡βˆ’π‘Œ(β„Ž),πœ‡+𝑋(β„Ž)βˆ§πœ‡+π‘Œξ€»ξƒ°,ξƒ―β„Ž(β„Ž)βˆ£β„Žβˆˆπ‘ˆπ‘‹βˆ©π‘Œ=ξ€Ίπœ‡βˆ’π‘‹(β„Ž)βˆ§πœ‡βˆ’π‘Œ(β„Ž),πœ‡+𝑋(β„Ž)βˆ¨πœ‡+π‘Œ(ξ€»ξƒ°,π‘‹β„Ž)βˆ£β„Žβˆˆπ‘ˆπ‘=ξƒ―β„Žξ€Ί1βˆ’πœ‡+𝑋(β„Ž),1βˆ’πœ‡βˆ’π‘‹ξ€»ξƒ°.(β„Ž)βˆ£β„Žβˆˆπ‘ˆ(2.1)

Definition 2.6 (see [31]). Let π‘ˆ be an initial universe, let 𝐸 be a set of parameters, and let π΄βŠ†πΈ. β„±(π‘ˆ) denotes the set of all interval-valued fuzzy sets of π‘ˆ. A pair (𝐹,𝐴) is an interval-valued fuzzy soft set over π‘ˆ, where 𝐹 is a mapping given by πΉβˆΆπ΄β†’β„±(π‘ˆ).

An interval-valued fuzzy soft set is a parameterized family of interval-valued fuzzy subsets of π‘ˆ. For each parameter π‘’βˆˆπ΄, 𝐹(𝑒) is actually an interval-valued fuzzy set of π‘ˆ, and it can be written as 𝐹(𝑒)={(β„Ž/πœ‡πΉ(𝑒)(β„Ž))βˆΆβ„Žβˆˆπ‘ˆ}, where πœ‡πΉ(𝑒)(β„Ž) is the interval-valued fuzzy membership degree that object β„Ž holds on parameter 𝑒.

Definition 2.7 (see [37]). A 𝑑-norm is an increasing, associative, and commutative mapping π‘‡βˆΆ[0,1]Γ—[0,1]β†’[0,1] that satisfies the boundary condition: 𝑇(π‘Ž,1)=π‘Ž for all π‘Žβˆˆ[0,1].

The commonly used continuous 𝑑-norms are 𝑇(π‘Ž,𝑏)=min(π‘Ž,𝑏), 𝑇(π‘Ž,𝑏)=max{0,π‘Ž+π‘βˆ’1}, and 𝑇(π‘Ž,𝑏)=π‘Žβ‹…π‘.

Definition 2.8 (see [37]). A 𝑑-conorm is an increasing, associative, and commutative mapping π‘†βˆΆ[0,1]Γ—[0,1]β†’[0,1] that satisfies the boundary condition: 𝑆(π‘Ž,0)=π‘Ž for all π‘Žβˆˆ[0,1].

The commonly used continuous 𝑑-conorms are 𝑆(π‘Ž,𝑏)=max(π‘Ž,𝑏), 𝑆(π‘Ž,𝑏)=π‘Ž+π‘βˆ’π‘Žβ‹…π‘, and 𝑆(π‘Ž,𝑏)=min{1,π‘Ž+𝑏}.

3. Generalised Interval-Valued Fuzzy Soft Set

Obviously, by combining generalised soft set and the interval-valued fuzzy set, it is natural to define the generalised interval-valued fuzzy soft set model. We first define two types of generalised interval-valued fuzzy soft set as follows.

Definition 3.1. Let π‘ˆ be an initial universe and 𝐸 a set of parameters, π΄βŠ†πΈ, ξ‚πΉβˆΆπ΄β†’β„±(π‘ˆ), and let 𝛼 be a fuzzy sets of 𝐴, that is, π›ΌβˆΆπ΄β†’[0,1]. Define a function ξ‚πΉπ›ΌβˆΆπ΄β†’β„±(π‘ˆ)Γ—[0,1] as 𝐹𝛼(𝑒)=(𝐹(𝑒)={β„Ž/πœ‡ξ‚πΉ(𝑒)(β„Ž)},𝛼(𝑒)), where πœ‡ξ‚πΉ(𝑒)(β„Ž)=[πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡+𝐹(𝑒)(β„Ž)] is an interval value is called the degree of membership an element β„Ž to 𝐹(𝑒), and 𝛼(𝑒) is called the degree of possibility of such belongness. Then 𝐹𝛼 is called type 1 generalised interval-valued fuzzy soft set over the soft universe (π‘ˆ,𝐸).

Here for each parameter 𝑒, 𝐹𝛼(𝑒) indicates not only the degree of belongingness of elements of π‘ˆ in 𝐹(𝑒) but also the degree of preference of such belongingness which is represented by 𝛼(𝑒).

Definition 3.2. Let π‘ˆ be an initial universe and 𝐸 a set of parameters, π΄βŠ†πΈ, ξ‚πΉβˆΆπ΄β†’β„±(π‘ˆ), and let 𝛼 be an interval-valued fuzzy sets of 𝐴; that is, π›ΌβˆΆπ΄β†’Int([0,1]), where Int([0,1]) stands for the set of all closed subintervals of [0,1]. Define a function ξ‚πΉπ›ΌβˆΆπ΄β†’β„±(π‘ˆ)Γ—Int([0,1]) as 𝐹𝛼(𝑒)=(𝐹(𝑒)={β„Ž/πœ‡ξ‚πΉ(𝑒)(β„Ž)},𝛼(𝑒)), where πœ‡ξ‚πΉ(𝑒)(β„Ž)=[πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡+𝐹(𝑒)(β„Ž)] and 𝛼(𝑒)=[π›Όβˆ’(𝑒),𝛼+(𝑒)] are interval values. Then 𝐹𝛼 is called type 2 generalised interval-valued fuzzy soft set over the soft universe (π‘ˆ,𝐸).

It is clear that if π›Όβˆ’(𝑒)=𝛼+(𝑒) holds for each π‘Žβˆˆπ΄, then the type 2 generalised interval-valued fuzzy soft set will degenerate to the type 1 generalised interval-valued fuzzy soft set. And if πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž)=πœ‡+𝐹(𝑒)(β„Ž) also holds for each π‘Žβˆˆπ΄, then type 1 generalised interval-valued fuzzy soft set will degenerate to generalised fuzzy soft set [32].

In this paper, the type 2 generalised interval-valued fuzzy soft set is denoted by GIVFS set in short. To illustrate this idea, let us consider the following example.

Example 3.3. Let π‘ˆ={β„Ž1,β„Ž2,β„Ž3} be a set of mobile telephones and 𝐴={𝑒1,𝑒2,𝑒3}∈𝐸 a set of parameters. The 𝑒𝑖(𝑖=1,2,3) stand for the parameters β€œexpensive”, β€œbeautiful”, and β€œmultifunctional”, respectively. Let ξ‚πΉπ›ΌβˆΆπ΄β†’π’«(π‘ˆ)Γ—Int([0,1]) be a function given as follows: 𝐹𝛼𝑒1ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.8,0.92[],β„Ž0.6,0.73[]ξ‚Ό,[]ξ‚Ά,𝐹0.5,0.60.7,0.8𝛼𝑒2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.7,0.82[],β„Ž0.3,0.43[]ξ‚Ό,[]ξ‚Ά,𝐹0.5,0.70.6,0.7𝛼𝑒3ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.5,0.62[],β„Ž0.5,0.73[]ξ‚Ό,[]ξ‚Ά.0.7,0.80.8,0.9(3.1) Then 𝐹𝛼 is a GIVFS set.

Definition 3.4. Let 𝐹𝛼 and 𝐺𝛽 be GIVFS sets over (π‘ˆ,𝐸). Then 𝐹𝛼 is called a GIVFS subset of 𝐺𝛽 if(1)π΄βŠ†π΅;(2)𝐹(𝑒) is an interval-valued fuzzy subset of 𝐺(𝑒) for any π‘’βˆˆπ΄; that is, πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž)β‰€πœ‡βˆ’ξ‚πΊ(𝑒)(β„Ž) and πœ‡+𝐹(𝑒)(β„Ž)β‰€πœ‡+𝐺(𝑒)(β„Ž) for any β„Žβˆˆπ‘ˆ and π‘’βˆˆπ΄;(3)𝛼 is an interval-valued fuzzy subset of 𝛽; that is, π›Όβˆ’(𝑒)β‰€π›½βˆ’(𝑒) and 𝛼+(𝑒)≀𝛽+(𝑒) for any π‘’βˆˆπ΄.
In this case, the above relationship is denoted by 𝐹𝛼⋐𝐺𝛽. And 𝐺𝛽 is said to be a GIVFS superset of 𝐹𝛼.

Definition 3.5. Let 𝐹𝛼 and 𝐺𝛽 be GIVFS sets over (π‘ˆ,𝐸). Then 𝐹𝛼 and 𝐺𝛽 are said to be GIVFS equal if and only if 𝐹𝛼⋐𝐺𝛽 and 𝐺𝛽⋐𝐹𝛼.

Definition 3.6. The relative complement of a GIVFS set 𝐹𝛼 is denoted by ξ‚πΉπ‘Ÿπ›Ό and is defined by ξ‚πΉπ‘Ÿπ›Όξ‚πΉ=(π‘Ÿ,π›Όπ‘Ÿ), where ξ‚πΉπ‘ŸβˆΆπ΄β†’β„±(π‘ˆ) is a mapping given by ξ‚πΉπ‘Ÿξ‚πΉ(𝑒)={β„Ž/πœ‡π‘Ÿ(𝑒)(β„Ž)} and π›Όπ‘ŸβˆΆπ΄β†’Int([0,1]) is a mapping given by π›Όπ‘Ÿ(𝑒) for all β„Žβˆˆπ‘ˆ,π‘’βˆˆπ΄, where πœ‡ξ‚πΉπ‘Ÿ(𝑒)(β„Ž)=[πœ‡βˆ’ξ‚πΉπ‘Ÿ(𝑒)(β„Ž),πœ‡+ξ‚πΉπ‘Ÿ(𝑒)(β„Ž)]=[1βˆ’πœ‡+𝐹(𝑒)(β„Ž),1βˆ’πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž)], π›Όπ‘Ÿ(𝑒)=[π›Όπ‘Ÿβˆ’(𝑒),π›Όπ‘Ÿ+(𝑒)]=[1βˆ’π›Ό+(𝑒),1βˆ’π›Όβˆ’(𝑒)].

Example 3.7. We consider the GIVFS set 𝐹𝛼 given in Example 3.3 and define a GIVFS set 𝐺𝛽 as follows: 𝐺𝛽𝑒1ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.7,0.82[],β„Ž0.4,0.53[]ξ‚Ό,[]ξ‚Ά,𝐺0.4,0.60.5,0.6𝛽𝑒2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.5,0.62[],β„Ž0.2,0.43[]ξ‚Ό,[]ξ‚Ά.0.5,0.60.3,0.4(3.2) Then 𝐺𝛽 is a GIVFS subset of 𝐹𝛼, and the relative complement of a GIVFS set 𝐺𝛽 is ξ‚πΊπ‘Ÿπ›½ξ€·π‘’1ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.2,0.32[],β„Ž0.5,0.63[]ξ‚Ό,[]ξ‚Ά,𝐺0.4,0.60.4,0.5π‘Ÿπ›½ξ€·π‘’2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.4,0.52[],β„Ž0.6,0.83[]ξ‚Ό,[]ξ‚Ά.0.4,0.50.6,0.7(3.3)

Definition 3.8. Let 1=[1,1]. A GIVFS set 𝐹𝛼 over (π‘ˆ,𝐸) is said to be relative absolute GIVFS set denoted by Ω𝐴, if πœ‡ξ‚πΉ(𝑒)(β„Ž)=1 and 𝛼(𝑒)=1 for all β„Žβˆˆπ‘ˆ and π‘’βˆˆπ΄.

Definition 3.9. Let 0=[0,0]. A GIVFS set 𝐹𝛼 over (π‘ˆ,𝐸) is said to be relative null GIVFS set, denoted by Φ𝐴, if πœ‡ξ‚πΉ(𝑒)(β„Ž)=0 and 𝛼(𝑒)=0 for all β„Žβˆˆπ‘ˆ and π‘’βˆˆπ΄.

Definition 3.10. The union of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸) denoted by 𝐹𝛼βˆͺ𝐺𝛽 is a GIVFS set 𝐻𝛾 and defined as ξ‚π»π›ΎβˆΆπ΄βˆͺ𝐡→ℱ(π‘ˆ)Γ—Int([0,1]) such that, for all β„Žβˆˆπ‘ˆ and π‘’βˆˆπ΄βˆͺ𝐡, 𝐻𝛾(⎧βŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽ©β„Žπ‘’)=ξƒ©ξƒ―πœ‡ξ‚πΉ(𝑒)ξƒ°ξƒͺβ„Ž(β„Ž),𝛼(𝑒),ifπ‘’βˆˆπ΄βˆ’π΅,ξƒ©ξƒ―πœ‡ξ‚πΊ(𝑒)ξƒ°ξƒͺβ„Ž(β„Ž),𝛽(𝑒),ifπ‘’βˆˆπ΅βˆ’π΄,ξƒ©ξƒ―πœ‡ξ‚π»(𝑒)ξƒ°ξƒͺ(β„Ž),𝛾(𝑒),ifπ‘’βˆˆπ΄βˆ©π΅,(3.4) where πœ‡ξ‚π»(𝑒)(β„Ž)=𝑆(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž))=[𝑆(πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒)(β„Ž)),𝑆(πœ‡+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒)(β„Ž))] and 𝛾(𝑒)=𝑆(𝛼(𝑒),𝛽(𝑒))=[𝑆(π›Όβˆ’(𝑒),π›½βˆ’(𝑒)),𝑆(𝛼+(𝑒),𝛽+(𝑒))].

Definition 3.11. The intersection of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸) denoted by 𝐹𝛼⋒𝐺𝛽 is a GIVFS set 𝐻𝛾 and defined as ξ‚π»π›ΎβˆΆπ΄βˆ©π΅β†’β„±(π‘ˆ)Γ—Int([0,1]) such that, for all β„Žβˆˆπ‘ˆ and π‘’βˆˆπ΄βˆ©π΅, 𝐻𝛾(𝑒)=({β„Ž/πœ‡ξ‚π»(𝑒)(β„Ž)},𝛾(𝑒)), where πœ‡ξ‚π»(𝑒)(β„Ž)=𝑇(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž))=[𝑇(πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒)(β„Ž)), 𝑇(πœ‡+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒)(β„Ž))] and 𝛾(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))=[𝑇(π›Όβˆ’(𝑒),π›½βˆ’(𝑒)),𝑇(𝛼+(𝑒),𝛽+(𝑒))].

Example 3.12. We consider the GIVFS sets 𝐹𝛼 and 𝐺𝛽 given in Examples 3.3 and 3.7, respectively, and consider 𝑆(π‘₯,𝑦)=max{π‘₯,𝑦} and 𝑇(π‘₯,𝑦)=min{π‘₯,𝑦}. Then 𝐹𝛼βˆͺ𝐺𝛽𝑒1ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.8,0.92[],β„Ž0.6,0.73[]ξ‚Ό,[]ξ‚Ά,𝐹0.5,0.60.7,0.8𝛼βˆͺ𝐺𝛽𝑒2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.7,0.82[],β„Ž0.3,0.43[]ξ‚Ό,[]ξ‚Ά,𝐹0.5,0.70.6,0.7𝛼βˆͺ𝐺𝛽𝑒3ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.5,0.62[],β„Ž0.5,0.73[]ξ‚Ό,[]ξ‚Ά,𝐹0.7,0.80.8,0.9𝛼⋒𝐺𝛽𝑒1ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.7,0.82[],β„Ž0.4,0.53[]ξ‚Ό,[]ξ‚Ά,𝐹0.4,0.60.5,0.6𝛼⋒𝐺𝛽𝑒2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.5,0.62[],β„Ž0.2,0.43[]ξ‚Ό,[]ξ‚Ά.0.5,0.60.3,0.4(3.5)

Proposition 3.13. Let 𝐹𝛼 be a GIVFS set over (π‘ˆ,𝐸). Then the following holds(1)𝐹𝛼⋒Ω𝐴=𝐹𝛼, (2)𝐹𝛼βˆͺΩ𝐴=Ω𝐴, (3)𝐹𝛼⋒Φ𝐴=Φ𝐴, (4)𝐹𝛼βˆͺΦ𝐴=𝐹𝛼.

Proof. It is easily obtained from Definitions 3.8–3.11.

Theorem 3.14. Let 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be GIVFS sets over (π‘ˆ,𝐸). Then the following holds(1)𝐹𝛼⋒𝐺𝛽=𝐺𝛽⋒𝐹𝛼, (2)𝐹𝛼𝐺⋒(𝛽⋒𝐻𝛾𝐹)=(𝛼⋒𝐺𝛽𝐻)⋒𝛾, (3)𝐹𝛼βˆͺ𝐺𝛽=𝐺𝛽βˆͺ𝐹𝛼, (4)𝐹𝛼𝐺βˆͺ(𝛽βˆͺ𝐻𝛾𝐹)=(𝛼βˆͺ𝐺𝛽)βˆͺ𝐻𝛾.

Proof. It is easily obtained from Definitions 3.10 and 3.11.

Definition 3.15. The restricted union of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸) denoted by 𝐹𝛼⋓𝐺𝛽 is a GIVFS set 𝐻𝛾 and defined as ξ‚π»π›ΎβˆΆπ΄βˆ©π΅β†’β„±(π‘ˆ)Γ—Int([0,1]) such that, for all β„Žβˆˆπ‘ˆ and π‘’βˆˆπ΄βˆ©π΅, 𝐻𝛾(𝑒)=({β„Ž/πœ‡ξ‚π»(𝑒)(β„Ž)},𝛾(𝑒)), where πœ‡ξ‚π»(𝑒)(β„Ž)=𝑆(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž))=[𝑆(πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒)(β„Ž)), 𝑆(πœ‡+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒)(β„Ž))] and 𝛾(𝑒)=𝑆(𝛼(𝑒),𝛽(𝑒))=[𝑆(π›Όβˆ’(𝑒),π›½βˆ’(𝑒)),𝑆(𝛼+(𝑒),𝛽+(𝑒))].

Definition 3.16. The extended intersection of two GVS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸), denoted by ξ‚πΉπ›Όξ‚βˆ©ξ‚πΊπ›½, is a GVS set ξ‚π»π›ΎβˆΆπ΄βˆͺ𝐡→ℱ(π‘ˆ)Γ—Int([0,1]) which is defined as, for all β„Žβˆˆπ‘ˆ,π‘’βˆˆπ΄βˆͺ𝐡,𝐻𝛾(⎧βŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽ©β„Žπ‘’)=ξƒ©ξƒ―πœ‡ξ‚πΉ(𝑒)ξƒ°ξƒͺβ„Ž(β„Ž),𝛼(𝑒),ifπ‘’βˆˆπ΄βˆ’π΅,ξƒ©ξƒ―πœ‡ξ‚πΊ(𝑒)ξƒ°ξƒͺβ„Ž(β„Ž),𝛽(𝑒),ifπ‘’βˆˆπ΅βˆ’π΄,ξƒ©ξƒ―πœ‡ξ‚π»(𝑒)ξƒ°ξƒͺ(β„Ž),𝛾(𝑒),ifπ‘’βˆˆπ΄βˆ©π΅,(3.6) where πœ‡ξ‚π»(𝑒)(β„Ž)=𝑇(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž))=[𝑇(πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒)(β„Ž)), 𝑇(πœ‡+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒)(β„Ž))] and 𝛾(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))=[𝑇(π›Όβˆ’(𝑒),π›½βˆ’(𝑒)),𝑇(𝛼+(𝑒),𝛽+(𝑒))].

Theorem 3.17. Let 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be three GIVFS sets over (π‘ˆ,𝐸). Then the following holds:(1)𝐹𝛼⋓𝐺𝛽=𝐺𝛽⋓𝐹𝛼, (2)𝐹𝛼𝐺⋓(𝛽⋓𝐻𝛾𝐹)=(𝛼⋓𝐺𝛽𝐻)⋓𝛾, (3)ξ‚πΉπ›Όξ‚βˆ©ξ‚πΊπ›½=ξ‚πΊπ›½ξ‚βˆ©ξ‚πΉπ›Ό, (4)ξ‚πΉπ›Όξ‚ξ‚πΊβˆ©(π›½ξ‚βˆ©ξ‚π»π›Ύξ‚πΉ)=(π›Όξ‚βˆ©ξ‚πΊπ›½)ξ‚βˆ©ξ‚π»π›Ύ.

Proof. It is easily obtained from Definitions 3.15 and 3.16.

Theorem 3.18. Let 𝐹𝛼 and 𝐺𝛽 be two GIVFS sets over (π‘ˆ,𝐸). Then the following holds: (1)(𝐹𝛼⋒𝐺𝛽)π‘Ÿξ‚πΉ=(𝛼)π‘Ÿξ‚πΊβ‹“(𝛽)π‘Ÿ,(2)(𝐹𝛼⋓𝐺𝛽)π‘Ÿξ‚πΉ=(𝛼)π‘Ÿξ‚πΊβ‹’(𝛽)π‘Ÿ.

Proof. (1) Suppose that 𝐹𝛼⋒𝐺𝛽=𝐻𝛾, then 𝐢=𝐴∩𝐡, and, forallπ‘’βˆˆπΆ,β„Žβˆˆπ‘ˆ, πœ‡ξ‚π»(𝑒)ξ‚€πœ‡(β„Ž)=𝑇𝐹(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)=ξ‚ƒπ‘‡ξ‚€πœ‡(β„Ž)βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒)ξ‚ξ‚€πœ‡(β„Ž),𝑇+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒),=ξ€Ί(β„Ž)𝛾(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))𝑇(π›Όβˆ’(𝑒),π›½βˆ’(𝛼𝑒)),𝑇+(𝑒),𝛽+(.𝑒)ξ€Έξ€»(3.7) Moreover, we have (𝐹𝛼⋒𝐺𝛽)π‘Ÿ=ξ‚π»π‘Ÿπ›Ύ, 𝐢=𝐴∩𝐡, and forallπ‘’βˆˆπΆ,β„Žβˆˆπ‘ˆ, πœ‡ξ‚π»π‘Ÿ(𝑒)ξ‚ƒξ‚€πœ‡(β„Ž)=1βˆ’π‘‡+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒)ξ‚ξ‚€πœ‡(β„Ž),1βˆ’π‘‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒),𝛾(β„Ž)ξ‚ξ‚„π‘Ÿξ€Ίξ€·π›Ό(𝑒)=1βˆ’π‘‡+(𝑒),𝛽+ξ€Έ(𝑒),1βˆ’π‘‡(π›Όβˆ’(𝑒),π›½βˆ’ξ€».(𝑒))(3.8) Assume that the parameters set of a GIVFS set 𝐽𝛿 is denoted 𝐷, and ξ‚πΉπ‘Ÿπ›Όβ‹“ξ‚πΊπ‘Ÿπ›½=𝐽𝛿. Then 𝐷=𝐴∩𝐡. Since πœ‡ξ‚πΉπ‘Ÿ(𝑒)(β„Ž)=1βˆ’πœ‡+𝐹(𝑒)(β„Ž),1βˆ’πœ‡βˆ’ξ‚πΉ(𝑒)ξ‚„(β„Ž),π›Όπ‘Ÿξ€Ί(𝑒)=1βˆ’π›Ό+(𝑒),1βˆ’π›Όβˆ’ξ€»,πœ‡ξ‚πΊ(𝑒)π‘Ÿ(𝑒)(β„Ž)=1βˆ’πœ‡+𝐺(𝑒)(β„Ž),1βˆ’πœ‡βˆ’ξ‚πΊ(𝑒)ξ‚„(β„Ž),π›½π‘Ÿξ€Ί(𝑒)=1βˆ’π›½+(𝑒),1βˆ’π›½βˆ’ξ€».(𝑒)(3.9) Then, for each π‘’βˆˆπ·,β„Žβˆˆπ‘ˆ, πœ‡ξ‚π½(𝑒)ξ‚€πœ‡ξ‚πΉ(β„Ž)=π‘†π‘Ÿ(𝑒)𝐺(β„Ž),πœ‡π‘Ÿ(𝑒)=𝑆(β„Ž)1βˆ’πœ‡+𝐹(𝑒)(β„Ž),1βˆ’πœ‡+𝐺(𝑒)(β„Ž),𝑆1βˆ’πœ‡βˆ’ξ‚πΉ(𝑒)(β„Ž),1βˆ’πœ‡βˆ’ξ‚πΊ(𝑒)=ξ‚ƒξ‚€πœ‡(β„Ž)1βˆ’π‘‡+𝐹(𝑒)(β„Ž),πœ‡+𝐺(𝑒)(ξ‚ξ‚€πœ‡β„Ž),1βˆ’π‘‡βˆ’ξ‚πΉ(𝑒)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑒)(ξ‚π»β„Ž)=πœ‡π‘Ÿ(𝑒)(β„Ž),𝛿(𝑒)=𝑆(π›Όπ‘Ÿ(𝑒),π›½π‘Ÿ=𝑆(𝑒))1βˆ’π›Ό+(𝑒),1βˆ’π›½+ξ€Έ(𝑒),𝑆(1βˆ’π›Όβˆ’(𝑒),1βˆ’π›½βˆ’ξ€»=𝛼(𝑒))1βˆ’π‘‡+(𝑒),𝛽+ξ€Έ(𝑒),1βˆ’π‘‡(π›Όβˆ’(𝑒),π›½βˆ’ξ€»(𝑒))=π›Ύπ‘Ÿ(𝑒).(3.10) Therefore, ξ‚π»π‘Ÿπ›Ύ and 𝐽𝛿 are the same GIVFS sets. Thus, (𝐹𝛼⋒𝐺𝛽)π‘Ÿξ‚πΉ=(𝛼)π‘Ÿξ‚πΊβ‹“(𝛽)π‘Ÿ.
(2) The proof is similar to that of (1).

Definition 3.19. The β€œAND” of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸), denoted by ξ‚πΉπ›ΌβŠΌξ‚πΊπ›½, is defined as ξ‚π»π›ΎβˆΆπ΄Γ—π΅β†’β„±(π‘ˆ)Γ—Int([0,1]) such that for all β„Žβˆˆπ‘ˆ and (π‘Ž,𝑏)βˆˆπ΄Γ—π΅, 𝐻𝛾(π‘Ž,𝑏)=({β„Ž/πœ‡ξ‚π»(π‘Ž,𝑏)(β„Ž)},𝛾(π‘Ž,𝑏)), where πœ‡ξ‚π»(π‘Ž,𝑏)(β„Ž)=𝑇(πœ‡ξ‚πΉ(π‘Ž)(β„Ž),πœ‡ξ‚πΊ(𝑏)(β„Ž))=[𝑇(πœ‡βˆ’ξ‚πΉ(π‘Ž)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑏)(β„Ž)), 𝑇(πœ‡+𝐹(π‘Ž)(β„Ž),πœ‡+𝐺(𝑏)(β„Ž))] and 𝛾(π‘Ž,𝑏)=𝑇(𝛼(π‘Ž),𝛽(𝑏))=[𝑇(π›Όβˆ’(π‘Ž),π›½βˆ’(𝑏)),𝑇(𝛼+(π‘Ž),𝛽+(𝑏))].

Definition 3.20. The β€œOR” of two GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸), denoted by ξ‚πΉπ›ΌβŠ»ξ‚πΊπ›½, is defined as ξ‚π»π›ΎβˆΆπ΄Γ—π΅β†’β„±(π‘ˆ)Γ—Int([0,1]) such that for all β„Žβˆˆπ‘ˆ and (π‘Ž,𝑏)βˆˆπ΄Γ—π΅, 𝐻𝛾(π‘Ž,𝑏)=({β„Ž/πœ‡ξ‚π»(π‘Ž,𝑏)(β„Ž)},𝛾(π‘Ž,𝑏)), where πœ‡ξ‚π»(π‘Ž,𝑏)(β„Ž)=𝑆(πœ‡ξ‚πΉ(π‘Ž)(β„Ž),πœ‡ξ‚πΊ(𝑏)(β„Ž))=[𝑆(πœ‡βˆ’ξ‚πΉ(π‘Ž)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑏)(β„Ž)), 𝑆(πœ‡+𝐹(π‘Ž)(β„Ž),πœ‡+𝐺(𝑏)(β„Ž))] and 𝛾(π‘Ž,𝑏)=𝑆(𝛼(π‘Ž),𝛽(𝑏))=[𝑆(π›Όβˆ’(π‘Ž),π›½βˆ’(𝑏)),𝑆(𝛼+(π‘Ž),𝛽+(𝑏))].

Theorem 3.21. Let 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be three GIVFS sets over (π‘ˆ,𝐸). Then the following holds (1)ξ‚πΉπ›Όξ‚πΊβŠ»(π›½βŠ»ξ‚π»π›Ύξ‚πΉ)=(π›ΌβŠ»ξ‚πΊπ›½ξ‚π»)βŠ»π›Ύ, (2)ξ‚πΉπ›Όξ‚πΊβŠΌ(π›½βŠΌξ‚π»π›Ύξ‚πΉ)=(π›ΌβŠΌξ‚πΊπ›½ξ‚π»)βŠΌπ›Ύ.

Proof. It is easily obtained from Definitions 3.19 and 3.20.

Theorem 3.22. Let 𝐹𝛼 and 𝐺𝛽 be two GIVFS sets over (π‘ˆ,𝐸). Then the following holds (1)(ξ‚πΉπ›ΌβŠ»ξ‚πΊπ›½)π‘Ÿξ‚πΉ=(𝛼)π‘Ÿξ‚πΊβŠΌ(𝛽)π‘Ÿ, (2)(ξ‚πΉπ›ΌβŠΌξ‚πΊπ›½)π‘Ÿξ‚πΉ=(𝛼)π‘Ÿξ‚πΊβŠ»(𝛽)π‘Ÿ.

Proof. (1) Suppose that ξ‚πΉπ›ΌβŠ»ξ‚πΊπ›½=𝐻𝛾, then 𝐢=𝐴×𝐡, and, forall(π‘Ž,𝑏)∈𝐢,β„Žβˆˆπ‘ˆ, πœ‡ξ‚π»(π‘Ž,𝑏)ξ‚€πœ‡(β„Ž)=𝑆𝐹(π‘Ž)(β„Ž),πœ‡ξ‚πΊ(𝑏)=ξ‚ƒπ‘†ξ‚€πœ‡(β„Ž)βˆ’ξ‚πΉ(π‘Ž)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑏)ξ‚ξ‚€πœ‡(β„Ž),𝑆+𝐹(π‘Ž)(β„Ž),πœ‡+𝐺(𝑏),=ξ€Ί(β„Ž)𝛾(π‘Ž,𝑏)=𝑆(𝛼(π‘Ž),𝛽(𝑏))𝑆(π›Όβˆ’(π‘Ž),π›½βˆ’(𝛼𝑏)),𝑆+(π‘Ž),𝛽+(.𝑏)ξ€Έξ€»(3.11) Moreover, we have (ξ‚πΉπ›ΌβŠ»ξ‚πΊπ›½)π‘Ÿ=ξ‚π»π‘Ÿπ›Ύ, 𝐢=𝐴×𝐡, and forall(π‘Ž,𝑏)∈𝐢,β„Žβˆˆπ‘ˆ, πœ‡ξ‚π»π‘Ÿ(π‘Ž,𝑏)ξ‚ƒξ‚€πœ‡(β„Ž)=1βˆ’π‘†+𝐹(π‘Ž)(β„Ž),πœ‡+𝐺(𝑏)ξ‚ξ‚€πœ‡(β„Ž),1βˆ’π‘†βˆ’ξ‚πΉ(π‘Ž)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑏),𝛾(β„Ž)ξ‚ξ‚„π‘Ÿξ€Ίξ€·π›Ό(π‘Ž,𝑏)=1βˆ’π‘†+(π‘Ž),𝛽+ξ€Έ(𝑏),1βˆ’π‘†(π›Όβˆ’(π‘Ž),π›½βˆ’ξ€».(𝑏))(3.12) Assume that the parameters set of a GIVFS set 𝐽𝛿 is denoted 𝐷, and ξ‚πΉπ‘Ÿπ›ΌβŠΌξ‚πΊπ‘Ÿπ›½=𝐽𝛿. Then 𝐷=𝐴×𝐡. Since forallπ‘Žβˆˆπ΄,π‘βˆˆπ΅,β„Žβˆˆπ‘ˆ, πœ‡ξ‚πΉπ‘Ÿ(π‘Ž)(β„Ž)=1βˆ’πœ‡+𝐹(π‘Ž)(β„Ž),1βˆ’πœ‡+𝐹(π‘Ž)ξ‚„(β„Ž),π›Όπ‘Ÿξ€Ί(π‘Ž)=1βˆ’π›Ό+(π‘Ž),1βˆ’π›Όβˆ’ξ€»,πœ‡ξ‚πΊ(π‘Ž)π‘Ÿ(𝑏)(β„Ž)=1βˆ’πœ‡+𝐺(𝑏)(β„Ž),1βˆ’πœ‡βˆ’ξ‚πΊ(𝑏)ξ‚„(β„Ž),π›½π‘Ÿξ€Ί(𝑏)=1βˆ’π›½+(𝑏),1βˆ’π›½βˆ’ξ€»,(𝑏)(3.13) then, for each (π‘Ž,𝑏)∈𝐷,β„Žβˆˆπ‘ˆ, πœ‡ξ‚π½(π‘Ž,𝑏)ξ‚€πœ‡ξ‚πΉ(β„Ž)=π‘‡π‘Ÿ(π‘Ž)𝐺(β„Ž),πœ‡π‘Ÿ(𝑏)=𝑇(β„Ž)1βˆ’πœ‡+𝐹(π‘Ž)(β„Ž),1βˆ’πœ‡+𝐺(𝑏)(β„Ž),𝑇1βˆ’πœ‡βˆ’ξ‚πΉ(π‘Ž)(β„Ž),1βˆ’πœ‡βˆ’ξ‚πΊ(𝑏)=ξ‚ƒξ‚€πœ‡(β„Ž)1βˆ’π‘†+𝐹(π‘Ž)(β„Ž),πœ‡+𝐺(𝑏)(ξ‚ξ‚€πœ‡β„Ž),1βˆ’π‘†βˆ’ξ‚πΉ(π‘Ž)(β„Ž),πœ‡βˆ’ξ‚πΊ(𝑏)(ξ‚π»β„Ž)=πœ‡π‘Ÿ(π‘Ž,𝑏)(β„Ž),𝛿(π‘Ž,𝑏)=𝑇(π›Όπ‘Ÿ(π‘Ž),π›½π‘Ÿ=𝑇(𝑏))1βˆ’π›Ό+(π‘Ž),1βˆ’π›½+ξ€Έ(𝑏),𝑇(1βˆ’π›Όβˆ’(π‘Ž),1βˆ’π›½βˆ’ξ€»=𝛼(𝑏))1βˆ’π‘†+(π‘Ž),𝛽+ξ€Έ(𝑏),1βˆ’π‘†(π›Όβˆ’(π‘Ž),π›½βˆ’ξ€»(𝑏))=π›Ύπ‘Ÿ(π‘Ž,𝑏).(3.14) Therefore, ξ‚π»π‘Ÿπ›Ύ and 𝐽𝛿 are the same GIVFS sets. Thus, (ξ‚πΉπ›ΌβŠ»ξ‚πΊπ›½)π‘Ÿξ‚πΉ=(𝛼)π‘Ÿξ‚πΊβŠΌ(𝛽)π‘Ÿ.
(2) The proof is similar to that of (1).

4. The Lattice Structures of GIVFS Sets

The lattice structures of soft sets have been studied by Qin and Hong in [14]. In this section, we will discuss the lattice structures of GIVFS sets. The following proposition shows the idempotent law with respect to operations βˆͺ and β‹’ does not hold in general.

Proposition 4.1. Let 𝐹𝛼 be a GIVFS sets over (π‘ˆ,𝐸). Then the following holds(1)𝐹𝛼𝐹⋐(𝛼βˆͺ𝐹𝛼), (2)(𝐹𝛼⋒𝐹𝛼𝐹)⋐𝛼.

To illuminate the above proposition, we give an example as follows.

Example 4.2. We consider the GIVFS set 𝐹𝛼 given in Example 3.3. We have that the following(1)If 𝑆(π‘Ž,𝑏)=π‘Ž+π‘βˆ’π‘Žβ‹…π‘, then (𝐹𝛼βˆͺ𝐹𝛼)(𝑒1)=({β„Ž1/[0.96,0.99],β„Ž2/[0.84,0.91], β„Ž3𝐹/[0.75,0.84]},[0.91,0.96])βŠ‡π›Ό(𝑒1), (𝐹𝛼βˆͺ𝐹𝛼)(𝑒2𝐹)βŠ‡π›Ό(𝑒2), and (𝐹𝛼βˆͺ𝐹𝛼)(𝑒3𝐹)βŠ‡π›Ό(𝑒3); that is, (𝐹𝛼βˆͺ𝐹𝛼𝐹)βŠ‡π›Ό.(2)If 𝑆(π‘Ž,𝑏)=min(1,π‘Ž+𝑏), then (𝐹𝛼βˆͺ𝐹𝛼)(𝑒1)=({β„Ž1/[1.0,1.0],β„Ž2/[1.0,1.0], β„Ž3𝐹/[1.0,1.0]},[1.0,1.0])βŠ‡π›Ό(𝑒1), (𝐹𝛼βˆͺ𝐹𝛼)(𝑒2𝐹)βŠ‡π›Ό(𝑒2), and (𝐹𝛼βˆͺ𝐹𝛼)(𝑒3𝐹)βŠ‡π›Ό(𝑒3); that is, (𝐹𝛼βˆͺ𝐹𝛼𝐹)βŠ‡π›Ό.(3)if 𝑇(π‘Ž,𝑏)=π‘Žβ‹…π‘, then (𝐹𝛼⋒𝐹𝛼)(𝑒1)=({β„Ž1/[0.64,0.81],β„Ž2/[0.36,0.49], β„Ž3𝐹/[0.25,0.36]},[0.49,0.64])βŠ†π›Ό(𝑒1), (𝐹𝛼⋒𝐹𝛼)(𝑒2𝐹)βŠ†π›Ό(𝑒2) and (𝐹𝛼⋒𝐹𝛼)(𝑒3𝐹)βŠ†π›Ό(𝑒3), that is, (𝐹𝛼⋒𝐹𝛼𝐹)⋐𝛼;(4)If 𝑇(π‘Ž,𝑏)=max(0,π‘Ž+π‘βˆ’1), then (𝐹𝛼⋒𝐹𝛼)(𝑒1)=({β„Ž1/[0.6,0.8],β„Ž2/[0.2,0.4], β„Ž3𝐹/[0.0,0.2]},[0.4,0.6])βŠ†π›Ό(𝑒1), (𝐹𝛼⋒𝐹𝛼)(𝑒2𝐹)βŠ†π›Ό(𝑒2) and (𝐹𝛼⋒𝐹𝛼)(𝑒3𝐹)βŠ†π›Ό(𝑒3); that is, (𝐹𝛼⋒𝐹𝛼𝐹)⋐𝛼.
For convenience, let 𝔖(π‘ˆ,𝐸) denote the set of all GIVFS sets over (π‘ˆ,𝐸); that is, 𝐹𝔖(π‘ˆ,𝐸)={π›Όβˆ£π΄βŠ†πΈ,πΉβˆΆπ΄β†’β„±(π‘ˆ),π›ΌβˆΆπ΄β†’Int([0,1])}.

From Proposition 4.1, we can see that (𝔖(π‘ˆ,𝐸),β‹’,βˆͺ) is not a lattice in general. However, if 𝑇(π‘Ž,𝑏)=min(π‘Ž,𝑏) and 𝑆(π‘Ž,𝑏)=max(π‘Ž,𝑏), then the idempotent law and absorption law with respect to operations βˆͺ and β‹’ hold. In the remainder of this section, we always consider 𝑇(π‘Ž,𝑏)=min(π‘Ž,𝑏) and 𝑆(π‘Ž,𝑏)=max(π‘Ž,𝑏).

Theorem 4.3. Let 𝐴,π΅βŠ†πΈ, 𝐹𝛼, and 𝐺𝛽 be GIVFS sets over (π‘ˆ,𝐸). Then the following hold: (1)(𝐹𝛼⋒𝐹𝛼𝐹)=𝛼, (2)(𝐹𝛼βˆͺ𝐹𝛼𝐹)=𝛼, (3)(𝐹𝛼βˆͺ𝐺𝛽𝐹)⋒𝛼=𝐹𝛼,(4)(𝐹𝛼⋒𝐺𝛽)βˆͺ𝐹𝛼=𝐹𝛼.

Proof. (1) and (2) are trivial to prove. We prove only (3) since (4) can be proved similarly.
Suppose that the parameter sets of two GIVFS sets 𝐽𝛿 and ξ‚πΎπœ‚ are denoted by 𝑀 and 𝑁, respectively. Let 𝐹𝛼βˆͺ𝐺𝛽=𝐽𝛿 and (𝐹𝛼βˆͺ𝐺𝛽𝐹)⋒𝛼=ξ‚πΎπœ‚. Then 𝑀=𝐴βˆͺ𝐡, 𝑁=(𝐴βˆͺ𝐡)∩𝐴=𝐴. And, for each π‘’βˆˆπ΄ and β„Žβˆˆπ‘ˆ,(i)if π‘’βˆ‰π΅, then πœ‡ξ‚πΎ(𝑒)(β„Ž)=𝑇(πœ‡ξ‚π½(𝑒)(β„Ž),πœ‡ξ‚πΉ(𝑒)(β„Ž))=min(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΉ(𝑒)(β„Ž))=πœ‡ξ‚πΉ(𝑒)(β„Ž), and πœ‚(𝑒)=𝑇(𝛼(𝑒),𝛼(𝑒))=min(𝛼(𝑒),𝛼(𝑒))=𝛼(𝑒),(ii)if π‘’βˆˆπ΅, then πœ‡ξ‚πΎ(𝑒)(β„Ž)=min(πœ‡ξ‚π½(𝑒)(β„Ž),πœ‡ξ‚πΉ(𝑒)(β„Ž))=min(max(πœ‡ξ‚πΉ(𝑒)(β„Ž), πœ‡ξ‚πΊ(𝑒)(β„Ž)),πœ‡ξ‚πΉ(𝑒)(β„Ž))=πœ‡ξ‚πΉ(𝑒)(β„Ž), and πœ‚(𝑒)=𝑇(𝑆(𝛼(𝑒),𝛽(𝑒)),𝛼(𝑒))=min(max(𝛼(𝑒),𝛽(𝑒)),𝛼(𝑒))=𝛼(𝑒).Thus ξ‚πΎπœ‚=𝐹𝛼; that is, (𝐹𝛼βˆͺ𝐺𝛽𝐹)⋒𝛼=𝐹𝛼.

Theorem 4.4. Let 𝐴,𝐡,πΆβŠ†πΈ, 𝐹𝛼, 𝐺𝛽, and 𝐻𝛾 be GIVFS sets over (π‘ˆ,𝐸). Then the following hold: (1)𝐹𝛼𝐺⋒(𝛽βˆͺ𝐻𝛾𝐹)=(𝛼⋒𝐺𝛽)𝐹βˆͺ(𝛼⋒𝐻𝛾),(2)𝐹𝛼𝐺βˆͺ(𝛽⋒𝐻𝛾𝐹)=(𝛼βˆͺ𝐺𝛽𝐹)β‹’(𝛼βˆͺ𝐻𝛾).

Proof. (1) Suppose that the parameter sets of two GIVFS sets 𝐽𝛿 and ξ‚πΎπœ‚ are denoted by 𝑀 and 𝑁, respectively. Let 𝐹𝛼𝐺⋒(𝛽βˆͺ𝐻𝛾𝐽)=𝛿 and (𝐹𝛼⋒𝐺𝛽)𝐹βˆͺ(𝛼⋒𝐻𝛾𝐾)=πœ‚. Then 𝑀=𝐴∩(𝐡βˆͺ𝐢)=(𝐴∩𝐡)βˆͺ(𝐴∩𝐢)=𝑁. And, for each π‘’βˆˆπ‘€,β„Žβˆˆπ‘ˆ, it follows that π‘’βˆˆπ΄ and π‘’βˆˆπ΅βˆͺ𝐢,(i)if π‘’βˆˆπ΄,π‘’βˆ‰π΅,π‘’βˆˆπΆ, then πœ‡ξ‚π½(𝑒)(β„Ž)=𝑇(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚π»(𝑒)(β„Ž))=min(πœ‡ξ‚πΉ(𝑒)(β„Ž), β€‰πœ‡ξ‚π»(𝑒)(β„Ž))=πœ‡ξ‚πΎ(𝑒)(β„Ž), and 𝛿(𝑒)=𝑇(𝛼(𝑒),𝛾(𝑒))=min(𝛼(𝑒),𝛾(𝑒))=πœ‚(𝑒),(ii)if π‘’βˆˆπ΄,π‘’βˆˆπ΅,π‘’βˆ‰πΆ, then πœ‡ξ‚π½(𝑒)(β„Ž)=𝑇(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž))=min(πœ‡ξ‚πΉ(𝑒)(β„Ž),   πœ‡ξ‚πΊ(𝑒)(β„Ž))=πœ‡ξ‚πΎ(𝑒)(β„Ž), and 𝛿(𝑒)=𝑇(𝛼(𝑒),𝛽(𝑒))=min(𝛼(𝑒),𝛽(𝑒))=πœ‚(𝑒),(iii) if π‘’βˆˆπ΄,π‘’βˆˆπ΅,π‘’βˆˆπΆ, then πœ‡ξ‚π½(𝑒)(β„Ž)=min(πœ‡ξ‚πΉ(𝑒)(β„Ž), max(πœ‡ξ‚πΊ(𝑒)(β„Ž),πœ‡ξ‚π»(𝑒)(β„Ž)))=max(min(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž)),   min((πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚π»(𝑒)(β„Ž)))=πœ‡ξ‚πΎ(𝑒)(β„Ž), and 𝛿(𝑒)=𝑇(𝛼(𝑒),𝑆(𝛽(𝑒),𝛾(𝑒)))=min(𝛼(𝑒),max(𝛽(𝑒),𝛾(𝑒)))=max(min(𝛼(𝑒),𝛽(𝑒)), min(𝛼(𝑒),𝛾(𝑒)))=𝑆(𝑇(𝛼(𝑒),𝛽(𝑒)),𝑇(𝛼(𝑒),𝛾(𝑒)))=πœ‚(𝑒).Thus 𝐽𝛿=ξ‚πΎπœ‚; that is, 𝐹𝛼𝐺⋒(𝛽βˆͺ𝐻𝛾𝐹)=(𝛼⋒𝐺𝛽)𝐹βˆͺ(𝛼⋒𝐻𝛾).
(2) The proof is similar to that of (1).

Theorem 4.5. (1)  (𝔖(π‘ˆ,𝐸),β‹’,βˆͺ) is a distributive lattice.
(2) Let ≀1 be the order relation in 𝔖(π‘ˆ,𝐸) and 𝐹𝛼,ξ‚πΊπ›½βˆˆξ‚π”–(π‘ˆ,𝐸). One has 𝐹𝛼≀1𝐺𝛽 if and only if π΄βŠ†π΅,πœ‡ξ‚πΉ(𝑒)(β„Ž)β‰€πœ‡ξ‚πΊ(𝑒)(β„Ž) and 𝛼(𝑒)≀𝛽(𝑒) for all π‘’βˆˆπ΄ and β„Žβˆˆπ‘ˆ.

Proof. (1) The proof is straightforward from Theorems 3.14, 4.3, and 4.4.
(2) Suppose that 𝐹𝛼≀1𝐺𝛽. Then 𝐹𝛼βˆͺ𝐺𝛽=𝐺𝛽. So by Definition 3.10, we have 𝐴βˆͺ𝐡=𝐡, max(πœ‡ξ‚πΉ(𝑒)(β„Ž),πœ‡ξ‚πΊ(𝑒)(β„Ž))=πœ‡ξ‚πΊ(𝑒)(β„Ž), and max(𝛼(𝑒),𝛽(𝑒))=𝛽(𝑒) for all π‘’βˆˆπ΄ and β„Žβˆˆπ‘ˆ. It follows that π΄βŠ†π΅, πœ‡ξ‚πΉ(𝑒)(β„Ž)β‰€πœ‡ξ‚πΊ(𝑒)(β„Ž) and 𝛼(𝑒)≀𝛽(𝑒) for all π‘’βˆˆπ΄ and β„Žβˆˆπ‘ˆ. Conversely, suppose that π΄βŠ†π΅, πœ‡ξ‚πΉ(𝑒)(β„Ž)β‰€πœ‡ξ‚πΊ(𝑒)(β„Ž) and 𝛼(𝑒)≀𝛽(𝑒) for all π‘’βˆˆπ΄ and β„Žβˆˆπ‘ˆ. We can easily verify that 𝐹𝛼βˆͺ𝐺𝛽=𝐺𝛽. Thus 𝐹𝛼≀1𝐺𝛽.

For operators β‹“ and ξ‚βˆ©, we can obtain similar results as follows.

Theorem 4.6. Let 𝐹𝛼 and 𝐺𝛽 be GIVFS sets over (π‘ˆ,𝐸). Then the following hold: (1)(𝐹𝛼⋓𝐹𝛼𝐹)=𝛼, (2)(ξ‚πΉπ›Όξ‚βˆ©ξ‚πΉπ›Όξ‚πΉ)=𝛼, (3)(𝐹𝛼⋓𝐺𝛽)ξ‚βˆ©ξ‚πΉπ›Ό=𝐹𝛼,(4)(ξ‚πΉπ›Όξ‚βˆ©ξ‚πΊπ›½ξ‚πΉ)⋓𝛼=𝐹𝛼.

Theorem 4.7. Let 𝐹𝛼, 𝐺𝛽 and 𝐻𝛾 be 𝐺𝐼𝑉𝐹𝑆 sets over (U,E). Then the following hold: (1)ξ‚πΉπ›Όξ‚ξ‚πΊβˆ©(𝛽⋓𝐻𝛾𝐹)=(π›Όξ‚βˆ©ξ‚πΊπ›½ξ‚πΉ)β‹“(π›Όξ‚βˆ©ξ‚π»π›Ύ), (2)𝐹𝛼𝐺⋓(π›½ξ‚βˆ©ξ‚π»π›Ύξ‚πΉ)=(𝛼⋓𝐺𝛽)ξ‚ξ‚πΉβˆ©(𝛼⋓𝐻𝛾).

Theorem 4.8. (1)(𝔖(π‘ˆ,𝐸),β‹“,∩) is a distributive lattice.
(2) Let ≀2 be the order relation in 𝔖(π‘ˆ,𝐸) and 𝐹𝛼,ξ‚πΊπ›½βˆˆξ‚π”–(π‘ˆ,𝐸). 𝐹𝛼≀2𝐺𝛽 if and only if π΅βŠ†π΄,πœ‡ξ‚πΉ(𝑒)(β„Ž)β‰€πœ‡ξ‚πΊ(𝑒)(β„Ž) and 𝛼(𝑒)≀𝛽(𝑒) for all π‘’βˆˆπ΅.

It is worth noting that (𝔖(π‘ˆ,𝐸),β‹’,β‹“) and (𝔖(π‘ˆ,𝐸),∩,βˆͺ) are not lattices, as the absorption laws of them do not hold necessarily. To illustrate this, we give an example as follows.

Example 4.9. Let π‘ˆ={β„Ž1,β„Ž2,β„Ž3} be the universe, 𝐸={𝑒1,𝑒2,𝑒3} the set of parameters, 𝐴={𝑒1,𝑒2}, 𝐡={𝑒2,𝑒3}. The GIVFS sets 𝐹𝛼 and 𝐺𝛽 over (π‘ˆ,𝐸) are given as𝐹𝛼𝑒1ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.5,0.72[],β„Ž0.3,0.43[]ξ‚Ό,[]ξ‚Ά,𝐹0.6,0.70.8,0.9𝛼𝑒2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.6,0.82[],β„Ž0.2,0.33[]ξ‚Ό,[]ξ‚Ά,𝐺0.7,0.90.4,0.5𝛽𝑒2ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.1,0.32[],β„Ž0.4,0.53[]ξ‚Ό,[]ξ‚Ά,𝐺0.5,0.60.6,0.8𝛽𝑒3ξ€Έ=β„Žξ‚΅ξ‚»1[],β„Ž0.3,0.42[],β„Ž0.5,0.83[]ξ‚Ό,[]ξ‚Ά.0.4,0.60.5,0.7(4.1)
Suppose that (𝐹𝛼⋓𝐺𝛽𝐹)⋒𝛼=𝐻𝛾. Then 𝐢=𝐴∩𝐡={𝑒2}≠𝐴. So 𝐻𝛾≠𝐹𝛼, that is, (𝐹𝛼⋓𝐺𝛽𝐹)⋒𝛼≠𝐹𝛼.
Again, suppose that the parameters set of a GIVFS set 𝐽𝛿 is denoted by 𝐷, and (ξ‚πΉπ›Όξ‚βˆ©ξ‚πΊπ›½)βˆͺ𝐹𝛼=𝐽𝛿. Then 𝐷=𝐴βˆͺ𝐡={𝑒1,𝑒2,𝑒3}≠𝐴, Therefore, 𝐽𝛿≠𝐹𝛼, that is, (ξ‚πΉπ›Όξ‚βˆ©ξ‚πΊπ›½)βˆͺ𝐹𝛼≠𝐹𝛼.

5. An Application of GIVFS Sets

In this section we present a simple application of GIVFS set in an interval-valued fuzzy decision making problem. We first give the following definition.

Definition 5.1. Let 𝐹𝛼 be a GIVFS set, β„Žπ‘–,β„Žπ‘—βˆˆπ‘ˆ,π‘’π‘˜βˆˆπ΄. One says membership value of β„Žπ‘— lowerly exceeds or equals to the membership value of β„Žπ‘– with respect to the parameter π‘’π‘˜ if πœ‡βˆ’ξ‚πΉ(π‘’π‘˜)(β„Žπ‘–)β‰€πœ‡βˆ’ξ‚πΉ(π‘’π‘˜)(β„Žπ‘—). The corresponding characteristic function is defined as follows: π‘“βˆ’π‘’π‘˜ξ€·β„Žπ‘–,β„Žπ‘—ξ€Έ=ξƒ―1,ifπœ‡βˆ’ξ‚πΉξ€·π‘’π‘˜ξ€Έξ€·β„Žπ‘–ξ€Έβ‰€πœ‡βˆ’ξ‚πΉξ€·π‘’π‘˜ξ€Έξ€·β„Žπ‘—ξ€Έ,0,otherwise.(5.1)

Definition 5.2. Let 𝐹𝛼 be a GIVFS set, β„Žπ‘–,β„Žπ‘—βˆˆπ‘ˆ,π‘’π‘˜βˆˆπ΄. One says membership value of β„Žπ‘— upperly exceeds or equals to the membership value of β„Žπ‘– with respect to the parameter π‘’π‘˜ if πœ‡+𝐹(π‘’π‘˜)(β„Žπ‘–)β‰€πœ‡+𝐹(π‘’π‘˜)(β„Žπ‘—). The corresponding characteristic function is defined as follows: 𝑓+π‘’π‘˜ξ€·β„Žπ‘–,β„Žπ‘—ξ€Έ=ξƒ―1,ifπœ‡+ξ‚πΉξ€·π‘’π‘˜ξ€Έξ€·β„Žπ‘–ξ€Έβ‰€πœ‡+ξ‚πΉξ€·π‘’π‘˜ξ€Έξ€·β„Žπ‘—ξ€Έ;0,otherwise.(5.2)

Remark 5.3. Let 𝐹𝛼 be a GIVFS set, β„Žπ‘–,β„Žπ‘—βˆˆπ‘ˆ,andπ‘’π‘˜βˆˆπ΄. For convenience, we denote the vectors (π‘“βˆ’π‘’π‘˜(β„Žπ‘–,β„Žπ‘—),𝑓+π‘’π‘˜(β„Žπ‘–,β„Žπ‘—)) and (π›Όβˆ’(π‘’π‘˜),𝛼+(π‘’π‘˜)) as ξ„¨π‘“π‘’π‘˜(β„Žπ‘–,β„Žπ‘—) and 𝛼(π‘’π‘˜), respectively.

Now we can define the generalised comparison table about GIVFS set 𝐹𝛼.

Definition 5.4. Let 𝐹𝛼 be a GIVFS set. The generalised comparison table about 𝐹𝛼 is a square table in which the number of rows and number of columns are equal. Both rows and columns are labeled by the object names of the universe such as β„Ž1,β„Ž2,…,β„Žπ‘›, and the entries are 𝐢𝑖𝑗, given as follows: 𝐢𝑖𝑗=π‘šξ“π‘˜=1ξ‚€ξ„¨π‘“π‘’π‘˜ξ€·β„Žπ‘–,β„Žπ‘—ξ€Έβ‹…ξƒ½π›Όξ€·π‘’π‘˜ξ€Έξ‚,𝑖,𝑗=1,2,…,𝑛.(5.3)

Clearly, for 𝑖,𝑗=1,…,𝑛, π‘˜=1,…,π‘š, 0≀𝐢𝑖𝑗≀2π‘š, and 𝐢𝑖𝑖=βˆ‘π‘šπ‘˜=1(π›Όβˆ’(π‘’π‘˜)+𝛼+(π‘’π‘˜)), where 𝑛 and π‘š are the numbers of objects and parameters present in a GIVFS set, respectively.

Remark 5.5. The generalised comparison table is different from the comparison table in [30]. First, the comparison in the generalised comparison table is between two interval values, instead of two single values. Second, the entries 𝐢𝑖𝑗 of the generalised comparison table are numbers of real interval [0,1] in general, instead of single values 0 and 1. Hence, the generalised comparison table is an extension of the comparison table in [30]. If each interval degenerates to a point and 𝛼(𝑒)=1 for each π‘’βˆˆπ΄, then the generalised comparison table will be degenerate to the comparison table in [30].

In the generalised comparison table, the row sum and the column sum of an object β„Žπ‘– are denoted by 𝑝𝑖 and π‘žπ‘–, respectively, and the score of an object β„Žπ‘– is denoted as 𝑆𝑖 which can be given by 𝑆𝑖=π‘π‘–βˆ’π‘žπ‘–. Now we present an algorithm as follows.

Algorithm 5.6. (1) Input the objects set π‘ˆ and the parameter set π΄βŠ†πΈ.
(2) Consider the GIVFS set 𝐹𝛼 in tabular form.
(3) By calculating the entries 𝐢𝑖𝑗, construct generalised comparison table.
(4) Compute the score of each β„Žπ‘– using row sum and the column sum.
(5) The optimal decision is to select β„Žπ‘˜ if the score of β„Žπ‘˜ is maximum.
(6) If π‘˜ has more than one value then any one of β„Žπ‘˜ may be chosen.

To illustrate the basic idea of the above algorithm, let us consider the following example.

Example 5.7. Let us consider a GIVFS set which describes the capability of the candidates who are wanted to fill a position for a company. Suppose that there are six candidates in the universe π‘ˆ={β„Ž1,β„Ž2,β„Ž3,β„Ž4,β„Ž5,β„Ž6} under consideration, and 𝐸={𝑒1,𝑒2,𝑒3,𝑒4,𝑒5,𝑒6} is the set of decision parameters, where 𝑒𝑖(𝑖=1,2,3,4,5,6) stands for the parameters β€œexperience”, β€œcomputer knowledge”, β€œyoung age”, β€œhigher education”, β€œgood health”, and β€œover-married”, respectively.

Here, the degree of possibility of belongingness of the parameter 𝑒𝑖 can be interpreted as the degree of importance of the parameter to the position. Our purpose is to find out the best candidate for the company based on her expected parameters. Suppose that the company do not consider the parameter β€œover-married”; that is, the degree of importance of parameter 𝑒6 is regarded as 0. In this case, let 𝐴={𝑒1,𝑒2,𝑒3,𝑒4,𝑒5}βŠ‚πΈ, and let π›ΌβˆΆπ΄β†’Int([0,1]) be an interval-valued fuzzy subset of 𝐴, which is given by the company as follows: 𝛼(𝑒1