# General Complex-Valued Grouping Functions

Authors: Ying Chen; Lvqing Bi; Bo Hu; Songsong Dai
Journal: Journal of Mathematics
 (2021)

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## Abstract

Grouping function is a special kind of aggregation function which measures the amount of evidence in favor of either of the two choices. Recently, complex fuzzy sets have been successfully used in many fields. This paper extends the concept of grouping functions to the complex-valued setting. We introduce the concepts of complex-valued grouping, complex-valued 0-grouping, complex-valued 1-grouping, and general complex-valued grouping functions. We present some interesting results and construction methods of general complex-valued grouping functions.


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## Body

## 1. Introduction

In 2012, Bustince et al. [1] introduced the concept of grouping function as a special type of aggregation functions [2]. A grouping function measures the amount of evidence in favor of either of two choices in decision making. It plays an important role in many aspects of applications such as image processing [1, 3], classification [4, 5], and decision making [6, 7].In this paper, we consider complex-valued grouping function, which is both a generation of real-valued grouping function and a special kind of complex fuzzy aggregation operator. In the literature, one can find many works on real-valued grouping functions. The concepts of general grouping functions [8] and interval-valued grouping functions [9, 10] have been proposed. Some properties incluing migrativity, homogeneity, idempotency, and distributivity of grouping functions have been studied [1, 11–14]. The multiplicative generators [15] and additive generators [16] of grouping functions have been investigated. (G, N)-implications [17] and QL-implications [18] derived from grouping functions have been constructed.Meanwhile, we can also find several works on complex fuzzy aggregation operators and related concepts. In 2002, Ramot et al. [19, 20] introduced the concepts of complex fuzzy sets and complex fuzzy aggregation operators, which have been successfully used in signal processing [20–22], time series forecasting [23–25], and decision making [26, 27]. Distance and entropy measures of complex fuzzy sets and its extensions have been proposed [28–30]. Moreover, some new concepts such as orthogonality and rotational invariance have been presented for complex fuzzy sets and complex fuzzy aggregation operators [31–33].Recently, Chen et al. [34] introduced the concept of complex-valued overlap function, which measures the overlapping degree between two objects with complex-valued information. In order to measure the amount of evidence in favor of either of two choices with complex-valued information, this paper extends traditional real-valued grouping functions to complex-valued grouping functions. As mentioned in [34], some features of complex fuzzy sets can lead to special properties for complex-valued overlap functions.Both real-valued grouping functions and complex fuzzy aggregation operators have gained a rapid development both in application and theory. However, as far as we know, nowadays, there are no corresponding discussions to propose the complex-valued grouping functions. Therefore, in this paper, we introduce the concepts of complex-valued grouping functions. This paper is organized as follows. In Section2, we recall the concepts of grouping functions. In Section 3, we introduce complex-valued grouping functions and their properties. In Section 4, we introduce construction methods of general complex-valued grouping functions. Conclusions are given in Section 5.


## 2. Preliminaries

In this section, we recall the concepts of bivariate grouping functions and n-dimensional grouping functions [1, 8, 12, 16].


### 2.1. Overlap Functions

Definition 1 (see [1]).
A mappingG:0,12⟶0,1 is a grouping function if it is symmetric, nondecreasing, continuous, and has the following properties:(G1)
Ga,b=0 if and only if a=b=0(G2)
Ga,b=1if and only ifa=1orb=1
As introduced in [16], a mapping G:0,12⟶0,1 is a 0-grouping function if we replace the property (G1) by the following: (G1’)a=b=0⟹Ga,b=0 without changing others.
Similarly, a mappingG:0,12⟶0,1 is a 1-grouping function if we replace the property (G2) by the following:(G2’)a=1orb=1⟹Oa,b=1 without changing others.Definition 2 (see [12]).
Ann-ary mapping Gn:0,1n⟶0,1 is an n-dimensional grouping function if it is commutative, nondecreasing, continuous, and has the following properties:Gn1Gna1,…,an=0 if and only if ai=0 for all i=1,2,…,nGn2Gna1,…,an=1 if and only if there exists i∈1,2,…,n such that ai=1
Analogously, ann-ary mapping Gn:0,1n⟶0,1 is an n-dimensional 0-grouping function if we replace the property Gn1 by the following:(Gn1’) If ai=0 for all i=1,2,…,n, then Gna1,…,an=0 without changing others.
Ann-ary mapping Gn:0,1n⟶0,1 is an n-dimensional 1-grouping function if we replace the property Gn2 by the following:(Gn2’) If there exists i∈1,2,…,n such that ai=1, then Gna1,…,an=1 without changing others.
Based on the concepts ofn-dimensional 0-grouping and 1-grouping functions, the general grouping functions is defined as follows.Definition 3 (see [8]).
A mappingGn:0,1n⟶0,1 is an n-dimensional general grouping function if it is commutative, nondecreasing, continuous, and has the following properties:Ggn1 If ∑i=1nai=0 then Gna1,…,an=0Ggn2 If there exists i∈1,2,…,n such that ai=1, then Gna1,…,an=1


## 2.1. Overlap Functions

Definition 1 (see [1]).
A mappingG:0,12⟶0,1 is a grouping function if it is symmetric, nondecreasing, continuous, and has the following properties:(G1)
Ga,b=0 if and only if a=b=0(G2)
Ga,b=1if and only ifa=1orb=1
As introduced in [16], a mapping G:0,12⟶0,1 is a 0-grouping function if we replace the property (G1) by the following: (G1’)a=b=0⟹Ga,b=0 without changing others.
Similarly, a mappingG:0,12⟶0,1 is a 1-grouping function if we replace the property (G2) by the following:(G2’)a=1orb=1⟹Oa,b=1 without changing others.Definition 2 (see [12]).
Ann-ary mapping Gn:0,1n⟶0,1 is an n-dimensional grouping function if it is commutative, nondecreasing, continuous, and has the following properties:Gn1Gna1,…,an=0 if and only if ai=0 for all i=1,2,…,nGn2Gna1,…,an=1 if and only if there exists i∈1,2,…,n such that ai=1
Analogously, ann-ary mapping Gn:0,1n⟶0,1 is an n-dimensional 0-grouping function if we replace the property Gn1 by the following:(Gn1’) If ai=0 for all i=1,2,…,n, then Gna1,…,an=0 without changing others.
Ann-ary mapping Gn:0,1n⟶0,1 is an n-dimensional 1-grouping function if we replace the property Gn2 by the following:(Gn2’) If there exists i∈1,2,…,n such that ai=1, then Gna1,…,an=1 without changing others.
Based on the concepts ofn-dimensional 0-grouping and 1-grouping functions, the general grouping functions is defined as follows.Definition 3 (see [8]).
A mappingGn:0,1n⟶0,1 is an n-dimensional general grouping function if it is commutative, nondecreasing, continuous, and has the following properties:Ggn1 If ∑i=1nai=0 then Gna1,…,an=0Ggn2 If there exists i∈1,2,…,n such that ai=1, then Gna1,…,an=1


## 3.N-Dimensional Complex-Valued Grouping Functions

LetD=α∈ℂ|α≤1, each a∈D is of the form raejθa, where j=−1, the amplitude term ra∈ℝ and the phase term θa∈0,2π.We definen-dimensional complex-valued grouping functions.Definition 4.
Ann-ary mapping Cgn:Dn⟶D is an n-dimensional complex-valued grouping function if it is commutative, continuous, and has the following properties:Cgn1Cgna1,…,an=0 if and only if ai=0 for all i=1,2,…,nCgn2Cgna1,…,an=1 if and only if there exists i∈1,2,…,n such that ai=1Cgn3Cgn is amplitude monotonic in the first variate: Cgna,a2,…,an≤Cgnb,a2,…,anwhena≤b
SinceCgn is commutative, it is also amplitude monotonic in any other variate based on the third property Cgn3. Obviously, these properties are analogous to those of Definition 1. When the domain is limited to [0, 1], it reduces to an n-dimensional real-valued grouping function of Definition 2.Example 1.
Nevertheless, there exists mapping such that is a grouping function in the domain [0, 1] but not a complex-valued grouping function. The function given as follows:(1)fa,b=1−1−a1−b,is a grouping function but not a complex-valued grouping function; for example, f0.5,−1=0, then Cgn1 does not hold.
Traditional real-valued grouping functions are the dual notion of overlap functions.fa,b is an overlap function on [0, 1], and ga,b=1−f1−a,1−b is a grouping function. However, this is not true for complex-valued overlap functions. For example, fa,b=ab is a complex-valued overlap function on D in [34], but ga,b=1−1−a1−b is not a complex-valued grouping function.
Similarly, we introduce some types of grouping functions, such as n-dimensional complex-valued 0-overlap and 1-overlap grouping functions.
Ann-ary mapping Cgn:Dn⟶D is an n-dimensional complex-valued 0-grouping function if we replace the property Cgn1 by the following:(Cgn1’) If ai=0 for all i=1,2,…,n, then Cgna1,…,an=0 without changing others.
Ann-ary mapping Cgn:Dn⟶D is an n-dimensional complex-valued 1-grouping function if we replace the property Cgn2 by the following:(Cgn2’) If there exists i∈1,2,…,n such that ai=1 then Cgna1,…,an=1 without changing others.
Based on these concepts, we define the concept ofn-dimensional general complex-valued grouping functions.Definition 5.
Ann-ary mapping GCgn:Dn⟶D is an n-dimensional general complex-valued grouping function if it is commutative, continuous, and has the following properties:GCgn1 If ai=0 for all i=1,2,…,n, then GCgna1,…,an=0GCgn2 If there exists i∈1,2,…,n such that ai=1 then GCgna1,…,an=1GCgn3GCgn is amplitude monotonic in the first variate: GCgna,a2,…,an≤GCgnb,a2,…,an when a≤b
The relations betweenn-dimensional complex-valued grouping functions, complex-valued 0-grouping functions, complex-valued 1-grouping functions, and general complex-valued grouping functions are given as follows.Proposition 1.
Letg:Dn⟶D be an n-ary mapping, then(1)
If it is ann-dimensional complex-valued grouping function, then it is an n-dimensional complex-valued 0-grouping and 1-grouping function(2)
If it is ann-dimensional complex-valued 0-grouping (or 1-grouping) function, then it is also a general complex-valued grouping function
This relation between complex-valued grouping functions is similar to that between interval-valued overlap functions [9].Now, we give several examples to demonstrate their relations of complex-valued grouping functions.Example 2.
The binary functiong:D2⟶D given by(2)ga,b=maxa,b⋅eθa∧θb,is a complex-valued overlap function.Example 3.
The binary functionf:D2⟶D given by(3)fa,b=a∨b,is a general complex-valued grouping function and complex-valued 1-grouping function, but neither a complex-valued grouping function nor a complex-valued 0-grouping function; for example, f−1,−1=1, then the property Cgn2 does not hold.Example 4.
The binary functionh:D2⟶D given by(4)ha,b=1,a=1orb=1,0,otherwise,is a general complex-valued grouping function and complex-valued 0-grouping function. However, h is not a complex-valued 1-grouping function; for example, f0.5,0.5=0, then the property Cgn1 does not hold.
Negative operation (–) is closed onD but not closed on [0, 1]. Then, we have following property only for complex-valued function on D.Definition 6.
Then-ary function g:Dn⟶D is symmetric with respect to the point 0, if(5)ga1,…,an=g−a1,−a2,…,−an,holds for any a1,…,an∈D.Example 5.
Then-ary function g:Dn⟶D given by(6)ga1,…,an=minn−∑i=1n1−ai,1⋅eθa1∧⋯∧θan,is a complex-valued grouping function, which is symmetric with respect to the point 0.


## 4. Construction of General Complex-Valued Overlap Functions

Proposition 2.
If ann-ary mapping φ:Dn⟶D is n-dimensional complex-valued grouping (0-grouping, 1-grouping, or general grouping) function which is expressed as(7)φa1,…,an=ϕra1,ra2,…,ranejψθa1,θa2,…,θan,then the function ϕ is an n-dimensional grouping (0-grouping, 1-grouping, or general grouping) function on [0, 1].Theorem 1.
If then-ary function ϕ:0,1n⟶0,1 is an n-dimensional 1-grouping function, the function ψ:0,2πn⟶0,2π satisfies the following properties:(i)
ψ is commutative(ii)
There existsi∈1,2,…,n such that θai=0 if and only if ψθa1,…,θan=0(iii)
ψ is continuous
Then, the functionφ:Dn⟶D defined by equation (7) is an n-dimensional complex-valued 1-grouping function.Proof.
It is immediate thatφ is commutative, amplitude monotonic, and continuous. Now, we prove the properties Cgn1 and (Cgn2’).Cgn1: ⇒ If φa1,…,an=ϕra1,ra2,…,ranejψθa1,θa2,…,θan=0, then ϕra1,ra2,…,ran=0. Then, rai=0 for all i=1,2,…,n since ϕ is a 1-grouping function. Then, ai=0 for all i=1,2,…,n.Cgn1: ⇐ If ai=0 for all i=1,2,…,n, this means rai=0 for all i=1,2,…,n, and then ϕra1,ra2,…,ran=0 since ϕ is a 1-grouping function. Then, φa1,…,an=ϕra1,ra2,…,ranejψθa1,θa2,…,θan=0⋅ejψθa1,θa2,…,θan=0.(Cgn2’): ⇐ If there exists i∈1,2,…,n such that ai=1, i.e., rai=1 and θai=0, then ϕra1,ra2,…,ran=1 since ϕ is a 1-grouping function, and ψθa1,θa2,…,θan=0 since ψ satisfies (ii). Then, φa1,…,an=ϕra1,ra2,…,ranejψθa1,θa2,…,θan=1⋅ej0=1.Corollary 1.
If then-ary function ϕ:0,1n⟶0,1 is an n-dimensional general grouping function, the function ψ:0,2πn⟶0,2π satisfies the following properties:(i)
ψ is commutative(ii)
There existsi∈1,2,…,n such that θai=0 if and only if ψθa1,…,θan=0(iii)
ψ is continuous
Then, the functionφ:Dn⟶D defined by equation (7) is an n-dimensional general complex-valued grouping function.However, this method cannot obtain complex-valued grouping function. For example, we getrai=1 from ϕ is a grouping function and θak=0 from property (ii), but i≠k; thus, we cannot get ai=1. So, we consider the construction of general complex-valued overlap functions. If the n-dimensional complex-valued function φ:Dn⟶D is defined by equation (7), then we can easily see that it is a key step to construct the function ψ:0,2πn⟶0,2π, which satisfies the condition (ii) of Corollary 1.Now, we give some examples of bivariate functionsψ:0,2π2⟶0,2π satisfying the condition (ii) of Corollary 1.Example 6.
Then-ary function ψ1:0,2πn⟶0,2π given by(8)ψ1θ1,θ2,…,θn=θ12π×θ22π×⋯×θn2π,satisfies the condition (ii) of Corollary 1.
Then-ary function ψ2:0,2πn⟶0,2π given by(9)ψ2θ1,θ2,…,θn=θ1∧θ2∧⋯∧θn,satisfies the condition (ii) of Corollary 1.
Based on these functions, we give the general complex-valued grouping functions.
The functionφ1:Dn⟶D given by(10)φ1a1,…,an=ra1∨ra2∨⋯∨ran⋅ejψ1θ1,θ2,…,θn,is a general complex-valued grouping function.
The functionφ2:Dn⟶D given by(11)φ2a1,…,an=minra1+ra2+⋯+ran,1⋅ejψ2θ1,θ2,…,θn,is a general complex-valued grouping function.
Now, we consider the way of constructing new complex-valued grouping functions from some given complex-valued grouping functions. In the case of real-valued grouping functions, it is possible to use some aggregation operations to obtain a new grouping function from some given grouping functions. Unfortunately, this approach fails for complex fuzzy arithmetic aggregation because it does not satisfy the property of amplitude monotonicity [35]. This means the weighted sum cannot be used to construct new complex-valued grouping functions.Example 7.
From Examples2 and 3, ga,b=maxa,b⋅eθa∧θb and fa,b=maxa,b are complex-valued general grouping functions. However, ha,b=ga,b+fa,b/2 is not a complex-valued general grouping function since it does not satisfy GCgn3. We give a example such that a1<a2 but ha1,b>ha2,b; let a1=0.5e2π/3, a2=0.9eπ, and b=0.5e2π/3, then(12)h0.5e2π/3,0.5e2π/3=0.5e2π/3+0.52=0.25,h0.9eπ,0.5e2π/3=0.9eπ+0.52=0.2.
Thus,h0.5e2π/3,0.5e2π/3>h0.9eπ,0.5e2π/3.
However, it is possible to use product operation to obtain a new grouping function from some given grouping functions. The product ofψ,φ:Dn⟶D is defined informally by(13)ψ⋅φa1,…,an=ψa1,…,an×φa1,…,an.Theorem 2.
Letψ,φ be two n-dimensional general complex-valued grouping functions, then t th power of ψ⋅φ also is an n-dimensional general complex-valued grouping function.Proof.
It is immediate thatφ is commutative and continuous. Now, we prove the properties GCgn1, GCgn2, and GCgn3.GCgn1: If ai=0 for all i=1,2,…,n, then ψa1,…,an=0 and φa1,…,an=0 since ϕ,φ are n-dimensional general complex-valued grouping functions. Then, ψta1,…,an=0…0=0.GCgn2: If there exists i∈1,2,…,n such that ai=1, then ϕa1,a2,…,an=φa1,a2,…,an=1 since ϕ,φ are n-dimensional general complex-valued grouping functions. Then, ψta1,…,an=1…1=1.GCgn3: If a≤b, then ψa,a2,…,an≤ψb,a2…,an and φa,a2,…,an≤φb,a2…,an since ϕ,φ are n-dimensional general complex-valued grouping function. Then, ψa,a2,…,an×φa,a2,…,an=ψa,a2,…,an×φa,a2,…,an≤ψb,a2,…,an×φb,a2,…,an=ψtb,a2,…,an×φb,a2,…,an.
Similarly, we can get the following result.Theorem 3.
Letψ,φ be two n-dimensional general complex-valued 1-grouping functions, then their product ψ⋅φ also is an n-dimensional complex-valued 1-grouping function.Proof.
From Theorem2 and the definition of n-dimensional complex-valued 1-grouping function, we only need to prove the following property:Cgn1: ⇒ If ψ⋅φa1,…,an=ψa1,…,an×φa1,…,an=0, then φa1,…,an=0 or ψa1,…,an=0. If ψa1,…,an=0, then ai=0 for all i=1,2,…,n since ϕ is an n-dimensional general complex-valued 1-grouping function. We have same result when φa1,…,an=0.
Now, we consider the power of complex-valued grouping function. First, for positive integerst, the t th power of ψ:Dn⟶D is defined informally by(14)ψta1,…,an=ψa1,…,an×⋯×ψa1,…,an︸ttimes.Theorem 4.
Letψ be an n-dimensional general complex-valued grouping function, then t th power of ψ also is an n-dimensional general complex-valued grouping function.Proof.
It is immediate thatφ is commutative and continuous. Now, we prove the properties GCgn1, GCgn2, and GCgn3.GCgn1: If ai=0 for all i=1,2,…,n, then ψa1,…,an=0 since ϕ is an n-dimensional general complex-valued grouping function. Then, ψta1,…,an=0…0=0.GCgn2: If there exists i∈1,2,…,n such that ai=1, then ϕa1,a2,…,an=1 since ϕ is an n-dimensional general complex-valued grouping function. Then, ψta1,…,an=1…1=1.GCgn3: If a≤b, then ψa,a2,…,an≤ψb,a2…,an since ϕ is an n-dimensional general complex-valued grouping function. Then, ψta,a2,…,an=ψa,a2,…,ant≤ψb,a2,…,ant=ψtb,a2,…,an.
Similarly, we can get the following result.Theorem 5.
Letψ be an n-dimensional general complex-valued 1-grouping function, then t th power of ψ also is an n-dimensional complex-valued 1-grouping function.Proof.
From Theorem4 and the definition of n-dimensional complex-valued 1-grouping function, we only need to prove the following property:Cgn1: ⇒ If ψta1,…,an=ψa1,…,ant=0, then ψa1,…,an=0. Then, ai=0 for all i=1,2,…,n since ϕ is an n-dimensional general complex-valued 1-grouping functionRemark 1.
In the unit interval [0, 1],a⋅b=1 if and only if a=b=1. However, it is not true for the complex unit disk D since −1⋅−1=1. We cannot establish a theorem similar to Theorem 2 or Theorem 4 for n-dimensional complex-valued grouping (or 0-grouping) function.


## 5. Conclusions

This paper introduced the concepts of complex-valued grouping, complex-valued 0-grouping, complex-valued 1-grouping, and general complex-valued grouping functions. Then, we presented some construction methods for general complex-valued grouping functions. Our method includes the construction of a general real-valued grouping function and a continuous, commutative functionψ:0,2πn⟶0,2π satisfying the following property:(i)
There existsi∈1,2,…,n such that θai=0 if and only if ψθa1,…,θan=0mod2πOf course, the application and properties of complex-valued overlap/grouping functions need further investigation.



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