# General Complex-Valued Overlap Functions

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

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## Abstract

Overlap function is a special type of aggregation function which measures the degree of overlapping between different classes. Recently, complex fuzzy sets have been successfully applied in many applications. In this paper, we extend the concept of overlap functions to the complex-valued setting. We introduce the notions of complex-valued overlap, complex-valued 0-overlap, complex-valued 1-overlap, and general complex-valued overlap functions, which can be regarded as the generalizations of the concepts of overlap, 0-overlap, 1-overlap, and general overlap functions, respectively. We study some properties of these complex-valued overlap functions and their construction methods.


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## Body

## 1. Introduction

Bustince et al. [1] introduced the concept of overlap function in order to express the overlapping degree between two different classes. Overlap functions are a special type of aggregation functions [2] that are used in many applications such as image processing [1, 3], classification [4, 5], and decision making [6, 7]. It has gained a rapid development with various forms. The concepts of Archimedean, general, 0-overlap, 1-overlap, n-dimensional, interval-valued overlap functions have been proposed [8–11]. Various properties including migrativity, distributivity, idempotency, and homogeneity of overlap functions have been investigated [7, 8, 12–17]. The additive generators [11, 18] and multiplicative generators [19] of overlap functions have been given. Implications derived from overlap functions have been studied [20, 21].Ramot et al. [22, 23] introduced the concept of complex fuzzy sets. It is an effective tool to handle uncertainty and periodicity simultaneously. It has been successfully applied in signal processing [23–25], image processing [26], time series forecasting [27–30], and decision making [31, 32]. Different measures including distance, similarity, and entropy of complex fuzzy sets have been proposed [33–38]. Various properties including δ-equality, parallelity, orthogonality, and rotational invariance of complex fuzzy sets have been investigated [39–42].Complex fuzzy sets have been successfully applied in many different fields. In some cases, overlapping degree is needed for complex-valued information of two or more objects. In this paper, we extend traditional real-valued overlap functions to complex-valued overlap function. The starting point is that complex-valued overlap differs from other real-valued overlap functions. For example,ej·x (j=−1) is a periodic function and negative operation (–) is closed in the range of complex unit circle. These features may lead to special properties and construction methods of complex-valued overlap functions and provide a good issue for generation of overlap functions. As far as we know, nowadays, there are no corresponding discussions to propose the complex-valued overlap functions. Therefore, in this paper, from the theoretical point of view, we propose the definitions and construction methods of complex-valued overlap functions.This paper is organized as follows. In Section2, we recall the concepts of overlap functions. In Section 3, we introduce complex-valued overlap functions and their properties. In Section 4, we propose some construction methods of complex-valued overlap functions. Conclusions are given in Section 5.


## 2. Preliminaries

In this section, we recall some concepts of bivariate overlap functions and n-dimensional overlap functions, which are largely studied [1, 10, 11].


### 2.1. Overlap Functions

Definition 1.
(see [1]). A mapping O:0,12⟶0,1 is an overlap function if, for all a,b∈0,1, it is commutative, nondecreasing, and continuous and satisfies the following conditions:(O1) Oa,b=0 if and only if ab=0;(O2) Oa,b=1 if and only if ab=1.As introduced in [11], a mapping O:0,12⟶0,1 is a 0-overlap function if we loose the condition (O1) to (O1’) ab=0⇒Oa,b=0 without changing any other condition.Similarly, a mappingO:0,12⟶0,1 is a 0-overlap function if we loose the condition (O2) to (O2’) ab=1⇒Oa,b=1 without changing any other condition.Definition 2.
(see [10]). A mapping On:0,1n⟶0,1 is a n-dimensional overlap function if, for all a1,…,an∈0,1, it is commutative, nondecreasing, and continuous and satisfies the following conditions:(On1) Ona1,…,an=0 if and only if ∏i=1nai=0;(On2) Ona1,…,an=1 if and only if ∏i=1nai=1.As introduced in [11], a mapping On:0,1n⟶0,1 is an n-dimensional 0-overlap function if we loose condition (On1) to (On1’) ∏i=1nai=0⇒Ona1,…,an=0 without changing any other condition.Analogously, a mappingOn:0,1n⟶0,1 is an n-dimensional 1-overlap function if we loose condition (On2) to (On2’) ∏i=1nai=1⇒Ona1,…,an=1without changing any other condition.Based on the concepts of n-dimensional 0-overlap and 1-overlap functions, the general overlap functions are defined as follows:Definition 3.
(see [10]). A mapping On:0,1n⟶0,1 is an n-dimensional general overlap function if, for all a1,…,an∈0,1, it is commutative, nondecreasing, and continuous and satisfies the following conditions:(GOn1) if ∏i=1nai=0, then Ona1,…,an=0;(GOn2) if ∏i=1nai=1, then Ona1,…,an=1.


## 2.1. Overlap Functions

Definition 1.
(see [1]). A mapping O:0,12⟶0,1 is an overlap function if, for all a,b∈0,1, it is commutative, nondecreasing, and continuous and satisfies the following conditions:(O1) Oa,b=0 if and only if ab=0;(O2) Oa,b=1 if and only if ab=1.As introduced in [11], a mapping O:0,12⟶0,1 is a 0-overlap function if we loose the condition (O1) to (O1’) ab=0⇒Oa,b=0 without changing any other condition.Similarly, a mappingO:0,12⟶0,1 is a 0-overlap function if we loose the condition (O2) to (O2’) ab=1⇒Oa,b=1 without changing any other condition.Definition 2.
(see [10]). A mapping On:0,1n⟶0,1 is a n-dimensional overlap function if, for all a1,…,an∈0,1, it is commutative, nondecreasing, and continuous and satisfies the following conditions:(On1) Ona1,…,an=0 if and only if ∏i=1nai=0;(On2) Ona1,…,an=1 if and only if ∏i=1nai=1.As introduced in [11], a mapping On:0,1n⟶0,1 is an n-dimensional 0-overlap function if we loose condition (On1) to (On1’) ∏i=1nai=0⇒Ona1,…,an=0 without changing any other condition.Analogously, a mappingOn:0,1n⟶0,1 is an n-dimensional 1-overlap function if we loose condition (On2) to (On2’) ∏i=1nai=1⇒Ona1,…,an=1without changing any other condition.Based on the concepts of n-dimensional 0-overlap and 1-overlap functions, the general overlap functions are defined as follows:Definition 3.
(see [10]). A mapping On:0,1n⟶0,1 is an n-dimensional general overlap function if, for all a1,…,an∈0,1, it is commutative, nondecreasing, and continuous and satisfies the following conditions:(GOn1) if ∏i=1nai=0, then Ona1,…,an=0;(GOn2) if ∏i=1nai=1, then Ona1,…,an=1.


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

LetD=α∈ℂ‖α|≤1, then we define n-dimensional complex-valued overlap functions.Definition 4.
A mappingCOn:Dn⟶D is an n-dimensional complex-valued overlap function if, for all a1,…,an∈D, it is commutative and continuous and satisfies the following conditions:(COn1) COna1,…,an=0 if and only if ∏i=1nai=0;(COn2) COna1,…,an=1 if and only if ∏i=1nai=1;(COn3) COn is amplitude monotonic in the first component: COna,a2,…,an≤|COnb,a2,…,an when a≤b.SinceCOn is commutative, n-dimensional complex-valued overlap functions also are amplitude monotonic in any other component based on (GCOn3). Obviously, these conditions are analogous to those of Definition 1. When the domain is limited to [0,1], n-dimensional complex-valued overlap function reduces to n-dimensional real-valued overlap function of Definition 1.Example 1.
Nevertheless, there are mappings that are overlap functions in the domain [0,1] but are not complex-valued overlap functions. The functionf:D2⟶D given by(1)fa,b=aba+b2is an overlap function but not a complex-valued overlap function.There are many types of real-valued overlap functions. Similarly, we extend the concept of 0-overlap and 1-overlap functions to n-dimensional complex-valued overlap functions.A mappingCOn:Dn⟶D is an n-dimensional complex-valued 0-overlap function if we loose condition (COn1) to (COn1’) ∏i=1nai=0⇒COna1,…,an=0 without changing any other condition.A mappingCOn:Dn⟶D is an n-dimensional 1-overlap complex-valued function if we loose condition (COn2) to (COn2’) ∏i=1nai=1⇒COna1,…,an=1 without changing any other condition.Based on the concepts of n-dimensional complex-valued 0-overlap and 1-overlap functions, we define the concept of n-dimensional general complex-valued overlap functionsDefinition 5.
A mappingCOn:Dn⟶D is an n-dimensional general complex-valued overlap function if, for all a1,…,an∈D, it is commutative and continuous and satisfies the following conditions:(GCOn1) if ∏i=1nai=0, then COna1,…,an=0;(GCOn2) if ∏i=1nai=1, then COna1,…,an=1;GCOnCOn is amplitude monotonic in the first component: COna,a2,…,an≤|COnb,a2,…,an when a≤b.The relations betweenn-dimensional complex-valued overlap functions, complex-valued 0-overlap functions, complex-valued 1-overlap functions, and general complex-valued overlap functions are shown in Figure 1. Asmus et al. [17] gave the relations between n-dimensional interval-valued overlap functions, interval-valued 0-overlap functions, interval-valued 1-overlap functions, and general interval-valued overlap functions, which are similar to that of Figure 1.Figure 1
Relations between complex-valued overlap functions, complex-valued 0-overlap functions, complex-valued 1-overlap functions, and general complex-valued overlap functions.Clearly, we have the following.Proposition 1.
If a mappingg:Dn⟶D is either an n-dimensional complex-valued overlap, complex-valued 0-overlap, or complex-valued 1-overlap function, then g is also a general complex-valued overlap function.We give some examples of these complex-valued overlap functions to demonstrate their relations.Example 2.
The functionπ:D2⟶D given by(2)πa,b=a·bis a complex-valued overlap function.Example 3.
The functionπ2:D2⟶D given by(3)π2a,b=a2·b2is a general complex-valued overlap function. Moreover, it is a complex-valued 1-overlap function but not a complex-valued 0-overlap function.Example 4.
The functiong:D2⟶D given by(4)ga,b=mina,b·a·bis a general complex-valued overlap function. Moreover, it is a complex-valued 1-overlap function but not a complex-valued 0-overlap function.Example 5.
The functionh:D2⟶D given by(5)ha,b=maxa+b−1,0·a·bis a general complex-valued overlap function. Moreover, it is a complex-valued 0-overlap function but not a complex-valued 1-overlap function.Note that the class of overlap functions is convex. But the convex sun is not amplitude monotonic [?], then the class of complex-valued overlap functions is not convex.Negative operation (–) is closed in the range of complex unit circle, but is not closed in [0,1]. Then, we have the following properties only for complex-valued overlap functions.Definition 6.
We say the complex-valued overlap functionCO:Dn⟶D satisfies the self-duality property, if it satisfies(6)COa1,…,an=−CO−a1,−a2,…,−an,for any a1,…,an∈D.Definition 7.
We say the complex-valued overlap functionCO:Dn⟶D is symmetric with respect to the point 0, if it satisfies(7)COa1,…,an=CO−a1,−a2,…,−an,for any a1,…,an∈D.There are complex-valued overlap functions satisfying the abovementioned properties.Example 8.
The functiong:Dn⟶D given by(8)ga1,…,an=max∑i=1nai−n+1,0·∏i=1naiis a complex-valued overlap function. Interestingly, it satisfies the self-duality property when n is an odd number. It is symmetric with respect to the point 0 when n is an even number.


## 4. Construction of Complex-Valued Overlap Functions

We assume that the complex numbers are used in the form of exponent, i.e.,a∈D is of the form raejθa, where j=−1, the amplitude term ra∈ℝ, and the phase term θa∈0,2π. In order to let the phase term within valid range, we compute the least positive residue modulo 2π of the phase term when it is out of range. For simplicity, we omit the symbol (mod2π).Proposition 2.
If a mappingf:Dn⟶D is n-dimensional complex-valued overlap (complex-valued 0-overlap, complex-valued 1-overlap, or general complex-valued overlap) function is expressed as(9)fa1,…,an=gra1,ra2,…,ranejhθa1,θa2,…θan,then the function g is an n-dimensional overlap (0-overlap, 1-overlap, or general overlap) function.Theorem 1.
If the functiong:0,1n⟶0,1 is an n-dimensional overlap function, the function h:0,2πn⟶0,2π satisfies the following properties:(i)
h is commutative;(ii)
∑i=1nθai=0 if and only if hθa1,…,θan=0;(iii)
h is continuous.
then, the functionf:Dn⟶D defined by equation (9) is an n-dimensional complex-valued overlap (complex-valued 0-overlap, complex-valued 1-overlap, or general complex-valued overlap) function.Proof.
It is immediate thatf is commutative, amplitude monotonic, and continuous, since g is nondecreasing, and g and h are both commutative and continuous.(COn1): (⇒) if fa1,…,an=gra1,ra2,…ranejhθa1,θa2,…θan=0, then gra1,ra2,…ran=0. Then, ∏i=1nrai=0 since g is an overlap function. Then, ∏i=1nai=∏i=1nrai·ej∑i=1nθai=0·ej∑i=1nθai=0.(COn1): (⇐) if ∏i=1nai=∏i=1nrai·ej∑i=1nθai=0, this means ∏i=1nrai=0, then gra1,ra2,…ran=0 since g is an overlap function. Then, fa1,…,an=gra1,ra2,…ranejhθa1,θa2,…θan=0·ejhθa1,θa2,…θan=0.(COn2): (⇒) if fa1,…,an=gra1,ra2,…ranejhθa1,θa2,…θan=1, then gra1,ra2,…ran=1 and hθa1,θa2,…θan=0. Then, ∏i=1nrai=1 since g is an overlap function, and ∑i=1nθai=0 since h satisfies (ii). Then, ∏i=1nai=∏i=1nrai·ej∑i=1nθai=1·ej0=1.(COn2): (⇐) if ∏i=1nai=∏i=1nrai·ej∑i=1nθai=1, this means ∏i=1nrai=1 and ∑i=1nθai=0, then gra1,ra2,…ran=1 since g is an overlap function, and hθa1,θa2,…θan=0 since h satisfies (ii). Then, fa1,…,an=gra1,ra2,…ranejhθa1,θa2,…θan=1·ej0=1. □Theorem 2.
If the functiong:0,1n⟶0,1 is an n-dimensional 0-overlap function, the function h:0,2πn⟶0,2π satisfies the following properties:(i)
h is commutative;(ii)
∑i=1nθai=0 if and only if hθa1,…,θan=0;(iii)
h is continuous.
Then, the functionf:Dn⟶D defined by equation (9) is an n-dimensional complex-valued 0-overlap function.Proof.
Analogous to the proof of Theorem1.Theorem 3.
If the functiong:0,1n⟶0,1 is an n-dimensional 1-overlap (or general overlap) function, the function h:0,2πn⟶0,2π satisfies the following properties:(i)
h is commutative;(ii)
If∑i=1nθai=0, then hθa1,…,θan=0;(iii)
h is continuous.
Then, the functionf:Dn⟶D defined by equation (9) is an n-dimensional complex-valued 1-overlap (or general complex-valued overlap) function.Proof.
Analogous to the proof of Theorem1.There are several construction methods of (general) overlap functions. Here, we consider the construction of (general) complex-valued overlap functions. If the n-dimensional complex-valued functionf:Dn⟶D is defined by equation (9), then we can easily see that it is a key step to construct the function h:0,2πn⟶0,2π, which satisfies the condition (ii) of Theorem 2 (or 3).Now, we give some examples of bivariate functionsh:0,2π2⟶0,2π satisfying condition (ii) of Theorem 2 (or 3).Example 6.
The functionh1:0,2π2⟶0,2π given by(10)h1a,b=a+bsatisfies condition (ii) of Theorem 2. The function h2:0,2π2⟶0,2π given by(11)h2a,b=−a−bsatisfies condition (ii) of Theorem 2. The function h3:0,2π2⟶0,2π given by(12)h3a,b=2a+bsatisfies condition (ii) of Theorem 3. But it does not satisfy condition (ii) of Theorem 2.Note that we omit the operation ofmod2π. If a=b=π/2, then h3a,b=2a+b=2π=0, but a+b=π≠0. So, h3 does not satisfy condition (ii) of Theorem 2. In general, we have the following results.Example 7.
The functionh4:0,2π2⟶0,2π given by(13)h4a,b=ka+b,k=±2,±3,…,satisfies condition (ii) of Theorem 3. But it does not satisfy condition (ii) of Theorem 2.Based on results of complex-valued overlap functions, we give the following examples.Example 8.
The functionhn,p,k:Dn⟶D given by(14)hn,p,k=ra1·ra2,…,ranp·ejkθa1+θa2+⋯+θanis a complex-valued overlap function when p>0 and k=±1.
The functionhn,p,k:Dn⟶D given by(15)hn,L,k=max∑i=1nrai−n+1,0·ejkθa1+θa2+⋯+θanis a complex-valued 0-overlap function when k=±1.
The functionhn,p,k:Dn⟶D given by(16)hn,∧,k=mini=1nrai·ejkθa1+θa2+⋯+θanis a complex-valued 1-overlap function when k=±2,±3,….


## 5. Conclusions

In this paper, we introduced the concepts of complex-valued overlap, complex-valued 0-overlap, complex-valued 1-overlap, and general complex-valued overlap functions. We gave the relationship between them and studied their properties. Different from the traditional real-valued overlap functions, we added the following properties for complex-valued overlap functions since the domain of each variable is the unit disk of complex plane:(17)fa1,…,an=−f−a1,−a2…,−an,fa1,…,an=f−a1,−a2…,−an.Then, we presented some construction methods for complex-valued overlap functions. Because of the periodicity of exponential functionejx, our method includes the construction of a continuous, commutative function h:0,2πn⟶0,2π satisfying the following property:(18)∑i=1nθai=0if and only ifhθa1,…,θan=0.Of course, we should note that complex-valued overlap functions have many differences with the traditional real-valued overlap functions. Some interesting properties are useful for complex-valued overlap functions but they do not appear in traditional real-valued overlap functions. As further works, we intend to investigate these special properties of complex-valued overlap functions.Complex-valued overlap functions can be viewed as a special class of complex fuzzy aggregation functions which have been widely used in many application fields. How to apply the complex-valued overlap functions is another problem of interest.



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