# Complex-Valued Migrativity of Complex Fuzzy Operations

Authors: Yingying Xu; Haifeng Song; Lei Du; Songsong Dai
Journal: Journal of Mathematics
 (2022)

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## Abstract

Complex fuzzy sets (CFSs), as an important extension of fuzzy sets, have been investigated in the literature. Operators of CFSs are of high importance. In addition,α−migrativity for various fuzzy operations on [0, 1] has been well discussed, where α is a real number and α∈0,1. Thus, this paper studies α−migrativity for binary functions on the unit circle of the complex plane O, where α is a complex number and α∈O. In particular, we show that a binary function is α−migrativity for all α∈O if and only if it is α−migrativity for all α∈0,1∪O¯, where O¯ is the boundary point subset of O. Finally, we discuss the relationship between migrativity and rotational invariance of binary operators on O.


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## Body

## 1. Introduction

Complex fuzzy sets (CFSs) were introduced by Ramot et al. [1, 2], whose membership degree is a complex number on the unit disc of the complex plane O, where O=α∈ℂ‖α|≤1. Operations are of high importance in the theory of CFSs. Various concepts and properties have been developed for complex fuzzy operations. Dick [3] introduced the rotational invariance of operators of CFSs. Dai [4, 5] generalized Dick’s works on rotational invariance and order induced by algebraic product operation. Zhang et al. [6] studied operation properties and δ-equalities of CFSs. Dick, Yager, and Yazdanbahksh [7] gave some complex fuzzy operations based on Pythagorean fuzzy operations, which was developed by Liu et al. [8]. Then Dick [9] considered complex fuzzy S-implications. Hu et al. [10–13] discussed orthogonality preserving operators and parallelity preserving operators of CFSs.Theα-migrativity [14] as an important property of binary fuzzy operators has been discussed in the cases of overlap/grouping functions [15, 16], uninorms [17–22], triangular subnorm [23], t-norms [24], nullnorm [25], copulas [26, 27], and aggregation functions [28–30]. In the aforementioned migrative functions, their research domain is limited to real numbers on [0, 1]. For example, a binary function f:I2⟶I is migrative if fαx,y=fμ,αν holds for all μ,ν∈I and α∈I, where I=0,1.This paper focuses on theα-migrativity of complex fuzzy binary operations, i.e., functions f:O2⟶O, where α∈O is a complex number. Moreover, since a CFS is composed of a magnitude term and a phase term, we consider magnitude-migrativity and phase-migrativity, which respectively limits α∈I and α∈O¯, where O¯ is the boundary point subset of O, i.e., O¯=α∈ℂ‖α|=1.As far as we know, migrativity including magnitude-migrativity and phase-migrativity of complex fuzzy operations have not been studied yet. Moreover, we note that phase-migrativity and rotational invariance [3, 4] of complex fuzzy operations are similar with respect to angle rotation operations. It is essential to straighten out the relationship between phase-migrativity and rotational invariance for complex fuzzy operations.This article is structured as follows: in Section2, we introduce the concepts of migrativity, magnitude-migrativity, and phase-migrativity for complex fuzzy binary operations. In Section 3, we give characterizations of these migrativity properties of complex fuzzy binary operations. In Section 4, the relationship between rotational invariance and migrativity is studied. In Section 5, concluding remarks are given.


## 2. Migrativity

Definition 1.
Consider a fixed pointα∈O, a binary operation f:O2⟶O is said to be α−migrative if(1)fαμ,ν=fμ,αν,for allμ,ν∈O.
Note thatα− migrativity refers to a fixed complex number α. This can be generalized as follows:Definition 2.
A binary operationf:O2⟶Ois said to be migrative if and only if (briefly, iff)(2)fαμ,ν=fμ,αν,for allμ,ν∈Oandα∈O.
A complex vector includes the amplitude term and the phase part. So, we introduce the following concepts:Definition 3.
A binary operationf:O2⟶Ois said to be amplitude-migrative iff(3)frμ,ν=fμ,rν,for allμ,ν∈Oandr∈I.Definition 4.
A binary operationf:O2⟶Ois said to be phase-migrative if and only if(4)fejθμ,ν=fμ,ejθν,for allμ,ν∈Oandθ∈ℝ,where j=−1.
Note that phase-migrativity meansα-migrativity for all α∈O¯.Theorem 1.
A binary operationf:O2⟶Ois migrative iff, for allr∈Iandθ∈ℝ, it holds that(5)fejθμ,ν=fμ,ejθνfrμ,ν=fμ,rν.Proof.
⇒ Trivial.
⇐ For any α∈O, denote α=rα⋅ejθα where rα∈I. Then fαμ,ν=frα⋅ejθαμ,ν=fejθαμ,rα⋅ν=fμ,ejθα⋅rαν=fμ,αν. □
For a complex fuzzy binary functionf, as shown in Figure 1(a) and 1(b), if it is phase-migrative, then we have β1=β2 for any θ and inputs μ,ν∈O.
A binary operation is migrative if and only if it is amplitude-migrative and phase-migrative. From this result, we have the following result:Figure 1
Phase-migrative function (a)fμ⋅ejθ,ν and (b) fμ,ν⋅ejθ.
(a)(b)Corollary 1.
Letf:O2⟶Obe a binary operation. Then the following statements are equivalent.(1)
fαμ,ν=fμ,αν, for all μ,ν∈O and α∈O;(2)
fαμ,ν=fμ,αν, for allμ,ν∈Oandα∈I∪O¯.Note thatf is α−migrative for all α∈O if and only if it is α−migrative for all α∈I∪O¯. This is very interesting because I∪O¯ is a proper subset of O, i.e., I∪O¯⫋O, and the size of I∪O¯ is much smaller than that of O. Obviously, in the above corollary, I∪O¯ could be replaced by other subsets, such as −1,0∪O¯.Example 1.
The operationsf1,f2,f3:O2⟶Oare respectively defined by(6)f1μ,ν=μ⋅ν,f2μ,ν=μ⋅ν,f3μ,ν=μ⋅μ⋅ν.Obviously,f1 is migrative. Interestingly, for all r∈I, we have f2rμ,ν=f2μ,rν. Thus f2 is amplitude-migrative. Similarly, for all θ∈ℝ, we have f3ejθμ,ν=f3μ,ejθν. Thus, f3 is phase-migrative. But f2 is not phase-migrative, f3 is not amplitude-migrative, thus, they are not migrative.


## 3. Characterization of Migrativity

One of the important results of migrative real-valued functions is the following theorem:Theorem 2 (see [28]).
A binary operationf:I2⟶Iis migrative iff there exists a functiong:I⟶Isuch thatfμ,ν=gxyfor allμ,ν∈I.
This result is not true for amplitude-migrative (or phase-migrative) functions (see Example1), but it is true for migrative complex-valued functions.Theorem 3.
A binary operationf:O2⟶Ois migrative iff there exists a functionf:O⟶Osuch thatfμ,ν=gxyfor allμ,ν∈O.Proof.
⇐ If g exists, then fαμ,ν=gαμν=fμ,αν.
⇒ If f is migrative, then fμ,ν=fμ⋅1,ν=f1,μν, thus, gμν=f1,μν is the function.
In this way, the functiong is the migrative generator of the migrative binary operation f.
The following result is immediate:Theorem 4.
Letf:O2⟶Obe a migrative binary operation. Then(1)
f1,1=1 if and only of g1=1;(2)
f0,0=0if and only ofg0=0;Example 2.
We give some migrative functions and their migrative generators.(1)
The migrative generator offμ,ν=μνisgμ=μ;(2)
The migrative generator offμ,ν=μνisgμ=μ;(3)
The migrative generator offμ,ν=1−μνisgμ=1−μ.
Moreover, we have the following results.Theorem 5.
Letf:O2⟶Obe a migrative function. Thenfis commutative, i.e.,fμ,ν=fν,μ.Proof.
Iff is migrative, then fμ,ν=fμ⋅1,ν=f1,μν=fν⋅1,μ=fν,μ for all μ,ν∈O.
This result is not true for amplitude-migrative (or phase-migrative) functions (see Example1). The following result is true even for amplitude-migrative (or phase-migrative) functions.Theorem 6.
If a binary operationf:O2⟶Ois amplitude-migrative (or phase-migrative), then for allμ,ν∈O,(1)
f−μ,−ν=fμ,ν;(2)
f−μ,ν=fμ,−ν.Proof.
Here we only give the proof of (1). Iff is amplitude-migrative, then(7)f−μ,−ν=fμ,−1−1ν=fμ,ν,for all μ,ν∈O.
Iff is phase-migrative, then(8)f−μ,−ν=fejπμ,ejπν=fμ,ejπejπν=fμ,ν,for all μ,ν∈O.Corollary 2.
If a binary operationf:O2⟶Ois migrative, thenf−μ,−ν=fμ,νfor allμ,ν∈O.Theorem 7.
A binary operationf:O2⟶Ois phase-migrative iff it is the convex sum of a finite family of phase-migrative functions.Proof.
⇒f is the convex sum of itself.
⇐ Let fμ,ν=∑i=1nwifiμ,ν with ∑i=1nwi and wi∈I. If fii=12,…,n is amplitude-migrative, then for any θ∈ℝ, fejθμ,ν=∑i=1nwifiejθμ,ν=∑i=1nwifiμ,ejθν=fμ,ejθν for all μ,ν∈O.
Similarly, we have the following results.Theorem 8.
A binary operationf:O2⟶Ois amplitude-migrative iff it is the convex sum of a finite family of amplitude-migrative functions.Corollary 3.
A binary operationf:O2⟶Ois migrative iff it is the convex sum of a finite family of migrative functions.


## 4. Migrativity and Rotational Invariance

Now we consider the relation between migrativity and rotational invariance [3, 4].Definition 5.
(see [3]). Let f:O2⟶O be a binary function, then f is rotationally invariant if(9)fμ⋅ejθ,ν⋅ejθ=fμ,ν⋅ejθ,for anyθ∈ℝandμ,ν∈O.
Dick’s concept of rotational invariance was generalized as follows:Definition 6.
(see [4]). Let f:O2⟶O be a binary function, then f is h-rotationally invariant if, for a function h:ℝ2⟶ℝ,(10)fμ⋅ejθ1,ν⋅ejθ2=fμ,ν⋅ejhθ1,θ2,for anyθ1,θ2∈ℝandμ,ν∈O.Theorem 9.
A binary operationf:O2⟶O is h-rotationally invariant iff it is the convex sum of a finite family of h-rotationally invariant functions.Proof.
⇒f is the convex sum of itself.
⇐ Let fμ,ν=∑i=1nwifiμ,ν with ∑i=1nwi and wi∈I. If fii=12,…,n is h-rotationally invariant, then for any θ1,θ2∈ℝ, fejθ1μ,ejθ2ν=∑i=1nwifiejθ1μ,ejθ2ν=∑i=1nwifiμ,νejhθ1,θ2=fμ,νejhθ1,θ2 for all μ,ν∈O. □Corollary 4.
A binary operationf:O2⟶Ois rotationally invariant iff it is the convex sum of a finite family of rotationally invariant functions.First, for binary operations, there is no direct relation between migrativity and Dick’s rotational invariance [3]. For example, fμ,ν=μν is migrative but not rotational invariance. fμ,ν=μ+ν/2 is rotational invariance but not migrative.Theorem 10.
Letf:O2⟶Obe a migrative binary operation andg:O⟶Obe its migrative generator, thengis rotationally invariant iff(11)fμ⋅ejθ1,ν⋅ejθ2=fμ,ν⋅ejθ1+θ2,for anyθ1,θ2∈ℝandμ,ν∈O.Proof.
⇒g is rotationally invariant, i.e., gμ⋅ejθ=gμ⋅ejθ for any θ∈ℝ and μ,ν∈O. Then fμ⋅ejθ1,ν⋅ejθ2=gμ⋅ejθ1⋅ν⋅ejθ2=ejθ1+θ2⋅gμν=ejθ1+θ2⋅fμ,ν.
⇐ If f satisfies equation (11), then for any θ∈ℝ and μ∈O, we have gμ⋅ejθ=f1,μ⋅ejθ=f1,μ⋅ejθ=gμ⋅ejθ.
Moreover, we consider the relation between phase-migrativity and conditional rotational invariance [4].Theorem 11.
A binary operationf:O2⟶Osatisfies(12)fμ⋅ejθ1,ν⋅ejθ2=fμ,ν⋅ejθ1+θ2,for any θ1,θ2∈ℝ and μ,ν∈O. Then it is phase-migrative. But the converse is not true.Proof.
For anyθ∈ℝ and μ,ν∈O, fejθμ,ν=fμ,ν⋅ejθ=fμ,ejθν. Moreover, fμ,ν=μν2 is phase-migrative but does not satisfy equation (12).Corollary 5.
A binary operationf:O2⟶Osatisfies(13)fμ⋅ejθ1,ν=fμ,ν⋅ejθ1fμ,ν⋅ejθ2=fμ,ν⋅ejθ2,for any θ1,θ2∈ℝ and μ,ν∈O. Then it is phase-migrative. But the converse is not true.Proof.
Because equation (12) is equivalent to equation (13),Theorem 12.
Letf:O2⟶O be a commutative binary operation, if it satisfies fejθμ,ν=fμ,ν⋅ejθ for all θ∈ℝ and μ,ν∈O. Then(1)
it is phase-migrative;(2)
it ish−rotationally invariant, where hθ1,θ2=θ1+θ2.Proof.
(1)
For anyθ∈ℝ and μ,ν∈O, we have fμ,ejθν=fejθν,μ=fν,μ⋅ejθ=fμ,ν⋅ejθ=fμ⋅ejθ,ν.(2)
For anyθ1,θ2∈ℝ and μ,ν∈O, we have fμ⋅ejθ1,ν⋅ejθ2=fμ,ν⋅ejθ2⋅ejθ1=fν⋅ejθ2,μ⋅ejθ1=fν,μ⋅ejθ1⋅ejθ2=fμ,ν⋅ejθ1+θ2.
We give a binary operationf without commutativity, fμ,ν=μν satisfies fejθμ,ν=fμ,ν⋅ejθ for all θ∈ℝ and μ,ν∈O. But it is neither commutative nor phase-migrative. Moreover, it is h′−rotationally invariant, where h′θ1,θ2=θ2.
The relations between complex-valued migrativity of complex fuzzy operations, amplitude migrativity of complex fuzzy operations, phase valued migrativity of complex fuzzy operations, rotational invariance of complex fuzzy operations, and the migrativity of fuzzy operations are shown in Figure2.Figure 2
Relations between complex-valued migrativity of complex fuzzy operations, amplitude migrativity of complex fuzzy operations, phase valued migrativity of complex fuzzy operations, rotational invariance of complex fuzzy operations, and the migrativity of fuzzy operations.Theorem 13.
Letf:O2⟶O be a commutative binary operation, if it satisfies frμ,ν=rfμ,ν for all r∈I and μ,ν∈O. Then(1)
it is amplitude-migrative;(2)
it satisfiesfr1μ,r2ν=r1r2fμ,ν. for all r1,r2∈I and μ,ν∈O.Proof.
For anyr∈I and μ,ν∈O, we have(1)
fμ,rν=frν,μ=r⋅fν,μ=r⋅fμ,ν=frμ,ν.(2)
fr1μ,r2ν=r1⋅fμ,r2ν=r1⋅fr2ν,μ=r2⋅r1⋅fν,μ=r1r2⋅fμ,ν.
We observe that it is homogeneous of order 2, i.e.,frμ,rν=r2fμ,ν when r1=r2.
We give a binary operationf without commutativity, fμ,ν=μ⋅ν⋅ν satisfies frμ,ν=r⋅fμ,ν for all r∈I and μ,ν∈O. But it is neither commutative nor amplitude-migrative. Moreover, it is homogeneous of order 3, i.e., frμ,rν=r3fμ,ν.Corollary 6.
Letf:O2⟶O be a commutative binary operation, if it satisfies fαμ,ν=α⋅fμ,ν for all α∈O and μ,ν∈O. Then it is migrativeTheorem 14.
If a binary operationf:O2⟶O is h−rotationally invariant where h1θ1,θ2=kθ1,θ2 for some k>0. Then f is phase-migrative,Proof.
For anyθ∈ℝ and μ,ν∈O, we have fμ⋅ejθ,ν=fμ,ν⋅ejkθ=fμ,ν⋅ejθ for some k>0.Theorem 15.
If a binary operationf:O2⟶O satisfies fr1μ,r2ν=r1⋅r2kfμ,ν for all r1,r2∈I, all μ,ν∈O, and some k>0. Then f is amplitude-migrative.Proof.
For anyr∈I and μ,ν∈O, we have frμ,ν=rkfμ,ν=fμ,rν for some k>0.Corollary 7.
If a binary operationf:O2⟶O satisfies fα1μ,α2ν=α1⋅α2kfμ,ν for all α1,α2∈O, all μ,ν∈O, and some k>0. Then f is migrative.


## 5. Conclusions

In this paper, we study the migrative binary complex fuzzy operators(14)fαμ,ν=fμ,αν,∀μ,ν∈O,for three cases α∈I, α∈O¯, and α∈O. Interestingly, this equation holds for all α∈O if and only if it holds for all α∈I∪O¯ (see Theorem 1). Note that the size of I∪O¯ is much smaller than that of O. Then we give the relationship among phase-migrativity, amplitude-migrativity, migrativity, and rotational invariance for complex fuzzy operations (see Figure 1). We show that phase-migrativity is a special case of conditional rotational invariance (see Theorem 12).Note that this paper focused on binary complex fuzzy operators. Future research should consider the migrativity ofn-dimensional complex fuzzy aggregation operators. Naturally, other properties of complex fuzzy operators are possible topics for future consideration.In [31], Yager and Abbasov used complex numbers of the form r⋅ejθ as Pythagorean membership grades, where r∈0,1 and θ∈0,π/2. These complex numbers are called π−i numbers, which belong to the upper-right quadrant of the unit disk in the complex plane. Viewed in this way, studying the migrativity of Pythagorean fuzzy operators is a special case of migrativity of complex fuzzy operators by limiting the domain to π−i numbers. Obviously, a more detailed discussion of the migrativity of Pythagorean fuzzy aggregation operators [32], Pythagorean t-norm [33], will be both necessary and interesting.



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