Complex-Valued Migrativity of Complex Fuzzy Operations

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 .


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]. en Dick [9] considered complex fuzzy S-implications. Hu et al. [10][11][12][13] discussed orthogonality preserving operators and parallelity preserving operators of CFSs.
is article is structured as follows: in Section 2, 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.

Migrativity
(1) Note that α− migrativity refers to a fixed complex number α. is can be generalized as follows: (2) A complex vector includes the amplitude term and the phase part. So, we introduce the following concepts: Proof. (⇒) Trivial.
For a complex fuzzy binary function f, as shown in Figure 1(a) and 1(b), if it is phase-migrative, then we have β 1 � β 2 for any θ and inputs μ, ] ∈ 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: Obviously, f 1 is migrative. Interestingly, for all r ∈ I, we have .
us, f 3 is phase-migrative. But f 2 is not phase-migrative, f 3 is not amplitude-migrative, thus, they are not migrative.

Characterization of Migrativity
One of the important results of migrative real-valued functions is the following theorem: Theorem 2 (see [28]). A binary operation f: I 2 ⟶ I is migrative iff there exists a function g: This result is not true for amplitude-migrative (or phasemigrative) functions (see Example 1), but it is true for migrative complex-valued functions.

Theorem 3. A binary operation
In this way, the function g is the migrative generator of the migrative binary operation f. e following result is immediate: Example 2. We give some migrative functions and their migrative generators.
Moreover, we have the following results.
is result is not true for amplitude-migrative (or phasemigrative) functions (see Example 1). e following result is true even for amplitude-migrative (or phase-migrative) functions.
Proof. Here we only give the proof of (1). If f is amplitudemigrative, then Similarly, we have the following results.

Migrativity and Rotational Invariance
Now we consider the relation between migrativity and rotational invariance [3,4].
Proof. For any θ ∈ R and .

Corollary 5. A binary operation f:
for any θ 1 , θ 2 ∈ R and μ, ] ∈ O. en it is phase-migrative. But the converse is not true.

Conclusions
In this paper, we study the migrative binary complex fuzzy operators 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 eorem 1). Note that the size of I ∪ O is much smaller than that of O. en 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 eorem 12). Note that this paper focused on binary complex fuzzy operators. Future research should consider the migrativity of n-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 · e jθ as Pythagorean membership grades, where r ∈ [0, 1] and θ ∈ [0, π/2]. ese 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.