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, migrativity for various fuzzy operations on [0, 1] has been well discussed, where is a real number and . Thus, this paper studies migrativity for binary functions on the unit circle of the complex plane , where is a complex number and . In particular, we show that a binary function is migrativity for all if and only if it is migrativity for all , where is the boundary point subset of . Finally, we discuss the relationship between migrativity and rotational invariance of binary operators on .

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 , where . 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 -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. [1013] discussed orthogonality preserving operators and parallelity preserving operators of CFSs.

The -migrativity [14] as an important property of binary fuzzy operators has been discussed in the cases of overlap/grouping functions [15, 16], uninorms [1722], triangular subnorm [23], t-norms [24], nullnorm [25], copulas [26, 27], and aggregation functions [2830]. In the aforementioned migrative functions, their research domain is limited to real numbers on [0, 1]. For example, a binary function is migrative if holds for all and , where .

This paper focuses on the -migrativity of complex fuzzy binary operations, i.e., functions , where 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 and , where is the boundary point subset of , i.e., .

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 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.

2. Migrativity

Definition 1. Consider a fixed point , a binary operation is said to be migrative ifNote that migrativity refers to a fixed complex number . This can be generalized as follows:

Definition 2. A binary operationis said to be migrative if and only if (briefly, iff)A complex vector includes the amplitude term and the phase part. So, we introduce the following concepts:

Definition 3. A binary operationis said to be amplitude-migrative iff

Definition 4. A binary operationis said to be phase-migrative if and only ifwhere .
Note that phase-migrativity means -migrativity for all .

Theorem 1. A binary operationis migrative iff, for alland, it holds that

Proof. Trivial.
For any , denote where . Then .
For a complex fuzzy binary function , as shown in Figure 1(a) and 1(b), if it is phase-migrative, then we have for any and inputs .
A binary operation is migrative if and only if it is amplitude-migrative and phase-migrative. From this result, we have the following result:

Corollary 1. Letbe a binary operation. Then the following statements are equivalent.(1), for all and ;(2), for alland.

Note that is migrative for all if and only if it is migrative for all . This is very interesting because is a proper subset of , i.e., , and the size of is much smaller than that of . Obviously, in the above corollary, could be replaced by other subsets, such as .

Example 1. The operationsare respectively defined by

Obviously, is migrative. Interestingly, for all , we have . Thus is amplitude-migrative. Similarly, for all , we have . Thus, is phase-migrative. But is not phase-migrative, 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 operation is migrative iff there exists a functionsuch thatfor all.
This result is not true for amplitude-migrative (or phase-migrative) functions (see Example 1), but it is true for migrative complex-valued functions.

Theorem 3. A binary operationis migrative iff there exists a functionsuch thatfor all.

Proof. If exists, then .
If is migrative, then , thus, is the function.
In this way, the function is the migrative generator of the migrative binary operation .
The following result is immediate:

Theorem 4. Letbe a migrative binary operation. Then(1) if and only of ;(2)if and only of;

Example 2. We give some migrative functions and their migrative generators.(1)The migrative generator ofis;(2)The migrative generator ofis;(3)The migrative generator ofis.Moreover, we have the following results.

Theorem 5. Letbe a migrative function. Thenis commutative, i.e.,.

Proof. If is migrative, then for all .
This result is not true for amplitude-migrative (or phase-migrative) functions (see Example 1). The following result is true even for amplitude-migrative (or phase-migrative) functions.

Theorem 6. If a binary operationis amplitude-migrative (or phase-migrative), then for all,(1);(2).

Proof. Here we only give the proof of (1). If is amplitude-migrative, thenfor all .
If is phase-migrative, thenfor all .

Corollary 2. If a binary operationis migrative, thenfor all.

Theorem 7. A binary operationis phase-migrative iff it is the convex sum of a finite family of phase-migrative functions.

Proof. is the convex sum of itself.
Let with and . If is amplitude-migrative, then for any , for all .
Similarly, we have the following results.

Theorem 8. A binary operationis amplitude-migrative iff it is the convex sum of a finite family of amplitude-migrative functions.

Corollary 3. A binary operationis 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 be a binary function, then is rotationally invariant iffor anyand.
Dick’s concept of rotational invariance was generalized as follows:

Definition 6. (see [4]). Let be a binary function, then is -rotationally invariant if, for a function ,for anyand.

Theorem 9. A binary operation is -rotationally invariant iff it is the convex sum of a finite family of -rotationally invariant functions.

Proof. is the convex sum of itself.
Let with and . If is -rotationally invariant, then for any , for all .

Corollary 4. A binary operationis 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, is migrative but not rotational invariance. is rotational invariance but not migrative.

Theorem 10. Letbe a migrative binary operation andbe its migrative generator, thenis rotationally invariant ifffor anyand.

Proof. is rotationally invariant, i.e., for any and . Then .
If satisfies equation (11), then for any and , we have .
Moreover, we consider the relation between phase-migrativity and conditional rotational invariance [4].

Theorem 11. A binary operationsatisfiesfor any and . Then it is phase-migrative. But the converse is not true.

Proof. For any and , . Moreover, is phase-migrative but does not satisfy equation (12).

Corollary 5. A binary operationsatisfiesfor any and . Then it is phase-migrative. But the converse is not true.

Proof. Because equation (12) is equivalent to equation (13),

Theorem 12. Let be a commutative binary operation, if it satisfies for all and . Then(1)it is phase-migrative;(2)it is rotationally invariant, where .

Proof. (1)For any and , we have .(2)For any and , we have .We give a binary operation without commutativity, satisfies for all and . But it is neither commutative nor phase-migrative. Moreover, it is rotationally invariant, where .
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 Figure 2.

Theorem 13. Let be a commutative binary operation, if it satisfies for all and . Then(1)it is amplitude-migrative;(2)it satisfies . for all and .

Proof. For any and , we have(1).(2).We observe that it is homogeneous of order 2, i.e., when .
We give a binary operation without commutativity, satisfies for all and . But it is neither commutative nor amplitude-migrative. Moreover, it is homogeneous of order 3, i.e., .

Corollary 6. Let be a commutative binary operation, if it satisfies for all and . Then it is migrative

Theorem 14. If a binary operation is rotationally invariant where for some . Then is phase-migrative,

Proof. For any and , we have for some .

Theorem 15. If a binary operation satisfies for all , all , and some . Then is amplitude-migrative.

Proof. For any and , we have for some .

Corollary 7. If a binary operation satisfies for all , all , and some . Then is migrative.

5. Conclusions

In this paper, we study the migrative binary complex fuzzy operatorsfor three cases , , and . Interestingly, this equation holds for all if and only if it holds for all (see Theorem 1). Note that the size of is much smaller than that of . 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 of -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 as Pythagorean membership grades, where and . These complex numbers are called 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 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

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the National Science Foundation of China (Grant nos. 62006168 and 62101375) and the Zhejiang Provincial Natural Science Foundation of China (Grant nos. LQ21A010001 and LQ21F020001).