The parallelity of complex fuzzy sets and parallelity preserving operators

Abstract

Complex fuzzy sets have been successfully applied to many domains. In some applications, complex fuzzy operators play an vital role, especially those results depend on the particular choice of the conjunction, disjunction and complement operators. This paper introduces an equivalence relation on complex fuzzy sets, called parallelity. Then we provide a criterion that operators should satisfy the property of parallelity preserving, for the choices of complex fuzzy operators. Finally, several parallelity preserving inference methods are discussed for complex fuzzy reasoning.

Keywords

Parallelity parallelity preserving operators complex fuzzy sets complex fuzzy operators complex fuzzy inference

1 Introduction

Since Ramot et al. [11] defined a complex fuzzy set as an important extension of traditional fuzzy set in which the range of the membership function was the unit disc of the complex plane. Various applications have been proposed based on the theory of complex fuzzy sets [1 , 12–15]. For each application, certain complex fuzzy operators are used to build a link between complex fuzzy inputs and complex fuzzy outputs. Then a corresponding problem arises, how to choose suitable operators in complex fuzzy systems? This is an important issue in complex fuzzy systems. For example, we can choose an operator from the criterion whether it has continuity property based on distances of complex fuzzy sets [15]. This is concerned with stability of complex fuzzysystems.

As mentioned in [3], complex fuzzy membership grades should be viewed as vectors in the plane, rather than scalar quantities. For a membership vector, there are many concepts and properties for the amplitude of the vector, which are defined in traditional fuzzy sets and operators. Some researchers made attempts to get the answers from geometric point of view, which mostly rely on the phase of the membership vector. There are two corresponding concepts on the choice of complex fuzzy operators.

One is introduced by Dick [3], who defined the rotational invariance of complex fuzzy operators. This concept is developed from the assertion “phase is relative” in [12]. Rotational invariance states that an operator is invariant under a simple rotation. Under this criterion, the algebraic product and negation are excluded from use in rotationally invariant lattices. Moreover, the removal of this criterion was also examined for the choices of operators based on the techniques of vector logic [9, 10].

The second concept is introduced by Hu et al. [4], who defined the orthogonality between complex fuzzy sets. This orthogonality relation is a dependency relation, i.e., is symmetric and reflexive, but is not transitive. Based on this relation, they presented a new criterion for the choice of complex operations, “orthogonality should be preserved for complex fuzzy operations”.

The method of this paper also relies on the phase of the membership vector. Similar to the orthogonality in geometry, the parallelity relation also is an important concept between two vectors. However, the orthogonality relation in complex fuzzy sets This orthogonality relation is a dependency relation, but is not an equivalence relation. Consequently, we make another attempt to the solution of the above problem in this paper. We propose an equivalence relation of complex fuzzy sets, denoted by the parallelity relation between complex fuzzy sets. Then we introduce the concepts of parallelity preserving operators. Orthogonality preserving property might serve as a certain criteria for choices of complex fuzzy operators in some of real applications.

The remainder of this paper is organized as follows. In Section 2, we review some related work. The concepts of parallelity between complex fuzzy sets are introduced in Section 3. Then, we discuss whether parallelity can be preserved for complex fuzzy operations and complex fuzzy inference in Sections 4 and 5, respectively. Finally, conclusions are presented in Section 6.

2 Related work

2.1 Rotational invariance

In order to develop the assertion “phase is purely relative” in [12], Dick [3] introduced the following definition,

Definition 1. Let D be the set of complex numbers whose modulus is less than or equal to 1, an operation f : D × D → D is rotationally invariant if and only if $f (R_{θ} (a), R_{θ} (b)) = {Rot}_{θ} (f (a, b))$ (1) for all elements a, b ∈ D. Where R_θ (x) = x · e^jθ for any x ∈ D, $j = \sqrt{- 1}$ .

Rotational invariance states that an operation is invariant under a simple rotation. According to this point of view, Dick discuss the choices of complex fuzzy operations under the criterion “complex fuzzy operations should satisfy the rotational invariance property”.

2.2 Orthogonality

A dependency relation between complex fuzzy sets was given by Hu et al. in [4], as follows,

Definition 2. Let U be a universe of discourse, A and B be two complex fuzzy sets on U, and μ_A (x) and μ_B (x) be their membership, respectively. Then A and B is said to be orthogonal, denoted by A ⊥ B, if $< μ_{A} (x), μ_{B} (x) > = 0$ (2) for each x ∈ U, where <μ_A (x), μ_B (x)> is the inner product of μ_A (x), μ_B (x) ∈ D.

Moreover, the concept of orthogonality preserving was presented as follows,

Definition 3. Let U be a universe of discourse, and CFS (U) be the set of all complex fuzzy sets on U. A function f : CFS (U) → CFS (U) is orthogonality preserving if $A ⊥ B \Rightarrow f (A) ⊥ f (B) .$ (3)

Then they discussed the choices of complex fuzzy operators under the new criterion “complex fuzzy operations should preserve the property of orthogonality”.

3 Parallelity

This section introduces the concept of parallelity betwen complex fuzzy sets. By intuition, this is analogous to parallelity of vectors in complex plane. In this way, the parallelity is a relation of two complex fuzzy sets which membership vectors of each element of universal set have the same (or opposite) direction. See Fig. 1 for an example, two membership vectors A and B are parallelity with same direction, and two membership vectors A and C are parallelity with opposite direction.

Fig.1

Parallelity between complex fuzzy sets.

Let U be a universe of discourse, and CFS (U) be the set of all complex fuzzy sets on U. A complex fuzzy set A on the universe U is given as $A = {(x, r_{A} (x) \cdot e^{j θ_{A} (x)} | x \in U}$ (4) where the amplitude term r_A (x) and the phase term θ_A (x) are real-valued numbers with r_A (x) ∈ [0, 1].

The inner product of r₁ · e^{jw
₁}, r₂ · e^{jw
₂} ∈ D is defined as $< r_{1} \cdot e^{j θ_{1}}, r_{2} \cdot e^{j θ_{2}} > = r_{1} \cdot r_{2} \cdot cos (| θ_{1} - θ_{2} |)$ (5)

Definition 4. Let U be a universe of discourse, A and B be two complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jθ_A(x) and μ_B (x) = r_B (x) · e^jθ_B(x) be their membership, respectively. Then A and B is said to be parallel, denoted by A ∥ B, if $< μ_{A} (x), μ_{B} (x) > = r_{A} (x) \cdot r_{B} (x)$ (6) for each x ∈ U.

Similarly, for subset $ℂ, D \subseteq C F S (U) A ∥ ℂ$ we write $A ∥ ℂ$ if $A ∥ ℂ$ for all $C \in ℂ$ , and $ℂ ∥ D$ if C ∥ D for all $C \in ℂ$ and all $D \in D$ .

Theorem 1. The parallelity of complex fuzzy sets has the following properties: for all A, B, C ∈ CFS (U),

0 ∥ A for all A ∈ CFS (U),

(Reflexivity): A ∥ A, for all A ∈ CFS (U),

(Symmetry): A ∥ B if and only if B ∥ A, for all A, B ∈ CFS (U),

(Transitivity): if A ∥ B, B ∥ C then A ∥ C, for all A, B, C ∈ CFS (U).

Proof. Trivial. ■

Remark 1. 0 means that membership vectors of each element is the zero vector on the universal set.

Remark 2. The orthogonality relation of complex fuzzy sets, introduced in [4], is a dependency relation ⊥, i.e., is symmetric and reflexive, but is not transitive. However, the parallelity of complex fuzzy sets is an equivalence relation.

4 Parallelity preserving

This section discusses whether parallelity can be preserved for complex fuzzy operations.

The concept of parallelity preserving is formally expressed as follows,

Definition 5. A function f : CFS (U) → CFS (U) is parallelity preserving if $A ∥ B \Rightarrow f (A) ∥ f (B)$ (7) for any A, B ∈ CFS (U).

Set rotation and reflection operations, described in [11], are parallelity preserving.

Let U be a universe of discourse, A be a complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jθ_A(x) be its membership functions.

The Rotation of A by θ radians, denoted by R_θ (A), is defined as $R_{θ} (μ_{A} (x)) = r_{A} (x) \cdot e^{j (θ_{A} (x) + θ)}$ (8)

The reflection of A, denoted by Ref (A), is defined as $Ref (μ_{A} (x)) = r_{A} (x) \cdot e^{- j θ_{A} (x)}$ (9)

Theorem 2. Let U be a universe of discourse, A and B be two complex fuzzy sets on U.

If A ∥ B then Ref (A) ∥ Ref (B).

If A ∥ B then R_θ (A) ∥ R_θ (B) for any θ radians.

Proof. Trivial. ■

Remark 3. It is easy to know that A ∥ B ⇔ R_π/2 (A) ⊥ B.

4.1 Complex fuzzy complement

Let U be a universe of discourse, A be a complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jw_A(x) its membership functions. The following three complex fuzzy complement ¬_i (i = 1, 2, 3), introduced by Ramot et al. [11], are given as $μ_{\neg_{1} A} (x) = (1 - r_{A} (x)) \cdot e^{j (- θ_{A} (x))},$ (10) $μ_{\neg_{2} A} (x) = (1 - r_{A} (x)) \cdot e^{j (θ_{A} (x))},$ (11) $μ_{\neg_{3} A} (x) = (1 - r_{A} (x)) \cdot e^{j (θ_{A} (x) + π)} .$ (12)

Theorem 3. Let U be a universe of discourse, A and B be two complex fuzzy sets on U, and r_A (x) · e^jθ_A(x) and r_B (x) · e^jθ_B(x) their membership functions, respectively. Support r_A (x) ≠0 and r_B (x) ≠0 for all x ∈ U. If A ∥ B, then ¬_iA ∥ ¬ _iB, i ∈ {1, 2, 3}.

Proof. For any r₁ · e^{jθ
₁}, r₂ · e^{jθ
₂} ∈ D with r₁ ≠ 0 and r₂ ≠ 0. If <r₁ · e^{jθ
₁}, r₂ · e^{jθ
₂} > = r₁ · r₂, then cos(|θ₁ - θ₂|) =1 $\begin{matrix} < (1 - r_{1}) \cdot e^{- j θ_{1}}, (1 - r_{2}) \cdot e^{- j θ_{2}} > \\ = (1 - r_{1}) \cdot (1 - r_{2}) \cdot cos (| - θ_{1} - (- θ_{2}) |) \\ = (1 - r_{1}) \cdot (1 - r_{2}) \cdot cos (| θ_{1} - θ_{2} |) \\ = (1 - r_{1}) \cdot (1 - r_{2}) \end{matrix}$

Thus, A ∥ B ⇒ ¬ ₁A ∥ ¬ ₁B. Similarly, we can prove A ∥ B ⇒ ¬ ₂A ∥ ¬ ₂B and A ∥ B ⇒ ¬ ₃A ∥ ¬ ₃B. ■

Example 1. Assume that we have the two complex fuzzy sets A and B in U = {x₁, x₂} as follows: $\begin{matrix} A & = & 0.2 e^{j 1.5 π} / x_{1} + 0.3 e^{j 0.6 π} / x_{2} \\ B & = & 0.6 e^{j 0.5 π} / x_{1} + 0.5 e^{j 0.6 π} / x_{2} \end{matrix}$

Then we have A ∥ B and $\begin{matrix} \neg_{1} A & = & 0.8 e^{j 0.5 π} / x_{1} + 0.7 e^{j 1.4 π} / x_{2} \\ \neg_{1} B & = & 0.4 e^{j 1.5 π} / x_{1} + 0.5 e^{j 1.4 π} / x_{2} \\ \neg_{2} A & = & 0.8 e^{j 1.5 π} / x_{1} + 0.7 e^{j 0.6 π} / x_{2} \\ \neg_{2} B & = & 0.4 e^{j 0.5 π} / x_{1} + 0.5 e^{j 0.6 π} / x_{2} \\ \neg_{3} A & = & 0.8 e^{j 0.5 π} / x_{1} + 0.7 e^{j 1.6 π} / x_{2} \\ \neg_{3} B & = & 0.4 e^{j 1.5 π} / x_{1} + 0.5 e^{j 1.6 π} / x_{2} \end{matrix}$

Then we can verify that ¬_iA ∥ ¬ _iB for any i ∈ {1, 2, 3}.

4.2 Complex fuzzy union

Let U be a universe of discourse, A and B be two complex fuzzy sets on U. The complex fuzzy union of A and B, introduced in [11], is given as $μ_{A \cup B} (x) = (r_{A} (x) \oplus r_{B} (x)) \cdot e^{j θ_{A \otimes B} (x)}$ (13) where ⊕ represents a t-conorm.

The following functions are possibilities of θ_A⊗B. $Sum : θ_{A \otimes B} = θ_{A} + θ_{B}$ (14) $Max : θ_{A \cup B} = max (θ_{A}, θ_{B})$ (15) $Min : θ_{A \cup B} = min (θ_{A}, θ_{B})$ (16) $Winner Take All : θ_{A \otimes B} = {\begin{matrix} θ_{A}, & r_{A} > r_{B} \\ θ_{B}, & r_{A} < r_{B} \end{matrix}$ (17) $Weighted Average : θ_{A \cup B} = \frac{r_{A} θ_{A} + r_{B} θ_{B}}{r_{A} + r_{B}}$ (18) $Average : θ_{A \cup B} = (θ_{A} + θ_{B}) / 2$ (19) $Product : θ_{A \cup B} = θ_{A} \cdot θ_{B}$ (20) $Difference : θ_{A \cup B} = θ_{A} - θ_{B}$ (21)

Remark 4. If the phase term is confined to the interval [0, 2π), so the above functions Sum, Product and Difference are addition modulo 2π, multiplication modulo 2π and subtraction modulo 2π, respectively.

Theorem 4. Let U be a universe of discourse, A, B and C be three complex fuzzy sets on U with r_A (x) ≠0, r_B (x) ≠0 and r_C (x) ≠0 for all x ∈ U. If θ_•⊗• is Sum or Difference function. Then A ∥ B ⇒ A ∪ C ∥ B ∪ C.

Proof. For any r₁ · e^{jθ
₁}, r₂ · e^{jθ
₂}, r₃ · e^{jθ
₃} ∈ D with r₁ ≠ 0 and r₂ ≠ 0.

If <r₂ · e^{jθ
₂}, r₃ · e^{jθ
₃} > = r₂ · r₃, then cos(|θ₂ - θ₃|) =1. $\begin{matrix} < (r_{1} \oplus r_{2}) \cdot e^{j θ_{1} \otimes θ_{2}}, (r_{1} \oplus r_{3}) \cdot e^{j θ_{1} \otimes θ_{3}}, > \\ = (r_{1} \oplus r_{2}) \cdot (r_{1} \oplus r_{3}) \cdot cos (| θ_{1} \otimes θ_{2} - θ_{1} \otimes θ_{3} |) \\ = (r_{1} \oplus r_{2}) \cdot (r_{1} \oplus r_{3}) \\ \cdot cos (| θ_{1} + θ_{2} - (θ_{1} + θ_{3}) |) \\ = (r_{1} \oplus r_{2}) \cdot (r_{1} \oplus r_{3}) \cdot cos (| θ_{2} - θ_{3} |) \\ = (r_{1} \oplus r_{2}) \cdot (r_{1} \oplus r_{3}) . \end{matrix}$

Thus, A ∥ B ⇒ A ∪ C ∥ B ∪ C when θ_•⊗• is Sum function. Similarly, we can prove the case of θ_•⊗• is Difference function. ■

Example 2. Assume that we have the three complex fuzzy sets A, B and C in U = {x₁, x₂, x₃} as follows: $\begin{matrix} A & = & 0.5 e^{j 0.1 π} / x_{1} + 0.3 e^{j 0.2 π} / x_{2} + 0.4 e^{j 1.2 π} / x_{3} \\ B & = & 0.8 e^{j 1.1 π} / x_{1} + 0.7 e^{j 0.2 π} / x_{2} + 0.8 e^{j 0.2 π} / x_{3} \\ C & = & 0.6 e^{j 1.2 π} / x_{1} + 0.2 e^{j 0.4 π} / x_{2} + 0.8 e^{j 0.4 π} / x_{3} \end{matrix}$

Then we have A ∥ B.

If ⊕ is max union function and θ_•⊗• is the Sum function, then $\begin{matrix} A \cup C & = & 0.6 e^{j 1.3 π} / x_{1} + 0.3 e^{j 0.6 π} / x_{2} \\ + 0.8 e^{j 1.6 π} / x_{3} \\ B_{2} \cup C & = & 0.8 e^{j 0.3 π} / x_{1} + 0.7 e^{j 0.6 π} / x_{2} \\ + 0.8 e^{j 0.6 π} / x_{3} \end{matrix}$

We can verify that A ∪ C ∥ B ∪ C.

If ⊕ is max union function and θ_•⊗• is the Max function, then $\begin{matrix} A \cup C & = & 0.6 e^{j 1.1 π} / x_{1} + 0.3 e^{j 0.2 π} / x_{2} \\ + 0.8 e^{j 1.2 π} / x_{3} \\ B_{2} \cup C & = & 0.8 e^{j 1.2 π} / x_{1} + 0.7 e^{j 0.4 π} / x_{2} \\ + 0.8 e^{j 1.2 π} / x_{3} \end{matrix}$

We can verify that A ∪ CnotparallelB ∪ C.

Theorem 5. Let U be a universe of discourse, A, B and C be three complex fuzzy sets on U with r_A (x) ≠0, r_B (x) ≠0 and r_C (x) ≠0 for all x ∈ U. If θ_•⊗• is Max, Min, or Winner Take All function, then $A ∥ B, A ∥ C \Rightarrow A ∥ B \cup C .$

Now we give a example with A ∥ B, A ∥ CnotRightarrowA ∥ B ∪ C when θ_•⊗• is Sum function.

Example 3. Assume that we have the three complex fuzzy sets A, B and C in U = {x₁, x₂, x₃} as follows: $\begin{matrix} A & = & 0.1 e^{j 0.1 π} / x_{1} + 0.2 e^{j 0.2 π} / x_{2} + 0.3 e^{j 1.2 π} / x_{3} \\ B & = & 0.1 e^{j 1.1 π} / x_{1} + 0.2 e^{j 1.2 π} / x_{2} + 0.3 e^{j 1.2 π} / x_{3} \\ C & = & 0.4 e^{j 0.1 π} / x_{1} + 0.7 e^{j 0.2 π} / x_{2} + 0.6 e^{j 0.7 π} / x_{3} \end{matrix}$

Then we have A ∥ B and A ∥ C.

If ⊕ is max union function and θ_•⊗• is the Sum function, then $\begin{matrix} B \cup C & = & 0.4 e^{j 1.2 π} / x_{1} + 0.7 e^{j 1.4 π} / x_{2} \\ + 0.6 e^{j 1.9 π} / x_{3} \end{matrix}$

We can verify that Anotparallel (B ∪ C).

Theorem 6. Let U be a universe of discourse, A, B, C and D be four complex fuzzy sets on U with r_A (x) ≠0,r_B (x) ≠0,r_C (x) ≠0 and r_D (x) ≠0 for all x ∈ U. If θ_•⊗• is the Sum or Difference function, then A ∥ B, C ∥ D ⇒ A ∪ C ∥ B ∪ D.

Proof. If r_A⊕C (x) =0 or r_B⊕D (x) =0. Then A ∥ B, C ∥ D ⇒ A ∪ C ∥ B ∪ D.

Similarly, we can prove the case of θ_•⊗• is Difference function. ■

Corollary 1. Let U be a universe of discourse, A_i, B_i ∈ CFS (U) i = 1, 2, …, n with r_{A
_i} (x) ≠0, r_{B
_i} (x) ≠0 for all and x ∈ U. If θ_•⊗• is Sum or Difference function, then $A_{i} ∥ B_{i} (i = 1, 2, . . ., n) \Rightarrow ⋃_{i = 1}^{n} A_{i} ∥ ⋃_{i = 1}^{n} B_{i}$ .

4.3 Complex fuzzy intersection

Let U be a universe of discourse, A and B be two complex fuzzy sets on U. The complex fuzzy intersection of A and B, introduced in [11], is given as $μ_{A \cup B} (x) = (r_{A} (x) ★ r_{B} (x)) \cdot e^{j θ_{A \otimes B} (x)}$ (22) where ★ represents a t-norm, θ_A⊗B is one of functions in (14)– (21).

Theorem 7. Let U be a universe of discourse, A, B and C be three complex fuzzy sets on U with r_A (x) ≠0, r_B (x) ≠0 and r_C (x) ≠0 for all x ∈ U. If θ_•⊗• is Sum or Difference function. Then A ∥ B ⇒ A ∩ C ∥ B ∩ C.

Proof. Similar to Theorem 4. ■

Theorem 8. Let U be a universe of discourse, A, B and C be three complex fuzzy sets on U with r_A (x) ≠0, r_B (x) ≠0 and r_C (x) ≠0 for all x ∈ U. If θ_•⊗• is Max, Min, or Winner Take All function, then $A ∥ B, A ∥ C \Rightarrow A ∥ B \cap C .$

Proof. Similar to Theorem 5. ■

Theorem 9. Let U be a universe of discourse, A, B, C and D be four complex fuzzy sets on U with r_A (x) ≠0,r_B (x) ≠0,r_C (x) ≠0 and r_D (x) ≠0 for all x ∈ U. If θ_•⊗• is the Sum or Difference function, then A ∥ B, C ∥ D ⇒ A ∩ C ∥ B ∩ D.

Proof. Similar to Theorem 6. ■

Corollary 2. Let U be a universe of discourse, A_i, B_i ∈ CFS (U) i = 1, 2, …, n with r_{A
_i} (x) ≠0, r_{B
_i} (x) ≠0 for all and x ∈ U. If θ_•⊗• is Sum or Difference function, then $A_{i} ∥ B_{i} (i = 1, 2, . . ., n) \Rightarrow ⋂_{i = 1}^{n} A_{i} ∥ ⋂_{i = 1}^{n} B_{i}$ .

5 Complex fuzzy inference

Next, we discuss whether parallelity can be preserved in a complex fuzzy inference system. Here, we only consider the complex fuzzy inference method of Ref. [12].

Let U and V be two universes of discourse. The form of fuzzy modus ponens(FMP) of complex fuzzy inference is given as follows:

Input:	X is A^*;
Rule:	IF X is A, THEN Y is B;
Output:	Y is B^* (denote FMP (A, B ; A^*)).

The sets A, A^* ∈ CFS (U) and B, B^* ∈ CFS (V) are complex fuzzy sets.

In [12], μ_A→B (x, y) is given as follows, $r_{A \to B} (x, y) = r_{A} (x) \cdot r_{B} (y)$ (23) $θ_{A \to B} (x, y) = θ_{A} (x) + θ_{B} (y)$ (24)

Then membership degree of B^* is given as follows, $r_{B^{*}} (y) = sup_{x \in U},$ (25) $θ_{B^{*}} (y) = f [g (θ_{A^{*}} (x), (θ_{A} (x) + θ_{B} (y)))],$ (26) where g is one of functions (14)– (21), f is one of the following functions $θ_{B^{*}} (y) = sup_{x \in U} [g (θ_{A^{*}} (x), (θ_{A} (x) + θ_{B} (y)))],$ (27) $θ_{B^{*}} (y) = inf_{x \in U} [g (θ_{A^{*}} (x), (θ_{A} (x) + θ_{B} (y)))],$ (28) $θ_{B^{*}} (y) = \sum_{x \in U} [g (θ_{A^{*}} (x), (θ_{A} (x) + θ_{B} (y)))] .$ (29)

Similar to Definition 5, the concept of parallelity preserving complex fuzzy inference is given as follows.

Definition 6. A complex fuzzy inference method f : CFS (U) → CFS (V) is a parallelity preserving complex fuzzy inference if for any A^*, A^*′ ∈ CFS (U), $A^{*} ∥ A^{*^{'}} \Rightarrow f (A^{*}) ∥ f (A^{*^{'}}) .$ (30)

Theorem 10. If f is the function of (29), and g is Sum or Difference function, then $A^{*} ∥ A^{*^{'}} \Rightarrow FMP (A, B; A^{*}) ∥ FMP (A, B, A^{*^{'}}) .$ (31)

Proof. Trivial from Theorem 7 and Theorem 9. ■

Example 4. Assume that we have the two complex fuzzy sets A and B in U = {x₁, x₂} and V = {y₁, y₂}, respectively, $\begin{matrix} A & = & 0.5 e^{j 0.1 π} / x_{1} + 0.6 e^{j 0.2 π} / x_{2}, \\ B & = & 0.4 e^{j 0.6 π} / y_{1} + 0.8 e^{j 0.7 π} / y_{2} . \end{matrix}$

Then we obtain $A \to B = [\begin{matrix} 0.2 e^{j 0.7 π} & 0.4 e^{j 0.8 π} \\ 0.24 e^{j 0.8 π} & 0.48 e^{j 0.9 π} \end{matrix}]$

If two inputs respectively are $\begin{matrix} A^{*} & = & 0.4 e^{j 0.1 π} / x_{1} + 0.2 e^{j 0.2 π} / x_{2} \\ A^{*^{'}} & = & 0.1 e^{j 1.1 π} / x_{1} + 0.7 e^{j 0.2 π} / x_{2} \end{matrix}$

We see that A^* ∥ A^*′.

If g is Sum function, ★ is Min t-norm, form of f is the function of (29). Then $\begin{matrix} FMP (A, B; A^{*}) & = & 0.2 e^{j 1.8 π} / y_{1} + 0.4 e^{j 2 π} / y_{2} \\ FMP (A, B; A^{*^{'}}) & = & 0.24 e^{j 0.8 π} / y_{1} + 0.48 e^{j π} / y_{2} \end{matrix}$

We can verify that FMP (A, B ; A^*) ∥ FMP (A, B ; A^*′).

If f is the function of (27). Then $\begin{matrix} FMP (A, B; A^{*}) = & 0.2 e^{j π} / y_{1} + 0.4 e^{j 1.1 π} / y_{2} \\ FMP (A, B; A^{*^{'}}) = & 0.24 e^{j 1.8 π} / y_{1} + 0.48 e^{j 1.9 π} / y_{2} \end{matrix}$

We can verify that FMP (A, B ; A^*) notparallelFMP (A, B ; A^*′).

Example 5. Assume that we have the two complex fuzzy sets A and B in U = {x₁, x₂} and V = {y₁, y₂}, respectively, $\begin{matrix} A & = & 0.5 e^{j 0.1 π} / x_{1} + 0.6 e^{j 0.2 π} / x_{2} + 1 e^{j 1.2 π} / x_{3}, \\ B & = & 0.4 e^{j 0.6 π} / y_{1} + 0.8 e^{j 0.7 π} / y_{2} . \end{matrix}$

Then we have $A \to B = [\begin{matrix} 0.2 e^{j 0.7 π} & 0.4 e^{j 0.8 π} \\ 0.24 e^{j 0.8 π} & 0.48 e^{j 0.9 π} \\ 0.4 e^{j 1.8 π} & 0.8 e^{j 1.9 π} . \end{matrix}]$

If two inputs respectively are $\begin{matrix} A^{*} & = & 0.4 e^{j 0.1 π} / x_{1} + 0.2 e^{j 0.2 π} / x_{2} + 0.3 e^{j 1.2 π} / x_{3} \\ A^{*^{'}} & = & 0.1 e^{j 1.1 π} / x_{1} + 0.7 e^{j 1.2 π} / x_{2} + 0.6 e^{j 1.2 π} / x_{3} \end{matrix}$

We see A^* ∥ A^*′.

If g is Sum function, ★ is Min t-norm, form of f is function in (29). Then $\begin{matrix} FMP (A, B; A^{*}) & = & 0.3 e^{j 0.8 π} / y_{1} + 0.6 e^{j 1.1 π} / y_{2} \\ FMP (A, B; A^{*^{'}}) & = & 0.4 e^{j 0.8 π} / y_{1} + 0.6 e^{j 1.1 π} / y_{2} . \end{matrix}$

We can verify that FMP (A, B ; A^*) ∥ FMP (A, B ; A^*′).

If form of g is Max function, then $\begin{matrix} FMP (A, B; A^{*}) & = & 0.3 e^{1.3 j π} / y_{1} + 0.6 e^{j 0.2 π} / y_{2} \\ FMP (A, B; A^{*^{'}}) & = & 0.4 e^{j 0.1 π} / y_{1} + 0.6 e^{j 0.2 π} / y_{2} \end{matrix}$

We can verify that FMP (A, B ; A^*) notparallelFMP (A, B ; A^*′).

6 Conclusion

In this paper, we introduced the parallelity relation between complex fuzzy sets. This is an equivalence relation. Then we explored the choices of complex fuzzy operators based on the criterion “complex fuzzy operators should be parallelity preserving”, and obtained the following results,

A ∥ B, ⇒ A ∘ C ∥ B ∘ C: Sum and Difference (See Theorems 4 and 7);

A ∥ B, A ∥ C ⇒ A ∥ B ∘ C: Max, Min and Winner Take All (See Theorems 5 and 8);

A ∥ B, C ∥ D ⇒ A ∘ C ∥ B ∘ D: Sum and Difference (See Theorems 6 and 9).

Our results are different from that obtained by Hu et al. [4], who defined the orthogonality preserving operators. The differences can be summarized as in Table 1.

Table 1

Comparison of the parallelity preserving with orthogonality preserving of complex fuzzy operators

	Sum	Difference	Max	Min	Winner Take All
A ∥ B ⇒ A ∘ C ∥ B ∘ C	√	√	×	×	×
A ⊥ B ⇒ A ∘ C ⊥ B ∘ C	√	×	×	×	×
A ∥ B, A ∥ C ⇒ A ∥ B ∘ C	×	×	√	√	×
A ⊥ B, A ⊥ C ⇒ A ⊥ B ∘ C	×	×	√	√	√
A ∥ B, C ∥ D ⇒ A ∘ C ∥ B ∘ D	√	√	×	×	×
A ⊥ B, C ⊥ D ⇒ A ∘ C ⊥ B ∘ D	×	×	×	×	×

√ and × represent the corresponding property holds and does not hold respectively.

The parallelity relation and the corresponding criterion maybe are very strict requirements for complex fuzzy sets and operators. As future work, we can consider a weaker parallelity relationship, denoted by “approximate” parallelity, and discuss approximately parallelity preserving operators in complex fuzzy sets.

Acknowledgments

This project was supported by the Science and Technology Foundation of Guizhou Province, China (LKS (2013) 35 and LKS (2012) 34), the Opening Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (Grant No. 2017CSOBDP0103) and the Guangxi University Science and Technology Research Project (Grant No. 2013YB193).

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