Classiﬁcation of Complex Fuzzy Numbers and Fuzzy Inner Products

: The paper is concerned with complex fuzzy numbers and complex fuzzy inner product spaces. In the classical complex number set, a complex number can be expressed using the Cartesian form or polar form. Both expressions are needed because one expression is better than the other depending on the situation. Likewise, the Cartesian form and the polar form can be deﬁned in a complex fuzzy number set. First, the complex fuzzy numbers (CFNs) are categorized into two types, the polar form and the Cartesian form, as type I and type II. The properties of the complex fuzzy number set of those two expressions are discussed, and how the expressions can be used practically is shown through an example. Second, we study the complex fuzzy inner product structure in each category and ﬁnd the non-existence of an inner product on CFNs of type I. Several properties of the fuzzy inner product space for type II are proposed from the modulus that is newly deﬁned. Specﬁcally, the Cauchy-Schwartz inequality for type II is proven in a compact way, not only the one for fuzzy real numbers. In fact, it was already discussed by Hasanhani et al; however, they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way from a general point of view. Finally, we introduce a complex fuzzy scalar product for the generalization of a complex fuzzy inner product and propose to study the condition for its existence on CFNs of type I.


Introduction
In classical complex analysis, we have two different types of notations in the complex number set C. One is the Cartesian form (z = a + bi), and the other one is the polar form (z = re i θ). Even though we have the Cartesian form, the polar form is needed depending on the situation. The Cartesian form is generally convenient, but if we have to use the angle such as in triangular functions, the polar form is better. Likewise, in fuzzy complex analysis (CF(C) or CF(R); see Section 3), we can first consider the Cartesian complex fuzzy number form, but depending on the situation, it is necessary to define the polar form to express some special cases (see Example 2). Sometimes, the polar form in CF(C) or CF(R) is better than the Cartesian form. For example, if we want to express some situation with a "periodic" time series that has a fuzzy amplitude, the polar form is better than the Cartesian form. As we see in Example 2, we can express the periodicity easily if the data include the monthly property or the seasonality. However, due to the complexity of CF(C) or CF(R), not all the properties in C hold in CF(C) or CF(R). Therefore, in this paper, we discuss similar properties in C that we can even have in CF(C) or CF(R), and we also discuss some properties that we cannot have in CF(C) or CF(R). For example, as we discuss in the paper, the complex fuzzy inner product does not exist based on the polar form. The Cauchy-Schwartz inequality holds using the Cartesian form in the complex fuzzy number set.
The definitions of complex fuzzy numbers have been introduced in several studies [1][2][3]. The complex fuzzy numbers can be applied to many real applications [3]. Buckley [1] first introduced a complex number approach to fuzzy numbers, which was named "fuzzy complex numbers". He defined the Cartesian form and the polar form of fuzzy complex numbers z based on their membership function, which is a mapping from the complex numbers into [0, 1]. On the other hand, Ramot et al. [3] introduced a complex-valued grade of fuzzy membership functions to the magnitude (modulus) and argument of the polar form to define complex fuzzy numbers. Fu and Shen [2] introduced a complex-valued grade of fuzzy membership functions to the real part and the imaginary part to define fuzzy complex numbers. There were some studies that dealt with the fuzzy inner product [4][5][6] and the fuzzy Hilbert space [7,8]. Recently, we [9] discussed the absence of non-trivial fuzzy inner product spaces and the Cauchy-Schwartz inequality in the fuzzy real number system.
In this paper, we discuss more properties of the complex fuzzy numbers as an extension of the previous study [1]. This approach is different from Buckley's [1]. We deal with complex fuzzy numbers, not "fuzzy complex numbers".
In Sections 3 and 4, the complex fuzzy numbers (CFNs) are categorized into two types, as CFNs of type I for the polar form and CFNs of type II for the Cartesian form. The basic properties of the operations in each category are checked.
A fuzzy Hilbert space has been introduced in [7,8]. This paper suggests a complex fuzzy inner product space based on our new approach. In Section 5, we discuss the complex fuzzy inner product on each category of CFN, which shows that, unlike the complex number, the complex fuzzy number of the polar form (type I) has no relation to that of the Cartesian form (type II). In fact, the non-existence of inner products on CFN of type I is proved. We introduce the definition of a complex fuzzy inner product space for type II. Also, several properties of the fuzzy inner product space for type II have been proposed from the modulus defined in Section 5. Especially, the Cauchy-Schwartz inequality for type II is proved in in a compact way, not only the one for fuzzy real numbers. In fact, it was already dealt with in [7], however they proved every case in a very complicated way. In this paper, we prove the Cauchy-Schwartz inequality in a much simpler way in a general point of view.
In Section 6, we introduce a complex fuzzy scalar product and investigate its properties.

Preliminaries
In this section, we provide basic definitions and notations for this study.
The set of all fuzzy real numbers is denoted by F(R). If η ∈ F(R) and η(t) = 0 whenever t < 0, then η is called a non-negative fuzzy real number and F * (R) denotes the set of all non-negative fuzzy real numbers. We note that real number η − α 0 for all η ∈ F * (R) and all α ∈ (0, 1]. Each r ∈ R can be considered as the fuzzy real numberr ∈ F(R) denoted bỹ hence it follows that R can be embedded in F(R).
which are special cases of Zadeh's extension principles.

Complex Fuzzy Numbers of Type I
Recall that the polar representation z = re iθ of a complex number, where r ≥ 0 and θ ∈ R [3]. In this section, we consider its extension in fuzzy category, called complex fuzzy numbers of type I, and investigate their basic properties under some structure.

Definition 5 ([3] (p. 171)).
A complex fuzzy set S of type I defined on a universe of discourse U, is characterized by a membership function that assigns a complex-valued grade of membership in S where r S (x) and θ S (x) are both real-valued and r S (x) is a fuzzy real number on U. Here, r S (x) = |µ S (x)| is called the amplitude of µ S (x).
The complex fuzzy set S may be represented as the set of ordered pairs (1) We now give examples of a complex fuzzy set S of type I. Example 1. Let us consider a complex fuzzy set defined by Then µ S (x) can be expressed in Figure 1.  (49))). Let U be the set of financial indicators or indexes of the American economy. Possible elements of this set are unemployment rate, inflation, interest rates, growth rate, GDP, Dow-Jones industrial average, etc. Let V be the set of financial indicators of the Japanese Economy. Let the complex fuzzy relation R(U, V) represent the relation of influence of American financial indexes on Japanese financial indexes: y is influenced by x", where x ∈ U and y ∈ V. The membership function for the relation R(U, V) and µ R (x, y), can be presented by complex valued, with an amplitude term and a phase term. The amplitude term indicates the degree of influence of an American financial index on a Japanese financial index. Consider, for example, µ R (Growth rate, export), i.e., the grade of membership associated with the statement: "American growth rate influences Japanese Export". Assume the interactions between American and Japanese financial indicators are measured in the limited time frame of 12 months, then it can be represented by

Example 2 ([3] (p. 184, Equation
Note that the amplitude term was selected to be 0.8, similar to the grade of membership of a traditional fuzzy set. The phase term was chosen to be 4 12 · 2π as an average of three-five months, normalized by 12 months. Consider the case both S = C and R respectively, which given CF I (C) and CF I (R). Definition 6. A mapping η : C → D 2 (or η : R → D 2 , respectively) is called a complex fuzzy number (CFN) on C( or R), respectively, whose α-level set is denoted by if it satisfies two axioms; (i) There exists z 0 ∈ C ( or R), respectively, such that |η(z 0 )| = 1.
It is obvious that both R and F(R) can be embedded in CF I (R). Furthermore, both C and CF I (R) can be embedded in CF I (C). In this section, CF I (R) and CF I (C) will be written by CF(R) and CF(C) for the simplicity, respectively.
Given r ∈ R, recall that the definition of r ∈ F(R) given by Let CF(U) be the set of all CFNs on U = R or C. Note that each z = re iθ ∈ C can be considered as the CFN z ∈ CF(C) defined by Then, z ∈ CF(U) is the extension of r ∈ F(R). Moreover, C can be embedded in CF(C).
Since each z ∈ R can be considered as the complex fuzzy number z ∈ CF(C) defined by it follows that R can be embedded in CF(C).
In this paper, we focus on CFN based on R.
Definition 7. Let η(t) = r η (t)e iθ η (t) and δ(t) = r δ (t)e iθ δ (t) be complex fuzzy numbers on R. Then Note that η(t) ∈ C for each t ∈ R so that |η(t)| ∈ R. As the square root of a polar form of a complex number, that of a complex fuzzy number is done as follows: Definition 8. For a CFN η(t) = r η (t)e iθ(t) on R, we define the square root of η(t) by where r(t) is the amplitude of η(t) and defined by r η (t) = |η(t)|.

Definition 9.
The absolute value |η| of η ∈ CF(R) of type I is defined by The Definition 9 implies |η| ∈ F(R).

Complex Fuzzy Numbers of Type II
Note that each z = a + bi ∈ C can be considered as the CFN of type II, z ∈ CF II (C), defined by where " i " is the imaginary unit of a complex fuzzy number of type II, ( not of a complex number). Then, z ∈ CF II (C) is the extension of r ∈ F(R). Moreover, C can be embedded in CF(C). In this section, CF II (C) will be written by CF(C) for the simplicity.

Definition 12.
A complex fuzzy set S of type I I defined on a universe of discourse U, is characterized by a membership function that assigns a complex-valued grade of fuzzy membership in S where both x S and y S are fuzzy real numbers on S.
Definition 13 ([2] (p. 1405)). Let z 1 = a ⊕ i b and z 2 = c ⊕ i d be complex fuzzy numbers where a, b, c, and d are fuzzy real numbers. The basic arithmetic operations on z 1 and z 2 are defined as follows; (i) Addition:
Note that the modulus of the Definition 14 is different from the absolute value of the Definition 9 even though they are equivalent in C. The triangle inequality on modulus may be shown in Theorem 1 with respect to type II. Theorem 1. Given two fuzzy real numbers ξ, η ∈ CF(R), it holds that |ξ ⊕ η| ≤ |ξ| ⊕ |η|.

Complex Fuzzy Inner Product Space
In this section, we introduce the definition and investigate some properties of a complex fuzzy inner product space for type II. We will use operations in [2] (p. 1405).
Definition 16. Let X be a vector space over C. A complex-valued fuzzy inner product on X is a mapping , : X × X → CF(R) such that for all vectors x, y, z ∈ X ans r ∈ C, we have (IP 1 ) x + y, z = x, z ⊕ y, z , The vector space X with a complex-valued fuzzy inner product is called a complex fuzzy inner product space.

Non-Existence of the Inner Product on CFNs of Type I
In this subsection, a complex fuzzy inner product in view of type I can not be defined. Let V be a given complex n-dimensional vector space. To show by a contradiction, assume that ·, · : V × V → CF(R) is a complex fuzzy inner product on V. It is well known that there is a basis β = {v 1 , · · · , v n , iv 1 , · · · , iv n } on a real vector space V. Let x = v 1 and y = iv 1 . Recall that, for z = re iθ , the CFNz ∈ CF(R) is given by Lemma 3. If r > 0, then re iθ =re iφ for some function φ : R → R satisfying e iφ(r) = e iθ .
Proof. Assume r > 0 and let re iθ = f e iφ for some fuzzy real number f ∈ F(R) and for a function φ : R → R. Then, Equation (6) says that, for t = r, which implies that f (t) = 0 for t = r. For t = r, it also does that which gives both f (r) = 1 and e iφ(r) = e iθ . Thus we get f =r.
The following example shows that the inner product of CFN of type I on a vector space cannot be defined.

Complex Fuzzy Inner Product Spaces Based on CFNs of Type II
A complex fuzzy inner product on X defines a fuzzy number for all x ∈ X. In fact, x = | x, x | from the the positive-definite property of an inner product.
To begin with, for real numbers a, b ∈ R, a + ib =ã ⊕ ib. Especially, ±i = i b with b = ±1.
Theorem 2. Given a complex fuzzy inner product space (V, ·, · ) and for given two element v, w ∈ V, if v, w = x ⊕ i y for some fuzzy real numbers x, y ∈ F(R), then x =ã and y =b for some real numbers a, b ∈ R.
Proof. Note that is a fuzzy real number. Thus y y =0 and so y =b for some b ∈ R. And, from v, w w, v = (x x) ⊕ i (2 ⊗ y), we get 0 ≤ iv + w, iv + w = iv, iv + w ⊕ w, iv + w which implies that x x =0, thus x =ã for some a ∈ R.
We give a simple application regarding the inner product of complex fuzzy numbers which is of type II. x, y (t) = 1 + i, t = a, 0, o.w.
Remark 3. Theorem 2 shows that an inner product complex fuzzy space is trivial. To find more meaningful complex fuzzy space, we will change the condition of positive definiteness (IP 4 ) in Definition 16 to that of non-degeneracy (see the Section 5).

Lemma 4 ([7] (Lemma 3.2)).
A real fuzzy inner product space X together with its corresponding norm · satisfy the Cauchy-Schwartz inequality for all x, y ∈ X.
From now on, the result of Theorem 2 is not used, which enables us to apply the arguments below to the real fuzzy inner product space by letting the imaginary part be0.

Remark 4.
Given a complex fuzzy number η = x ⊕ iy, where both x and y are fuzzy real numbers, and given α ∈ (0, 1], the equalities Remark 5. For a vector v in a fuzzy inner product space (V, ·, · ) and for t ≥ 0, The following lemma is easily checked: Lemma 5. Given a fuzzy real number x ∈ F(R), Theorem 3. For vectors v, w, and for each α ∈ (0, 1], we have Hence, it holds that | v, w | ≤ v ⊗ w .
Proof. Since all of | v, w |, v and w are fuzzy real numbers, it suffices to show that the inequality | v, w | + α ≤ v − α w − α holds for each α ∈ (0, 1]. If w is a zero vector 0, then which implies | v, w | + α = 0 and so the theorem holds. Assume that w is not a zero vector. Then from the Definition 16 (IP5), w − α = 0. Denote v, w = x ⊕ iy by fuzzy real numbers x and y. Let and put b, related to y − α and y + α , in a similar way. Consider γ = a + ib ∈ C. Then, the inequalitỹ holds andγ ⊗ v, w ⊕γ ⊗ v, w becomes a fuzzy real number. Thus we can rewrite the equality as follows: From Lemma 5, Similarly, we get

Complex Fuzzy Scalar Product
We already saw that no inner product can exist on the complex fuzzy numbers of type I. In Linear Algebra, recall the concept of a scalar product, which is a weaker version of the concept on an inner product. We introduce a complex fuzzy scalar product for a generalization of a complex fuzzy inner product. In this section, CF(R) does not restrict to the case of type II, except Theorem 4.
Definition 17. Let X be a vector space over C. A complex-valued fuzzy scalar product on X is a mapping , : X × X → CF(R) such that for all vectors x, y, z ∈ X ans r ∈ C, we have For s = ( | v, v |) − α and t = ( | w, w |) − α , both | v, v |(s) and | w, w |(t) are greater than or equal to α, thus (SP 2 ) says that Therefore, The Equation (8) A complex fuzzy scalar product on X defines a fuzzy number for all x ∈ X. Lemmas 6 and 7 imply that Theorem 5. For vectors x, y, we have the following inequalities (i) | x, y | ≤ x ⊗ y .

Conclusions
In this study, we defined complex fuzzy numbers (CFNs) of the Cartesian form and the polar form and took a look at the advantage for each type, providing examples. To study them, we proposed some operations and properties such as triangular inequality in fuzzy real number set. Based on proposed definitions and properties, complex fuzzy inner product space has been proposed based on CFN of the Cartesian form. Especially, we made a more clear definition of modulus, which gave an easy proof for Cauchy-Schwartz inequality. We also showed the non-existence of the inner product on CFNs of the polar form. This implies that the polar form cannot substitute Cartesian form, so we need to apply each type in the right situation. And we proposed a new concept, called complex fuzzy scalar product, and proved its some basic properties such as both Cauchy-Schwartz inequality and triangular inequality. Regarding this, the condition for the existence of a scalar product on CFN of type I will be discussed in our further study.