Two Extensions of the Sugeno Class and a Novel Constructed Method of Strong Fuzzy Negation for the Generation of Non-Symmetric Fuzzy Implications

: In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form.


Introduction
In recent years, there has been a growing body of research on fuzzy sets, systems, and fuzzy logic and their applications in practice, as well as the construction of new fuzzy negations, implications, and copulas.The production of a new fuzzy implication and copula is required.Fuzzy implications are the generalization of classical (Boolean) inference in the interval of [0, 1].They are widely known to play an important role in the fields of fuzzy logic, decision theory, and fuzzy control.For this reason, the generation of new fuzzy implications has created the need to generate new fuzzy negations.Extensive research has been conducted in the literature on the production of fuzzy negations [1][2][3][4][5][6].We know that we can generate fuzzy implications from aggregation functions and fuzzy negations [7][8][9][10][11][12][13][14][15].Other methods of generating fuzzy implications can be achieved using additive generating functions or by some initial implications [16][17][18][19][20][21][22].Thus, fuzzy implications are useful in fuzzy relational equations and fuzzy mathematical morphology, fuzzy measures and image processing [23], data mining [24], and computing with words and fuzzy partitions.On the other hand, functions with two variables, named copulas, have attracted the interest of many researchers because they are used in many fields.Copulas [25][26][27][28] are functions of two variables with specific properties based on probability theory and are often used in conditions to produce the copula.We present examples of the quadratic form of the copula and Archimedean copulas.

Preliminaries
To help the reader to become familiar with the theory, here, we outline some of the concepts and results employed in the rest of the paper.Definition 1. (see [30] Definition 1.4.1).The function N : [0, 1] → [0, 1] is a fuzzy negation if the following properties are applied: N: is decreasing. ( Definition 2. (see [30] Definition 1.4.2(i)).A fuzzy negation N is called strict if the following properties are applied: N is strictly decreasing.
N is continuous.
Definition 3. (see [30] Definition 1.4.2(ii)).A fuzzy negation N is called strong if Definition 4. (see [30] Definition 1.4.2(ii)).The solution of the equation N(x) = x is called the equilibrium point of N. If the function N is continuous, the equilibrium point is unique.
Table 1 below shows some basic fuzzy negations used in this article.
Table 1.Examples of basic fuzzy negations.

Name Fuzzy Negations
Threshold class The least fuzzy negation The greatest fuzzy negation N D2 (x) = 0, i f x = 1 1, i f x ∈ [0, 1)   Yager class Fuzzy implications have probably become the most important operations in fuzzy logic, approximate reasoning, and fuzzy control.These operators not only model fuzzy conditionals, but also make inferences in any fuzzy rule-based system.These operators are defined as follows.
Definition 7. (see [30] Definitions 1.3.1,1.5.1).A fuzzy implication I is said to satisfy: i. the left neutrality property if: ii. the exchange principle if: iii. the identity principle if: iv. the ordering property if: v. the law of contraposition with respect to N if: vi. the law of left contraposition with respect to N if: vii. the law of right contraposition with respect to N if: Definition 8. Let I be a nonempty interval of R. A function f from I to R is convex if and only if The C-volume of a rectangle must be not negative, e.g., for each x 1 ≤ x 2 and y 1 ≤ y 2 where 0 ≤ x 1 , x 2 , y 1 , y 2 ≤ 1.
Symmetry 2024, 16, 317 5 of 18 Definition 10. ( [28]).If the function C is a copula, then the function in form f or each 0 ≤ x, y ≤ 1 is also a copula, and it is called a survival copula.

Results
In this section, we give definitions and proofs of the newly generated fuzzy negations.The utility of fuzzy negations is known because, with the help of a new negation, we can construct a family of fuzzy implications.Also, we give examples of the new negations and the new fuzzy implications that are produced.Using the above equilibrium points and generalizing a known formula, we construct the branching functions and we generate strong fuzzy negations.Moreover, we combine the new fuzzy negations with the quadratic form of the copula.
Remark 4. The function of Theorem 2 is a strict fuzzy negation.
Remark 5.The function of Theorem 2 is an evolution if and only if g is a constant function.
Remark 6.If we choose g(x) = − 1 x+1 , then the fuzzy negation that is produced from Theorem 2 is a well-known N g (x) = 1 − x 2 .Proposition 2. Let g : R → (−1, +∞), the increasing function of the Theorem 2. We define the following increasing function, K : R → (−1, +∞), given by: If K is a concave function, then the fuzzy negation N g of the Theorem 2 is a convex function.
Proof.If K is an increasing function, then ∂x 2 ≤ 0. We will calculate the second partial derivate of K.
We know from Theorem 3 that N g is a strictly decreasing function, so: In Figure 1 we present the graphs of three functions, N 3,5 (x) = 1−x 3x 2 +5x+1 (the black one), N 5 (x) = 1−x 1+5x (the green one), and the identity function f (x) = x.(the blue one).N 3,5 belongs to fuzzy negations of Theorem 1 and N 5 belongs to fuzzy negation N δ of the Table 1.
Let  1 ,  2 be the equilibrium points of  , and   , respectively.Suppose that In Figure 1 we present the graphs of three functions,  3,5 () =  As we can see, the equilibrium point of  , ( = 3 > 0) is to the left of the point of the   function.
We can produce strong branching fuzzy negations [1] where one branch is a rational function.If N is a fuzzy negation, which is not necessary, there is a strong negation, and () = , where ε is the equilibrium point of .Thus, if  is any continuous fuzzy negation in the interval [0, 1], then the following form [12] produces strong fuzzy negations  1 , and, in our case, rational fuzzy negations.As we can see, the equilibrium point of N γ,δ ( γ = 3 > 0) is to the left of the point of the N δ function.
The distance between these two equilibrium points is , where ε 1 , ε 2 are the equilibrium points of functions N γ , N δ , respectively.
We can produce strong branching fuzzy negations [1] where one branch is a rational function.If N is a fuzzy negation, which is not necessary, there is a strong negation, and N(ε) = ε, where ε is the equilibrium point of N. Thus, if N is any continuous fuzzy negation in the interval [0, 1], then the following form [12] produces strong fuzzy negations N 1 , and, in our case, rational fuzzy negations.Below, in Figure 2 we present the graph of the function N 1  We will generalize the above formula using two decreasing functions, , .

𝑔𝑔(𝑥𝑥)•𝜀𝜀 𝑔𝑔(𝜀𝜀)
For every  1 ,  2 ∈ [0, ] where Thus, we conclude that   is decreasing when  > .□ Now, we will present some examples of new fuzzy negations.We will define some values for parameters γ and δ of Theorem 1, and we will produce new fuzzy negations.We will generalize the above formula using two decreasing functions, f , g. Theorem 3. Let ε be the equilibrium point of N s .f : R → (−∞, 1] and g : R → [0, +∞), two decreasing functions with the following conditions: f −1 = g, f (0) = 1 and g(1) = 0. Then the following form is a strong fuzzy negation: Thus, we conclude that N s is decreasing when x ≤ ε.
For every Thus, we conclude that N s is decreasing when x > ε. □ Symmetry 2024, 16, 317 Now, we will present some examples of new fuzzy negations.We will define some values for parameters γ and δ of Theorem 1, and we will produce new fuzzy negations.
In Figure 3  (the black one).
In Figure 3  Then, the produced strong fuzzy negation has the following form and graph: is the equilibrium point of   .
In the Figure 4 we present the graph of function   .Particularly, the black graph is the first branch of   and the green graph is the second branch of   .Then, the produced strong fuzzy negation has the following form and graph: where ε = 3 (the black one).
In Figure 3  Then, the produced strong fuzzy negation has the following form and graph: is the equilibrium point of   .
In the Figure 4 we present the graph of function   .Particularly, the black graph is the first branch of   and the green graph is the second branch of   .3.2.Fuzzy Implications Generated from Fuzzy Negations of Theorems 1 and 2 Fuzzy negations are very useful in the construction of both fuzzy implications and copulas.Firstly, we will use fuzzy negations that were generated from the above methods for the construction of fuzzy implications.▼N 1 (y) , N 1 , N 2 are two fuzzy negations and d : [0, 1] → [0, ∞) is an increasing and continuous function with g(0) = 0. From Theorem 1, if we set γ = 1, δ = 1, and from Theorem 3, if g(x) = x 2 , then the produced negations are N 1,1 (x) = 1−x x 2 +x+1 and N g (x) = 1−x x 3 +1 .If we set N 1 = N 1,1 , N 2 = N g , and d(x) = √ x, we generate the following fuzzy implication: And its natural negation is Moreover, we can construct parametric fuzzy implications using the produced fuzzy negations of Theorems 1 and 2.
Remark 7. If we choose a strong fuzzy negation from relation (32), then the produced fuzzy implication satisfies the neutrality property.Proposition 4. Let N γ,δ , N g be two fuzzy negations of Theorems 1, 2 with the form N γ,δ (x) = ▼N 1 (y) , and the produced implication has the form: Proof.Proof is obvious.□ With this combination, we can make many rational fuzzy implications.We can make a family of them.Also, we can use fuzzy negations as decreasing functions to construct copulas.The following form helps us to understand this.
In the literature, various methods of manufacturing copulas have been presented.Here, we will deal with the Archimedean copulas.
Example 5.According to Theorem 1, if we choose for γ = −3, δ = 7, then we take the fuzzy negation: Fuzzy negation N −3,7 is strict and convex.Its inverse function has the form: And its pseudoinverse has the form: If we define: then, by the form (18), a new copula is generated.
Proof.If we calculate the second derivate of N γ,δ , we have the following: And the discriminant of the polynomial If we examine the case, γ < 0 and 2γ + δ > 0; then, ∆ = 36γ 3 (1 + δ + γ) < 0, which means that In this case (γ < 0), we already know that γ + δ + 1 > 0. So, In the case that γ > 0 and δ 2 + δ − γ > 0, we have the following: , is an increasing function.Thus, Copulas with Quadratic Sections We will analyze one of the linear sections of copulas, the quadratic form of the copula.In the book of Nelsen [28], we can find the quadratic and cubic sections of copulas.
The quadratic section of the copula is defined by the form: And the cubic section of the copula is defined by the form: Before that, we present a proof of the equivalency ,y) ∂xy ≥ 0 and the last condition of the copula.We will use the main value theorem twice.Proposition 6.We know that, for a function to be 2-increasing, it must satisfy the inequality when C is a differentiable function.
Proof.We apply the mean value theorem for the function C(x, y 1 ) in the interval [x 1 , x 2 ]: We apply the mean value theorem for the function C(x, y 2 ) in the interval [x 1 , x 2 ]: Let us suppose that C(x 1 , y 1 ) + C(x 2 , y 2 ) − C(x 1 , y 2 ) − C(x 2 , y 1 ) ≥ 0, then: The other implication is analogous, but proceeds in reverse order.□ Proof.Employing the boundary conditions, we have the following: After the boundary conditions, the quadratic form of the copula takes the following form We know that, in order for a function to be 2-increasing, it must satisfy the following inequality: This inequality is equivalent to ∂xy ≥ 0 when the function C is differentiable.We will use this inequality to set some conditions for the function a(y).

∂C(x, y)
∂x = 2xa(y) + y − a(y) In the case of ∂a(y) ∂y ≥ 0, we have: In the case of ∂a(y) ∂y ≤ 0, we have: We conclude that the function a(y) must satisfy the following condition: □ Function a(y) has at least two roots: the numbers 0 and 1.Using root 1, and because it is also a root of the fuzzy negations, we can give it the following form: a(y) = y•N(y), where N is a fuzzy negation.Here, we can make combinations with the above generated fuzzy negations.If we choose the Sugeno fuzzy negation, we will study what conditions must be satisfied by coefficient δ.Theorem 4. Let N δ : [0, 1] → [0, 1], the Sugeno class fuzzy negation N δ (x) = 1−x δχ+1 , and a : [0, 1] → R with the form a(y) = y•N δ (y).
If we take the quadratic section of the copula C(x, y) = a(y)x 2 + ( y − a(y))x for δ = 2 we can set a(y) = y•N δ (y) = y•(1−y) 2y+1 , then the produced copula is: The survival copula ( 15) is defined by: which is true for every x ∈ [0, 1].Thus, we can produce another copula if we set a(y) = yN g (y) The survival copula is defined by:

A Presentation of a Hypothetical Scenario
Let X, Y represented linguistic variables, i.e., fuzzy sets.Let us also suppose that X ⇒ Y .For each, x ∈ X and y ∈ Y correspond to a value pair (x i , y i ).If we collect a "good sample" (x i , y i ), i = 1, . . ., n, then we have: x n ⇒ y n But since we have a "good sample", Then, using our implication, which is produced by a convex negation with a parameter a, we have the parametric implication J(x, y, a).
x n ⇒ y n = J(x n , y n , a) Now, we can select the "best implication" as that which has the shortest distance from 1.That is, we see this when the number (1 − J(x 1 , y 1 , a)) 2 + (1 − J(x 2 , y 2 , a)) 2 + . . .+ (1 − J(x n , y n , a)) 2  (51) becomes the minimum.For further reading and other research applications, the development method in [31,32] could also be applied.

Conclusions
The main goal of our construction of fuzzy negations is the generation of fuzzy implications.The symmetry or lack thereof of the generated fuzzy implications plays a key role in the application.For example, if the generated implications are symmetric, then the cause and the causality are mixed.In our construction, the cause and the causality are distinct.Moreover, new fuzzy implications give us new copulas.In this work, we have proposed some novel construction methods of fuzzy negations.Firstly, we presented a new class of rational fuzzy negations inspired by the Sugeno class fuzzy negation.Secondly, we replaced the parameter δ of the Sugeno class with an increasing function g with specific conditions.We generalized a form which generates strong fuzzy negations by using two decreasing functions f , g.Also, we gave some extra conditions so that the new fuzzy negations would be convex and we gave many examples of the new generated fuzzy negations (see Figures 1-5).Finally, we dealt with the quadratic section of the copula, trying to find the appropriate function a(x) using the new fuzzy negations of the Theorems 1 and 2. As a future work, we can investigate some other methods that produce copulas.We will produce parametric copulas and research what conditions the coefficients must satisfy.At this point, we must emphasize the fact that our suggested method could also be applied by using a method that is amazing, in our opinion, on the cubic section of the copula: C(x, y) = a(y)x 3 + b(y)x 2 + c(y)x + d(y) which we can find in the book of Nelsen [28].With this method, the case of the cubic section of the copula, which has two functions, a and b, with common roots of the numbers 0 and 1, is investigated.Using the new fuzzy negation of the first method, we can find which conditions its coefficients must satisfy in order to produce a copula.We could accomplish the same result using the extension of the Sugeno class.In this class, we investigate a function g in which we must determinate properties to be satisfied such that a cubic copula can be produced.On the other hand, we can continue to extend other well-known fuzzy negations by using appropriate functions instead of parameters.In addition to this, we can investigate the extension of the Yager class of fuzzy negation, where instead of the parameter w, we can use a function with the appropriate properties.

18 Example 2 .
equilibrium point of N s .In the Figure4we present the graph of function N s .Particularly, the black graph is the first branch of N s and the green graph is the second branch of N s .Symmetry 2024, 16, x FOR PEER REVIEW 9 of If we define  = 1,  = 0, the produced negation is  1,0 () =

Example 4 . 2 d
According to the formula [8] I(x, y) = N
The fuzzy negation of the Theorem 1 is strong if and only if γ = 0, which means it is the Sugeno class.