A method of constructing fuzzy implications from the FIφ -construction

Abstract

In the recent years, several new construction methods of fuzzy implications have been proposed. However, these construction methods actually care about that the new implication could preserve more properties. In this paper, we introduce a new method for constructing fuzzy implications based on an aggregation function with F (1, 0) =1, a fuzzy implication I and a non-decreasing function φ, called FIφ-construction. Specifically, some logical properties of fuzzy implications preserved by this construction are studied. Moreover, it is studied how to use the FIφ-construction to produce a new implication satisfying a specific property. Furthermore, we produce two new subclasses of fuzzy implications such as UIφ-implications and G_pIφ-implications by this method and discuss some additional properties. Finally, we provide a way to generate fuzzy subsethood measures by means of FIφ-implications.

Keywords

Fuzzy implication aggregation function uninorm nullnorm fuzzy subsethood measure

1 Introduction

Fuzzy implications attract wide attention of researches occupied in fuzzy logic both because of their interesting properties and many applications, not only in fuzzy control and approximate reasoning, but also in many other areas they have proved to be valuable like composition of fuzzy relations [7 , 20], fuzzy relational equations [22, 23], fuzzy mathematical morphology [11, 14], fuzzy neural networks [38, 39], fuzzy rough sets [12 , 41] and data mining [43].

This fact has led more and more people to a systematic research of many fuzzy implication functions in theory (see the recent survey [4] and the book [3]). However, in these theoretical studies, the new methods of constructing fuzzy implications have become a hot topic.

From these studies, more and more construction methods have been introduced over time, some interesting methods construct fuzzy implications from given ones. Specifically, there are many well-known construction methods, for which we highlight the following ones:

The classical ones like for instance the φ-conjugate or the N-reciprocal, lower and medium contrapositivisations, and the minimum, maximum or any convex linear combinations of fuzzy implication functions (see [3] for details).

Other construction methods such as the horizontal and vertical threshold generation methods [30 –32], some new types of contrapositivisations [2] and the ⊛-composition [40].

More recent construction methods for fuzzy implications have been introduced like the rotation construction [36], the FNI-method [1], the quadratic construction method [26] and the n increasing functions and n + 1 negations-composition [35].

In fact, when finding new construction methods to generate fuzzy implication functions, we need to consider two main important points. Firstly, the new construction method of implication functions should be as simple as possible so that the new ones can be easily express and compute. Secondly, the new obtained implication functions should preserve as many desired properties as possible.

In this sense, we introduce a new method for constructing fuzzy implications based on an aggregation function F with F (1, 0) =1, an implication function I and a non-decreasing function φ, called FIφ-construction. On the one hand, we will show how to use this method to modify a given implication such that the new obtained implication function can be satisfied some desired properties. On the other hand, we find our method can be applied to generate fuzzy subsethood measures, which is useful in several areas, such as fuzzy decision making [27], fuzzy relational databases [28], image processing [9], clustering [44] and intelligent systems [34].

The importance of our research can be divided into the following two aspects:

The first aspect is that the importance of new construction methods is related to the preservation of the properties of the initial fuzzy implications to the final one. Although all the construction methods can preserve certain properties, none of the previous construction methods from two implications ensure a resulting implication satisfying the exchange principle even when the two initial ones satisfy it. Therefore, it is necessary and interesting to study how to modify a given fuzzy implication such that the new obtained implication can be satisfied some desired properties, such as, identity principle, exchange principle and strong negation principle.

The second one is that the construction methods of fuzzy implications have been used in many practical applications. For instance, Dimuro et al. [16] generated fuzzy subsethood and entropy measures by means of QL-implications constructed from tuples (O, G, N_⊤), where O is an overlap function, G is an grouping function and N_⊤ is the greatest fuzzy negation. Later, Pinheiro et al. [33] introduced a method to construct fuzzy subsethood measures using (T, N)-implications. Su et al. [36] proposed the rotation construction, which provides a wide spectrum of choices for fuzzy connectives. Recently, Zanotelli et al. [45] introduced n-Dimensional fuzzy (S, N)- implication and explored it in approximate reasoning.

The paper is organized as follows: In Section 2, we recall some basic definitions and related properties which are necessary for the development of this paper. In Section 3, we study the new construction method for fuzzy implications. Firstly, we analyse the logical properties of the newly constructed fuzzy implications, and then we discuss how to use the FIφ-construction to modify a given implication and we also compare our method with several existing methods. In Section 4, we produce two subclasses of fuzzy implications such as UIφ- and G_pIφ-implications through this method and discuss some additional properties. In Section 5, we provide a way to generate fuzzy subsethood measures using FIφ-implications. Section 6 concludes the paper with our final remarks, and outlines some further work.

2 Preliminaries

In this section, we recall here some basic definitions and related properties which are necessary for the development of this paper. More details about t-norms and t-conorms can be found in [24].

Definition 2.1. ([24]). A function T : [0, 1] ² → [0, 1] is called a t-norm, if it satisfies the following conditions for all x, y, z ∈ [0, 1]:

Commutativity: T (x, y) = T (y, x) ;

Associativity: T (T (x, y) , z) = T (x, T (y, z)) ;

Monotonicity: T (x, y) ⩽ T (x, z) for y⩽ z ;

1-identity: T (x, 1) =1.

Definition 2.2. ([24]). A function S : [0, 1] ² → [0, 1] is called a t-conorm, if it satisfies the following conditions for all x, y, z ∈ [0, 1] :

Commutativity: S (x, y) = S (y, x) ;

Associativity: S (S (x, y) , z) = S (x, S (y, z)) ;

Monotonicity: S (x, y) ⩽ S (x, z) for y⩽ z ;

0-identity: S (x, 0) = x.

Definition 2.3. ([6]). A function F : [0, 1] ² → [0, 1] is called an aggregation function, if it satisfies the following conditions:

F is increasing in each variable;

F (0, 0) =0 and F (1, 1) =1.

Definition 2.4. ([10]). An aggregation function U is a uninorm, if it satisfies, for all x, y, z ∈ [0, 1], the following conditions:

Commutativity: U (x, y) = U (y, x) ;

Associativity: U (U (x, y) , z) = U (x, U (y, z)) ;

There is a neutral element e ∈ [0, 1], such that U (x, e) = x for all x ∈ [0, 1].

It is clear that U is a t-norm if e = 1 and is a t-conorm if e = 0. Moreover, for any e ∈ (0, 1), the uninorm U works as a t-norm in the [0, e] ² square, and as a t-conorm in [e, 1] ². We usually denote a uninorm U with neutral element e, underlying t-norm T_U, and underlying t-conorm S_U by U ≡ 〈T_U, e, S_U〉.

Definition 2.5. ([10]). An aggregation function G is a nullnorm, if it satisfies, for all x, y, z ∈ [0, 1], the following conditions:

Commutativity: G (x, y) = G (y, x) ;

Associativity: G (G (x, y) , z) = G (x, G (y, z)) ;

There is an absorbing element k ∈ [0, 1], such that G (k, x) = k for all x ∈ [0, 1], G (0, x) = x for all x ∈ [0, k] and G (1, x) = x for all x ∈ [k, 1].

Evidently, when k = 0 and 1, then G is respectively a t-norm and a t-conorm. It is worth noting that the absorbing element k is given by k = G (0, 1) = G (1, 0). Similarly, for any nullnorm G, we usually denote it by G ≡ 〈S_G, k, T_G〉.

Theorem 2.6. ([10]). If the nullnorm G with the absorbing element k = G (0, 1) ≠0, 1, then $\begin{matrix} G (x, y) = \\ {\begin{matrix} {kS}_{G} (\frac{x}{k}, \frac{y}{k}), & x, y \in [0, k], \\ k + (1 - k) T_{G} (\frac{x - k}{1 - k}, \frac{y - k}{1 - k}), & x, y \in [k, 1], \\ k, & otherwise, \end{matrix} \end{matrix}$ where T_G and S_G are represent a t-norm and a t-conorm, respectively.

Definition 2.7. ([3]). A function N : [0, 1] → [0, 1] is called a fuzzy negation if it satisfies the following conditions:

N (0) =1 and N (1) =0 ;

If x ⩽ y, then N (x) ⩾ N (y) for all x, y ∈ [0, 1].

In addition, we highlight several properties of fuzzy negation:

strict, if it is continuous and strict decreasing;

strong, if it is an involution, i.e., N (N (x)) = x for all x ∈ [0, 1].

The most frequently used strong negation is the standard negation N_s (x) =1–x for all x ∈ [0, 1].

Definition 2.8. ([3]). A function I : [0, 1] ² → [0, 1] is called a fuzzy implication, if it satisfies:

I is non-increasing with respect to the first variable;

I is non-decreasing with respect to the second variable;

I (0, 0) = I (1, 1) =1 and I (1, 0) =0.

It is worth noting that from the above definition, we can deduce that I (0, x) =1 and I (x, 1) =1 for all x ∈ [0, 1]. The family of all fuzzy implications will be denoted by $F I$ .

Definition 2.9. ([3]). Let $I \in F I$ . The function N_I : [0, 1] → [0, 1] defined by $N_{I} (x) : = I (x, 0) for all x \in [0, 1]$ (1) is called the natural negation of I.

Definition 2.10. ([3]). A fuzzy implication I is said to satisfy:

The identity principle, if $\forall x \in [0, 1], I (x, x) = 1;$ (2)

The consequent boundary, if $\forall x, y \in [0, 1], I (x, y) ⩾ y;$ (3)

The neutrality property, if $\forall y \in [0, 1], I (1, y) = y;$ (4)

The continuity condition, if $I (x, y) is a continuous mapping;$ (5)

The order principle, if $\forall x, y \in [0, 1], I (x, y) = 1 \Leftrightarrow x ⩽ y;$ (6)

The strong negation principle, if $N_{I} (x) is a strong negation;$ (7)

The exchange principle, if $\forall x, y, z \in [0, 1], I (x, I (y, z)) = I (y, I (x, z));$ (8)

The lowest falsity property, if $\forall x, y \in [0, 1], I (x, y) = 0 \Leftrightarrow x = 1 and y = 0;$ (9)

The lowest truth property, if $\forall x, y \in [0, 1], I (x, y) = 1 \Leftrightarrow x = 0 or y = 1;$ (10)

3 The FIφ-construction

In this section, we introduce a new method, called FIφ-construction, to produce implication functions. Firstly, we give the definition of new generated fuzzy implications and then we examine the logical properties preserved by this construction. Finally, we compare the results of the FIφ-construction with several existing methods.

3.1 Fuzzy implications generated by the FIφ-construction

Theorem 3.1. Let F be an aggregation function with F (1, 0) =1, I a fuzzy implication, and a non-decreasing function with φ (0) =0 and φ (1) =1. Then the function I_FIφ defined by $I_{FI φ} (x, y) = F (I (x, y), φ (y)), x, y \in [0, 1]$ (11) is a fuzzy implication.

Proof. I_FIφ satisfies (I1) since for all x₁, x₂, y ∈ [0, 1] with $\begin{matrix} x_{1} ⩽ x_{2} \Rightarrow I (x_{1}, y) ⩾ I (x_{2}, y) \Rightarrow F (I (x_{1}, y), φ (y)) ⩾ \\ F (I (x_{1}, y), φ (y)) \Rightarrow I_{FI φ} (x_{1}, y) ⩾ I_{FI φ} (x_{2}, y) . \end{matrix}$

I_FIφ satisfies (I2) since for all x, y₁, y₂ ∈ [0, 1] with $\begin{matrix} y_{1} ⩽ y_{2} \Rightarrow {\begin{matrix} I (x, y_{1}) ⩽ I (x, y_{2}) \\ φ (y_{1}) ⩽ φ (y_{2}) \end{matrix} \\ \Rightarrow F (I (x, y_{1}), φ (y_{1})) ⩽ F (I (x, y_{2}), φ (y_{2})) \\ \Rightarrow I_{FI φ} (x, y_{1}) ⩽ I_{FI φ} (x, y_{2}) . \end{matrix}$

I_FIφ satisfies (I3) since ${\begin{matrix} I_{FI φ} (0, 0) = F (I (0, 0), φ (0)) = F (1, 0) = 1, \\ I_{FI φ} (1, 1) = F (I (1, 1), φ (1)) = F (1, 1) = 1, \\ I_{FI φ} (1, 0) = F (I (1, 0), φ (0)) = F (0, 0) = 0 . \end{matrix}$

Therefore, $I_{FI φ} \in F I$ .□

Remark 3.2. Fuzzy implications constructed as Equation (11) are called FIφ-implications, and denoted by I_FIφ. The family of all FIφ-implications will be denoted by $F I J$ . In particular, when the aggregation functions F are t-conorms S, generate SIφ-implications, and denoted by I_SIφ.

Next, let us give some interesting examples of the FIφ-construction to produce fuzzy implications.

Example 3.3.

(i) Consider the probabilistic sum t-conorm, that is, S_P (x, y) = x + y - xy, I the Goguen implication $I_{GG} (x, y) = {\begin{matrix} 1, & x ⩽ y, \\ \frac{y}{x}, & x > y, \end{matrix}$ and φ a non-decreasing function. Then the corresponding SIφ-implications are given by $I_{SI φ} (x, y) = {\begin{matrix} 1, x ⩽ y, \\ \frac{(1 - φ (y)) y}{x} + φ (y), x > y . \end{matrix}$

(ii) Take a t-conorm S, φ a non-decreasing function, and I the Rescher implication $I_{RS} (x, y) = {\begin{matrix} 1, & x ⩽ y, \\ 0, & x > y . \end{matrix}$

Then we generate the following SIφ-implications: $I_{SI φ} (x, y) = {\begin{matrix} 1, & x ⩽ y, \\ φ (y), & x > y . \end{matrix}$

(iii) Taking a t-conorm S, I arbitrary implication function, and φ a non-decreasing function given by $φ (y) = {\begin{matrix} 0, y < 1, \\ 1, y = 1 . \end{matrix}$

Then we obtain I_SIφ (x, y) = I (x, y) for all x, y ∈ [0, 1].

(iv) Taking a t-conorm S, φ a non-decreasing function, and I the Gödel implication $I_{GD} (x, y) = {\begin{matrix} 1, & x ⩽ y, \\ y, & x > y . \end{matrix}$

Then we obtain the SIφ-implications are given by $I_{SI φ} (x, y) = {\begin{matrix} 1, & x ⩽ y, \\ S (y, φ (y)), & x > y . \end{matrix}$

On the other hand, if we take S = max and φ (y) = y, then we retrieve the Gödel implication.

(v) Similarly, take again a t-conorm S, a non-decreasing function φ (y) = y, and I the Reichenbach implication, that is, I_RC (x, y) =1 - x + xy, then we obtain $I_{SI φ} (x, y) = S (1 - x + xy, y), x, y \in [0, 1] .$

If we take again S = max, then we retrieve Reichenbach implication.

(vi) Take I = I_RC, φ any non-decreasing function such that φ (y) ⩽ y for all y ∈ [0, 1], and F the aggregation function is given by $F_{mM} (x, y) = min {1, \frac{max {x, y}}{1 - min {x, y}}}, x, y \in [0, 1] .$

Then we obtain the FIφ-implications are given by $\begin{matrix} I_{FI φ} (x, y) = F (1 - x + xy, φ (y)) \\ = {\begin{matrix} 1, x ⩽ \frac{φ (y)}{1 - y}, \\ \frac{1 - x + xy}{1 - φ (y)}, x > \frac{φ (y)}{1 - y}, \end{matrix} \end{matrix}$ which depend on the non-decreasing function φ. For example, take φ (y) = y, then we obtain $I_{FI φ} (x, y) = {\begin{matrix} 1, x ⩽ \frac{y}{1 - y}, \\ \frac{1 - x + xy}{1 - y}, x > \frac{y}{1 - y} . \end{matrix}$ (12)

The structure of the corresponding FIφ-implication can be viewed in Fig. 1.

Fig. 1

The structure of implication given in (12).

(vii) Take a disjunctive uninorm U with e = 0.5, and it is given by $U_{mM} (x, y) = {\begin{matrix} min {x, y}, & y < 1 - x, \\ max {x, y}, & y ⩾ 1 - x . \end{matrix}$

Take again I = I_RC, and φ any non-decreasing function such that φ (y) ⩽ y for all y ∈ [0, 1]. Then we have the corresponding FIφ-implications are given by $\begin{matrix} I_{FI φ} (x, y) = U (1 - x + xy, φ (y)) \\ = {\begin{matrix} 1 - x + xy, & x ⩽ \frac{φ (y)}{1 - y}, \\ φ (y), & x > \frac{φ (y)}{1 - y}, \end{matrix} \end{matrix}$ (13) which depend on the non-decreasing function φ. If we take φ (y) = y, then we obtain $I_{FI φ} (x, y) = {\begin{matrix} 1 - x + xy, x ⩽ \frac{y}{1 - y}, \\ y, x > \frac{y}{1 - y} . \end{matrix}$ (14)

The structure of the corresponding FIφ-implication can be viewed in Fig. 2.

Fig. 2

The structure of implication given in (14).

Inspired by the FIφ-construction, it is proved that all fuzzy implications are FIφ-implications as the following result.

Theorem 3.4. Let F be an aggregation function with neutral element 0 and I any implication function. If φ satisfies that φ (y) =0 iff y < 1, then I_FIφ = I.

Proof. It follows that: $\begin{matrix} I_{FI φ} (x, y) = \\ {\begin{matrix} F (I (x, y), 0) = I (x, y), if y < 1, \\ F (I (x, 1), 1) = F (1, 1) = 1, if y = 1 . \end{matrix} \end{matrix}$

Thus, we have I_FIφ = I. □

Remark 3.5. This fact also implies that both the family $F I J$ of all FIφ-implications and the family $F I$ of all fuzzy implications are the same, i.e., $F I J = F I$ .

By Theorem 3.4, we know that there exist a case such that I_FIφ = I. Thus, an open problem arises:

(OP) Can we characterize all the cases a fuzzy implication remains invariant by the FIφ-construction?

3.2 Properties of FIφ-implications

In this subsection, we study the logical properties that are preserved by the FIφ-construction, and we obtain the following results.

Proposition 3.6. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. If I satisfies the identity principle (2), then I_FIφ also satisfies it.

Proof. If I satisfies (2), then it is immediate that $I_{FI φ} (x, x) = F (I (x, x), φ (x)) = F (1, φ (x)) .$

Moreover, by (F1) and (F2), it follows that: $\begin{matrix} 1 = F (1, 0) ⩽ F (1, φ (x)) ⩽ F (1, 1) = 1 \\ \Rightarrow F (1, φ (x)) = 1, \end{matrix}$ which implies that I_FIφ (x, x) =1 and thus I_FIφ satisfies (2). □

Proposition 3.7. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. If I satisfies the consequent boundary (3) and F satisfies that F (y, 0) ⩾ y for all y ∈ [0, 1], then I_FIφ satisfies (3).

Proof. For all x, y ∈ [0, 1], since I satisfies (3) and by (F1) we deduce that: $\begin{matrix} I_{FI φ} (x, y) = F (I (x, y), φ (y)) ⩾ F (y, φ (y)) \\ ⩾ F (y, 0) ⩾ y, \end{matrix}$ which implies that I_FIφ (x, y) ⩾ y and thus I_FIφ satisfies (3). □

Proposition 3.8. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. If F satisfies that F (x, y) = x for all x, y ∈ [0, 1], then I satisfies the neutrality property (4) if and only if I_FIφ satisfies (4).

Proof. Since F satisfies that F (x, y) = x, then it holds that: $\begin{matrix} I (1, y) = y \Leftrightarrow I_{FI φ} (1, y) = F (I (1, y), φ (y)) \\ = F (y, φ (y)) = y . \end{matrix}$

Thus, I satisfies (4) iff I_FIφ satisfies (4). □

Proposition 3.9. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. Suppose that F and φ are continuous. If I satisfies (5), then I_FIφ also satisfies it.

Proof. It follows directly verified from the definition of I_FIφ. □

Next, we further study the identity principle (2) and the order principle (6). Note that from the monotonicity of F and the condition F (1, 0) =1, we deduce that F (1, y) =1 for all y ∈ [0, 1].

Proposition 3.10. Let I_FIφ be the fuzzy implication given by (11) in Theorem 3.1 and I an implication function. Suppose that F (x, y) =1 if and only if x = 1, then it holds that:

I satisfies (2) if and only if I_FIφ satisfies (2).

I satisfies (6) if and only if I_FIφ satisfies (6).

Proof. (i) If I satisfies (2), then it follows from Proposition 3.6 that I_FIφ satisfies (2). Conversely, if I_FIφ satisfies (2), then it follows that: $I_{FI φ} (x, x) = F (I (x, x), φ (x)) = 1 for all x \in [0, 1],$ which means that I (x, x) =1, and thus I satisfies (2).

(ii) If I satisfies (6), then for all x, y ∈ [0, 1], we have the following equivalence: $I_{FI φ} (x, y) = F (I (x, y), φ (y)) = 1 \Leftrightarrow I (x, y) = 1,$ and thus I satisfies (6) if and only if I_FIφ also satisfies it. □

Example 3.11. It worth noting that without the supposition of Proposition 3.10, both statements (i) and (ii) fail.

Consider the aggregation function F_mM in Example 3.3 (vi), the non-decreasing function φ (y) = y, and the Reichenbach implication I = I_RC. It is well known that I_RC does not satisfy (2), but we generate the corresponding I_FIφ given in Equation (12) can satisfy (2).

Consider the Łukasiewicz t-conorm, given by S_LK (x, y) = min(x + y, 1), the non-decreasing function φ (y) = y, and the Gödel implication I = I_GD. It is known that I_GD satisfies (6), but we generate

I_{SI φ} (x, y) = {\begin{matrix} 1, & x ⩽ y, \\ \min (2 y, 1), & x > y, \end{matrix}

it does not satisfy (6), since I_SIφ (0.6, 0.5) =1.

Proposition 3.12. Let I_FIφ be a fuzzy implication given by Equation (11) in Theorem 3.1, then the natural negation of I_FIφ is $N_{I_{FI φ}} (x) = F (N_{I} (x), 0), x \in [0, 1] .$ (15)

Proof. It is deduced by Definition 2.9 and Theorem 3.1. □

Proposition 3.13. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. If F has neutral element 0, then I satisfies the strong negation principle (7) if and only if I_FIφ satisfies (7).

Proof. If F has 0 as neutral element, then by Equation (15) we deduce that: $N_{I_{FI φ}} (x) = F (N_{I} (x), 0) = N_{I} (x), x \in [0, 1] .$

Thus, I satisfies (7) if and only if I_FIφ satisfies (7). □

Again, note that without the precondition stated in the proposition above, the corresponding I_FIφ does not satisfy (7) even when the initial fuzzy implication I satisfies (7) as it is presented to the following example.

Example 3.14. Consider the uninorm U_mM in Example 3.3 (vii) and I = I_RC the Reichenbach implication. Observe that the natural negation of I_RC is a strong negation, but we obtain the corresponding $N_{I_{FI φ}} (x) = U (1 - x, 0) = N_{D 1} (x) = {\begin{matrix} 1, & x = 0, \\ 0, & x > 0 . \end{matrix}$

It is clear that the least fuzzy negation N_⊥ is not a strong negation, and thus I_FIφ does not satisfy (7).

Corollary 3.15. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I any fuzzy implication satisfying the exchange principle (8). If F has neutral element 0 and φ satisfies that φ (y) =0 iff y < 1, then I_FIφ satisfies (8).

Proof. It is deduced by Theorem 3.4. □

However, if we consider other non-decreasing functions φ, then it is not ensured that the corresponding I_FIφ satisfies (8) even when the initial fuzzy implication I satisfies (8) as it is shown in the following example.

Example 3.16. Consider the probabilistic sum t-conorm S_P (x, y) = x + y - xy, the increasing function φ (y) = y, the Kleene-Dienes implication given by I_KD (x, y) = max(1 - x, y) and I = I_GD the Gödel implication. It is known that both I_KD and I_GD satisfy (8), but we generate the corresponding I_SIφ does not satisfy (8), since $\begin{matrix} I_{S_{P}, I_{KD}, φ} (0.1, I_{S_{P}, I_{KD}, φ} (0.6, 0.7)) = 0.9919 \\ \neq 0.9991 = I_{S_{P}, I_{KD}, φ} (0.6, I_{S_{P}, I_{KD}, φ} (0.1, 0.7)), \end{matrix}$ and $\begin{matrix} I_{S_{P}, I_{GD}, φ} (0.7, I_{S_{P}, I_{GD}, φ} (0.5, 0.4)) = 0.8704 \\ \neq 1 = I_{S_{P}, I_{GD}, φ} (0.5, I_{S_{P}, I_{GD}, φ} (0.7, 0.4)) . \end{matrix}$

3.3 Modifying fuzzy implications through the FIφ-construction

In this subsection, we study how to use the FIφ-construction to produce a new fuzzy implication satisfying a concrete property like (2), (3) or (5) even when the original implication I does not satisfy them. In this direction, we must consider adequate aggregation functions F and non-decreasing functions φ to get the objective. Firstly, let us further study the identity principle (2).

Proposition 3.17. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. Supposed that F such that F (x, φ (x)) =1 for every x ∈ (0, 1]. If I satisfies that I (x, x) ⩾ x for all x ∈ [0, 1], then I_FIφ satisfies (2).

Proof. If x = 0, then clearly I_FIφ (x, x) = I_FIφ (0, 0) =1. If x ∈ (0, 1], then we have the following inequalities: $\begin{matrix} I_{FI φ} (x, x) = F (I (x, x), φ (x)) ⩽ F (1, 1) = 1, \\ I_{FI φ} (x, x) = F (I (x, x), φ (x)) ⩾ F (x, φ (x)) = 1, \end{matrix}$ which implies that I_FIφ (x, x) =1 for all x ∈ [0, 1], and thus the corresponding I_FIφ satisfies (2). □

Example 3.18. Take a t-conorm S and φ a non-decreasing function given by $φ (x) = {\begin{matrix} 0, x = 0, \\ 1, x > 0 . \end{matrix}$

It is clear that S such that S (x, φ (x)) =1 for every x ∈ (0, 1]. Consider now I = I_KD the Kleene-Dienes implication which satisfies I_KD (x, x) ⩾ x, but it does not satisfy the identity principle (2). The corresponding I_SIφ is given by $\begin{matrix} I_{SI φ} (x, y) & = S (max (1 - x, y), φ (y)) \\ = {\begin{matrix} 1 - x, y = 0, \\ x, y > 0 . \end{matrix} \end{matrix}$

Then we have I_SIφ (x, x) =1 for all x ∈ [0, 1], and thus I_SIφ satisfies (2).

Proposition 3.19. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and I an implication function. Supposed that F satisfies that F (0, φ (y)) ⩾ yfor all y ∈ [0, 1], then the corresponding I_FIφ satisfies (3) even if the original implication I does not satisfy (3).

Proof. Supposed that F (0, φ (y)) ⩾ y for all y ∈ [0, 1], then by (F1) and Equation (11) we deduce that: $I_{FI φ} (x, y) = F (I (x, y), φ (y)) ⩾ F (0, φ (y)) ⩾ y,$ which implies that I_FIφ satisfies (3) even if the original implication I does not satisfy (3). □

Example 3.20. Consider the aggregation function F_mM in Example 3.3 (vi), the increasing function φ (y) = y and I = I_RS the Rescher implication. Observe that I_RS does not satisfy (3), but we generate the corresponding I_FIφ = I_GD the Gödel implication which satisfies (3).

Proposition 3.21. Let I_FIφ be a fuzzy implication given by (11) in Theorem 3.1 and F (0, 1) =1. If F and φ are continuous, then the corresponding I_FIφ satisfies (5) even when the original implication I is discontinuous at point (1, 1) .

Proof. By Proposition 3.9, we know that I_FIφ satisfies (5) except at point (1, 1). Thus, we only need to prove that I_FIφ is continuous at point (1, 1). Again from the monotonicity of F and Equation (11), we deduce that: $\begin{matrix} F (0, φ (y)) ⩽ F (I (x, y), φ (y)) \\ = I_{FI φ} (x, y) ⩽ F (1, φ (y)) . \end{matrix}$

If F and φ are continuous, then we have $lim_{(x, y) \to (1, 1)} F (0, φ (y)) = lim_{(x, y) \to (1, 1)} F (1, φ (y)) = 1,$ which implies that $lim_{(x, y) \to (1, 1)} I_{FI φ} (x, y) = 1 = I_{FI φ} (1, 1)$ , and thus I_FIφ is continuous at point (1, 1). Therefore, I_FIφ satisfies (5). □

Example 3.22. Take F to be the maximum t-conorm S_M = max(x, y), the increasing function φ (y) = y and the Rescher implication given in Example 3.3 (ii). Observe that I_RS is discontinuous at point (1, 1), but we generate the corresponding I_SIφ = I_GD, which satisfies (5).

3.4 Comparing with the existing construction methods

In this subsection, we compare our method with the existing methods from two aspects, including the main characteristics and application areas of the construction methods. Especially, we provide Table 1 to show the differences between the existing construction methods and our method.

Table 1
Comparisons with existing construction methods

Construction methods / implications Main characteristics Application areas

The FIφ-construction (our method) Based on an aggregation function, an implication function and a non-decreasing function Fuzzy subsethood measures

Horizontal threshold generation methods [30, 31] Constructed from two given implications Unknown

Vertical threshold generation methods [32] Constructed from two given implications Unknown

New types of contrapositivisations [2] With respect to any fuzzy negation Approximate reasoning, deductive systems, decision support systems and formal methods of proof

The ⊛-composition [40] From a given pair or fuzzy implications Unknown

The rotation construction [36] An infinite number of fuzzy implications can be generated Fuzzy connectives in fuzzy set theory

The FNI-method [1] Based on an aggregation function, a fuzzy negation and an implication function Unknown

The quadratic construction method [26] Based on polynomial ternary functions Unknown

The n increasing functions and n + 1 negations-composition [35] Combine a lot of fuzzy negations and increasing functions In a fuzzy environment, to find a large number of implications

Yager’s f- and g- generated implications [42] By additive generators of continuous Archimedean t-norms and t-conorms Approximate reasoning

Balasubramaniam’s h-generated implications [5] By continuous multiplicative generators of t-conorms Fuzzy logic

h-implications [29] By additive generators of representable uninorms Approximate reasoning

Zhou’s k-generated implications [46] By continuous multiplicative generators of t-norms Fuzzy preference modelling and fuzzy reasoning

Ordinal sum implications [37] By means of the ordinal sum of a family of given implications Approximate reasoning and fuzzy control

(T, N)-implications [33] Obtained from the composition of t-norms and fuzzy negations Fuzzy subsethood measures

R_O-implications [15] Derived from overlap functions, preserving the residuation property Decision making and fuzzy control of the agent’s intentions

(G, N)-implications [17] Derived from grouping functions and fuzzy negations Performing inferences and decision making

QL-implications constructed from tuples (O, G, N) [16] Derived from overlap and grouping functions and the greatest fuzzy negations Fuzzy subsethood and entropy measures

Construction methods / implications	Main characteristics	Application areas
The FIφ-construction (our method)	Based on an aggregation function, an implication function and a non-decreasing function	Fuzzy subsethood measures
Horizontal threshold generation methods [30, 31]	Constructed from two given implications	Unknown
Vertical threshold generation methods [32]	Constructed from two given implications	Unknown
New types of contrapositivisations [2]	With respect to any fuzzy negation	Approximate reasoning, deductive systems, decision support systems and formal methods of proof
The ⊛-composition [40]	From a given pair or fuzzy implications	Unknown
The rotation construction [36]	An infinite number of fuzzy implications can be generated	Fuzzy connectives in fuzzy set theory
The FNI-method [1]	Based on an aggregation function, a fuzzy negation and an implication function	Unknown
The quadratic construction method [26]	Based on polynomial ternary functions	Unknown
The n increasing functions and n + 1 negations-composition [35]	Combine a lot of fuzzy negations and increasing functions	In a fuzzy environment, to find a large number of implications
Yager’s f- and g- generated implications [42]	By additive generators of continuous Archimedean t-norms and t-conorms	Approximate reasoning
Balasubramaniam’s h-generated implications [5]	By continuous multiplicative generators of t-conorms	Fuzzy logic
h-implications [29]	By additive generators of representable uninorms	Approximate reasoning
Zhou’s k-generated implications [46]	By continuous multiplicative generators of t-norms	Fuzzy preference modelling and fuzzy reasoning
Ordinal sum implications [37]	By means of the ordinal sum of a family of given implications	Approximate reasoning and fuzzy control
(T, N)-implications [33]	Obtained from the composition of t-norms and fuzzy negations	Fuzzy subsethood measures
R_O-implications [15]	Derived from overlap functions, preserving the residuation property	Decision making and fuzzy control of the agent’s intentions
(G, N)-implications [17]	Derived from grouping functions and fuzzy negations	Performing inferences and decision making
QL-implications constructed from tuples (O, G, N) [16]	Derived from overlap and grouping functions and the greatest fuzzy negations	Fuzzy subsethood and entropy measures

4 Using uninorms and nullnorms in the FIφ-construction

In this suction, we will use uninorms and nullnorms in the FIφ-construction to produce two subclasses of fuzzy implications such as UIφ- and G_pIφ-implications and discuss some additional properties.

4.1 UIφ-implications and G_pIφ-implications

Corollary 4.1. Let U be a disjunctive uninorm with e ∈ (0, 1), I a fuzzy implication, and φ : [0, 1] → [0, 1] a non-decreasing function with φ (0) =0 and φ (1) =1. Then the function I_UIφ defined by $I_{UI φ} (x, y) = U (I (x, y), φ (y)), x, y \in [0, 1]$ (16) is a fuzzy implication.

Definition 4.2. Let G ≡ 〈S_G, k, T_G〉 be a nullnorm with k = G (0, 1) ≠0, 1. Then, the function that is defined by $G_{p} (x, y) = {\begin{matrix} 1, x = 1, \\ G (x, y), x \neq 1, \end{matrix}$ (17) will be called a p-nullnorm.

Remark 4.3. Similarly, we denote a p-nullnorm with k = G_p (0, 1), underlying t-conorm S_G, and underlying t-norm T_G by G_p ≡ 〈S_G, k, T_G〉_p.

Corollary 4.4. Let G_p be a p-nullnorm, I a fuzzy implication, and φ : [0, 1] → [0, 1] a non-decreasing function with φ (0) =0 and φ (1) =1. Then the function I_{G
_p
Iφ} defined by $I_{G_{p} I φ} (x, y) = G_{p} (I (x, y), φ (y)), x, y \in [0, 1]$ (18) is a fuzzy implication.

Corollary 4.5. Let I_{G
_p
Iφ} be a fuzzy implication given by Equation (18) and I an implication function. Then I satisfies (2) (resp. (6)) if and only if I_{G
_p
Iφ} satisfies it.

Proof. It is deduced by Proposition 3.10 since G_p (x, y) =1 ⇔ x = 1. □

Proposition 4.6. Let I_UIφ be a fuzzy implication given by Equation (16) in Corollary 4.1 and I an implication function. If φ satisfies that φ (y) =1 if and only if y = 1 and U satisfies that U (x, y) =1 if and only if max(x, y) =1, then I satisfies (2) (resp. (6)) if and only if I_UIφ satisfies it.

Proof. For all x, y ∈ [0, 1], we have the following equi-valences: $\begin{matrix} I_{UI φ} (x, y) = 1 \Leftrightarrow U (I (x, y), φ (y)) = 1 \\ \Leftrightarrow max (I (x, y), φ (y)) = 1 . \end{matrix}$

Note that the last equivalence holds, and thus I_UIφ satisfies (2) (resp. (6)) if and only if I satisfies it. □

4.2 The property: I (x, e) = e

In this subsection, we want to study a property of fuzzy implications: I (x, e) = e for all x > 0, introduced in [31] and in the same paper, they were named e-threshold generated implication with e ∈ (0, 1). In particular, this kind of implication functions take values depending on the threshold e. In this case, we will see that the corresponding UIφ- and G_pIφ-implications can satisfy this property.

Proposition 4.7. Let U ≡ 〈T_U, e, S_U〉 be a disjunctive uninorm with e ∈ (0, 1), I an implication function and φ a non-decreasing function satisfies that φ (y) = e if and only if y = e. If I satisfies that I (x, e) = e for all x > 0, then I_UIφ also satisfies this property, and moreover, $\begin{matrix} I_{UI φ} (x, y) = \\ {\begin{matrix} 1, & x = 0, \\ {eT}_{U} (\frac{I (x, y)}{e}, \frac{φ (y)}{e}), & x > 0, y ⩽ e, \\ (1 - e) S_{U} (\frac{I (x, y) - e}{1 - e}, \frac{φ (y) - e}{1 - e}) + e, & x > 0, y > e . \end{matrix} \end{matrix}$

Proof. If x = 0, then clearly I_UIφ (0, y) =1. Thus, supposed that x > 0, we consider the following two cases:

Case 1, if y ⩽ e, then by (I2) we deduce that I (x, y) ⩽ e. Moreover, note that in this case, φ (y) ⩽ e. Thus, in points (I (x, y) , φ (y)), the uninorm U is given by T_U.

Case 2, if y > e, then by (I2) we deduce that I (x, y) ⩾ e. Moreover, note that in this case, φ (y) ⩾ e. Thus, in points (I (x, y) , φ (y)), the uninorm U is given by S_U. □

Proposition 4.8. Let G_p ≡ 〈S_G, k, T_G〉_p be a p-nullnorm with G_p (0, 1) = k ∈ (0, 1), I an implication function, and φ a non-decreasing function satisfies that φ (y) = k if and only if y = k. If I satisfies that I (x, k) = k for all x > 0, then I_{G
_p
Iφ} also satisfies this property, and moreover, $\begin{matrix} I_{G_{p} I φ} (x, y) = \\ {\begin{matrix} 1, & x = 0, \\ {kT}_{U} (\frac{I (x, y)}{k}, \frac{φ (y)}{k}), & x > 0, y ⩽ k, \\ (1 - k) S_{U} (\frac{I (x, y) - k}{1 - k}, \frac{φ (y) - k}{1 - k}) + k, & x > 0, y > k . \end{matrix} \end{matrix}$

Proof. By a similar argument as in the proof of Proposition 4.7, so we omit the details here. □

Example 4.9. Since the values of the corresponding UIφ- and G_pIφ-implications are related to the underlying operators, and consequently, any uninorm with the same associated T_U and S_U will produce the same effect. For instance, if we consider any disjunctive uninorm U with e = 0.5, T_U the product t-norm, and S_U the probabilistic sum t-conorm. Take the increasing function. φ (y) = y and the e-threshold generated implication with e = 0.5 (see [31] Example 6 (i)): $I (x, y) = {\begin{matrix} 1, & x = 0, \\ max {\frac{1}{2} - \frac{x}{2}, y}, & x > 0, y ⩽ 0.5, \\ max {1 - \frac{x}{2}, y}, & x > 0, y > 0.5, \end{matrix}$ it satisfies that I (x, 0.5) =0.5 for all x > 0. Moreover, we generate $\begin{matrix} I_{UI φ} (x, y) = \\ {\begin{matrix} 1, & x = 0, \\ 2 y max {\frac{1}{2} - \frac{x}{2}, y}, & x > 0, y ⩽ 0.5, \\ (2 - 2 y) max {1 - \frac{x}{2}, y} + 2 y - 1, & x > 0, y > 0.5, \end{matrix} \end{matrix}$ which is a new fuzzy implication satisfying the property I_UIφ (x, & 0.5) =0.5 for all x > 0 .

4.3 The property: I (x, N (x)) = N (x)

In this subsection, we consider an important property: I (x, N (x)) = N (x) for all x ∈ [0, 1], studied in [8] and recently received a lot of attention because it proved to be valuable in the fuzzy indices. In this case, we will see that the corresponding UIφ- and G_pIφ-implications can be used in this property as follows.

Proposition 4.10. Let U be a disjunctive idempotent uninorm with e ∈ (0, 1), I an implication function, N a fuzzy negation, and φ an increasing function φ (y) = y. If I satisfies that I (x, N (x)) = N (x) for all x ∈ [0, 1], then I_UIφ also satisfies it.

Proof. By Corollary 4.1, we deduce that: $I_{UI φ} (x, N (x)) = U (I (x, N (x)), φ (N (x))) .$

Furthermore, since U is idempotent and φ (y) = y, thus we have I_UIφ (x, N (x)) = U (N (x) , N (x)) = N (x). □

Example 4.11. Consider the disjunctive idempotent uninorm given in Equation (13), I = I_KD the Kleene-Dienes implication and N = N_s the standard fuzzy negation. Observe that I_KD and N_s such that I (x, N (x)) = N (x) for all x ∈ [0, 1]. On the other hand, we generate $\begin{matrix} I_{U, I_{KD}, φ} (x, N_{S} (x)) = U (I_{KD} (x, N_{S} (x)), φ (N_{S} (x))) \\ = U (N_{S} (x), φ (N_{S} (x))) \\ = U (N_{S} (x), N_{S} (x)) \\ = {\begin{matrix} \max {N_{S} (x), N_{S} (x)}, & x ⩽ 0.5, \\ \min {N_{S} (x), N_{S} (x)}, & x > 0.5, \end{matrix} \\ = N_{S} (x) . \end{matrix}$ it satisfies that the property I (x, N (x)) = N (x) for all x ∈ [0, 1].

Proposition 4.12. Let G_p ≡ 〈S_G, k, T_G〉_p be a p-nullnorm with k ∈ (0, 1) and underlying operators S_G = max and T_G = min. Let I be an implication function, N a fuzzy negation satisfies that N (x) = k if and only if x = k, and φ an increasing function φ (y) = y. If I satisfies that I (x, N (x)) ≠1 for every x ∈ [0, 1] and I (k, y) = y for all y ∈ [0, 1], then I_{G
_p
Iφ} satisfies the property I_{G
_p
Iφ} (x, N (x)) = N (x) for all x ∈ [0, 1].

Proof. By Corollary 4.4 and φ (y) = y, we deduce that: $I_{G_{p} I φ} (x, N (x)) = G_{p} (I (x, N (x)), N (x)) .$

Now, we shall consider the following two cases:

Case 1, if x ⩽ k, then by (N2), it is N (x) ⩾ N (k) = k. Moreover, by (I1) we deduce that I (x, N (x)) ⩾ N (x). So in this case, we have that G_p (I (x, N (x)) , N (x)) = min {I (x, N (x)) , N (x)} = N (x).

Case 2, if x ⩾ k, then by (N2), it is N (x) ⩽ N (k) = k. Moreover, by (I1) we deduce that I (x, N (x)) ⩽ N (x). So in this case, we have that G_p (I (x, N (x)) , N (x)) = max {I (x, N (x)) , N (x)} = N (x).

Therefore, the corresponding I_{G
_p
Iφ} satisfies the property I_{G
_p
Iφ} (x, N (x)) = N (x) for all x ∈ [0, 1]. □

5 Applying FIφ-implications to construct fuzzy subsethood measures

Fuzzy subsethood measures [44] allow one to determine up what extent a given fuzzy set included into another one, which is useful in some applications, such as fuzzy decision making [27], fuzzy relational databases [28], image processing [9], clustering [44] and intelligent systems [34].

Fuzzy subsethood measures can be applied to generate fuzzy entropy measures, which determine the amount of fuzziness, or vagueness in a given fuzzy set [13 , 25].

It is well known that fuzzy implications play an important role in the generation of fuzzy subsethood measures. In this section, we provide a way to construct fuzzy subsethood measures by means of FIφ-implications. In order to achieve our goals, first we recall Pinheiro-Bedregal’s fuzzy subsethood measure, which is defined as follows:

Definition 5.1. ([33]). The Pinheiro-Bedregal’s fuzzy subsethood measure is a mapping σ_PB : F (X) ² → [0, 1] satisfying the following conditions, for all A, B, C∈ F (X):

σ_PB (A, B) =1 ⇔ A (x) =0 or B (x) =1, ∀ x ∈ X ;

σ_PB (A, B) =0 ⇔ A = X and B =∅ ;

(i) If A ⩽ B ⩽ C, then σ_PB (C, A)⩽ σ_PB (B, A) ;

(ii) If A ⩽ B, then σ_PB (C, A) ⩽ σ_PB (C, B).

In [33], Pinheiro et al. provided a way to construct fuzzy subsethood measures in the sense of Definition 5.1 by means of (T, N)-implications. Following this line, we study how to generate Pinheiro-Bedregal’s fuzzy subsethood measures using FIφ-implications. It is necessary here to recall the notions of n-ary aggregation functions.

Definition 5.2. ([6]). An n-ary aggregation function is a mapping $A : [0, 1]^{n} \to [0, 1]$ with n ⩾ 2 satisfying the following conditions:

$(A 1)$ $A$ is increasing in each variable;

$(A 2)$ $A (0, \dots, 0) = 0$ and $A (1, \dots, 1) = 1$ .

Definition 5.3. ([6]). A 0-positive (1-positive) n-ary aggregation function is an n-ary aggregation function $A : [0, 1]^{n} \to [0, 1]$ also satisfying the following condition, respectively:

$\begin{matrix} (A - 0) A (x_{1}, \dots, x_{n}) = 0 \Leftrightarrow x_{i} = 0 for all i \in {1, \dots, n}; \\ (A - 1) A (x_{1}, \dots, x_{n}) = 1 \Leftrightarrow x_{i} = 1 for all i \in {1, \dots, n} . \end{matrix}$

In the following, we present a proposition that generates Pinheiro-Bedregal’s fuzzy subsethood measure by aggregating the FIφ-implication I_FIφ.

Proposition 5.4. Let $A$ be a 0- and 1-positive n-ary aggregation function, F a binary aggregation function such that F (1, 0) =1, I a fuzzy implication and φ a non-decreasing function with φ (0) =0 and φ (1) =1. If F satisfies that F (x, y) =1 ⇔ x = 1 and F (x, y) =0⇔ x = 0 and I satisfies the lowest falsity property (9) and the lowest truth property (10). Then the mapping σ_FIφ : F (X) ² → [0, 1], defined for all A, B ∈ F (X), by $\begin{matrix} σ_{FI φ} (A, B) = A (I_{FI φ} (A (x_{1}), B (x_{1})), \dots, \\ I_{FI φ} (A (x_{n}), B (x_{n}))), \end{matrix}$ where I_FIφ is a FIφ-implication defined by Equation (11), is a Pinheiro-Bedregal’s fuzzy subsethood measure.

Proof. We verify that σ_FIφ satisfies the conditions of Definition 5.1 as follows.

(PB1) For all A, B ∈ F (X), since A is 1-positive, F satisfies that F (x, y) =1 ⇔ x = 1 and I satisfies that I (x, y) =1 ⇔ x = 0 or y = 1, then it follows that: $\begin{matrix} σ_{FI φ} (A, B) = 1 \\ \Leftrightarrow A (I_{FI φ} (A (x_{1}), B (x_{1})), \dots, I_{FI φ} \\ (A (x_{n}), B (x_{n}))) = 1 \\ \Leftrightarrow I_{FI φ} (A (x_{i}), B (x_{i})) = 1, for all i \in {1, \dots, n} \\ \Leftrightarrow F (I (A (x_{i}), B (x_{i})), φ (B (x_{i}))) = 1 \\ \Leftrightarrow I (A (x_{i}), B (x_{i})) = 1 \\ \Leftrightarrow A (x_{i}) = 0 or B (x_{i}) = 1 . \end{matrix}$

(PB2) For all A, B ∈ F (X), since $A$ is 0-positive, F satisfies that F (x, y) =0 ⇔ x = 0 and I satisfies that I (x, y) =0 ⇔ x = 1 and y = 0, then it holds that: $\begin{matrix} σ_{FI φ} (A, B) = 0 \\ \Leftrightarrow A (I_{FI φ} (A (x_{1}), B (x_{1})), \dots, I_{FI φ} \\ (A (x_{n}), B (x_{n}))) = 0 \\ \Leftrightarrow I_{FI φ} (A (x_{i}), B (x_{i})) = 0, forall i \in {1, \dots, n} \\ \Leftrightarrow F (I (A (x_{i}), B (x_{i})), φ (B (x_{i}))) = 0 \\ \Leftrightarrow I (A (x_{i}), B (x_{i})) = 0 \\ \Leftrightarrow A (x_{i}) = 1 and B (x_{i}) = 0 \\ \Leftrightarrow A = X and B = \emptyset . \end{matrix}$

(PB3) (i) For any A, B, C ∈ F (X) with A ⩽ B ⩽ C, it follows from the monotonicity of I, F and $A$ that $\begin{matrix} B (x_{i}) ⩽ C (x_{i}), forall i \in {1, \dots, n} \\ \Rightarrow I (C (x_{i}), A (x_{i})) ⩽ I (B (x_{i}), A (x_{i})) \\ \Rightarrow F (I (C (x_{i}), A (x_{i})), φ (A (x_{i}))) ⩽ \\ F (I (B (x_{i}), A (x_{i})), φ (A (x_{i}))) \\ \Rightarrow I_{FI φ} (C (x_{i}), A (x_{i})) ⩽ I_{FI φ} (B (x_{i}), A (x_{i})) \\ \Rightarrow A (I_{FI φ} (C (x_{i}), A (x_{i}))) ⩽ A (I_{FI φ} (B (x_{i}), A (x_{i}))) \\ \Rightarrow σ_{FI φ} (C, A) ⩽ σ_{FI φ} (B, A) . \end{matrix}$

(ii) Similarly, for any A, B ∈ F (X) with A ⩽ B, it follows from the monotonicity of I, φ, F and A that $\begin{matrix} A (x_{i}) ⩽ B (x_{i}), for all i \in {1, \dots, n} \\ \Rightarrow F (I (C (x_{i}), A (x_{i})), φ (A (x_{i}))) ⩽ F (I (C (x_{i}), \\ B (x_{i})), φ (B (x_{i}))) \\ \Rightarrow I_{FI φ} (C (x_{i}), A (x_{i})) ⩽ I_{FI φ} (C (x_{i}), B (x_{i})) \\ \Rightarrow A (I_{FI φ} (C (x_{i}), A (x_{i}))) ⩽ A (I_{FI φ} (C (x_{i}), B (x_{i}))) \\ \Rightarrow σ_{FI φ} (C, A) ⩽ σ_{FI φ} (C, B) . \end{matrix}$

Therefore, σ_FIφ is a Pinheiro-Bedregal’s fuzzy subsethood measure. □

Example 5.5. Let I be a fuzzy implication satisfying the lowest falsity property (9) and the lowest truth property (10). Consider the binary aggregation function F given by $F (x, y) = {\begin{matrix} 1, & x = 1 and 0 ⩽ y ⩽ 1, \\ \begin{matrix} 0, \\ x, \end{matrix} & \begin{matrix} x = 0 and 0 ⩽ y ⩽ 1, \\ otherwise, \end{matrix} \end{matrix}$ the non-decreasing function φ (x) = x and the 0- and 1-positive n-ary aggregation function $A$ given by $A (x_{1}, \dots, x_{n}) = \frac{x_{1} + \dots + x_{n}}{n} .$

Then, for all A, B ∈ F (X), we have $\begin{matrix} σ_{FI φ} (A, B) = A (I_{FI φ} (A (x_{1}), B (x_{1})), \dots, \\ I_{FI φ} (A (x_{n}), B (x_{n}))) \\ = \frac{\sum_{i = 1}^{n} I_{FI φ} (A (x_{i}), B (x_{i}))}{n} \\ {\begin{matrix} 1, & A (x_{i}) = 0 or B (x_{i}) = 1, \\ \begin{matrix} 0, \\ \frac{\sum_{i = 1}^{n} I (A (x_{i}), B (x_{i}))}{n}, \end{matrix} & \begin{matrix} A (x_{i}) = 1 and B (x_{i}) = 0, \\ otherwise, \end{matrix} \end{matrix} \end{matrix}$ is a Pinheiro-Bedregal’s fuzzy subsethood measure.

6 Conclusion

It is necessary and interesting to study how to modify a given fuzzy implication such that the new obtained implication can be satisfied some desired properties. In this sense, a new method for constructing fuzzy implications based on an aggregation function with F (1, 0) =1, a fuzzy implication I and a non-decreasing function φ, called FIφ-construction, is introduced. Our method can be used to generate a new implication satisfying a specific property. Moreover, the logical properties preserved by the new fuzzy implications generated by the FIφ-construction are studied. However, an open problem is still unsolved:

(OP) Can we characterize all the cases a fuzzy implication remains invariant by the FIφ-construction?

As further work, we intend to further study the FIφ-construction method, and we believe that many different directions related to this construction also deserve to be investigated. For instance, one can use the FIφ-construction to generate a new fuzzy implication, which depending on only one operator. If we consider the probabilistic sum t-conorm and the non-decreasing function φ (y) = y, then we generate $I_{SI φ} (x, y) = (1 - y) I (x, y) + y x, y \in [0, 1],$ which depend on the initial implication function I, and the properties preserved by the new fuzzy implications are also worth to be studied.

Footnotes

Acknowledgments

The authors would like to express gratitude to the Editor-in-Chief and anonymous reviewers for their valuable comments and suggestions. This research is partially supported by the Natural Science Foundation of Hebei Province (Grant No. F2018201060).

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A method of constructing fuzzy implications from the FIφ -construction

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 Fuzzy implications generated by the FIφ-construction

3.4 Comparing with the existing construction methods

4.1 UIφ-implications and GpIφ-implications

4.3 The property: I (x, N (x)) = N (x)

5 Applying FIφ-implications to construct fuzzy subsethood measures

6 Conclusion

Footnotes

Acknowledgments

References

4.1 UIφ-implications and G_pIφ-implications