A Study of Generalized QL^′-Implications

Abstract

In this paper, we introduce and study the GQL${}^{\prime }$-operations. We prove that this class is a hyper class of the known class of QL${}^{\prime }$-operations. Similar to QL${}^{\prime }$-operations, GQL${}^{\prime }$-operations are not always fuzzy implications. On the other hand, we present and prove a necessary but not sufficient condition that leads to the generation of a GQL${}^{\prime }$-implication. Our study is completed by studying the satisfaction or the violation of some basic properties of fuzzy implications, such as the left neutrality property, the exchange principle, the identity principle and the left ordering property. Our study also completes the study of the aforementioned basic properties for QL${}^{\prime }$-implications and leads to a new connection between QL-operations and D${}^{\prime }$-operations.

1. Introduction

Many definitions and properties in fuzzy theory have been generated from generalizations of classical tautologies [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Moreover, fuzzy implications have been used for the construction of many applications [14,24,25] and are applied in many different scientific and applicable areas [26,27,28,29,30,31,32,33] such as approximate reasoning, control theories and expert systems, robotics, decision-making theories, fuzzy mathematical morphology, image processing, and others. The necessity of construction methods, or different classes, or different models of fuzzy implications is remarked by Mas et al. [14]. According to Mas et al. [14], new fuzzy implication functions are necessary, since they can be adequate in specific applications.

In [4], a construction method of new classes of fuzzy implications via known classes of fuzzy implications was presented, and their connection has also been studied (between the preliminary and the induced class). As a result, the third application of this method in [4] leads to the class of QL${}^{\prime }$-implications, which is the generalization of the following classical tautology

$$(p\Rightarrow q)\equiv {[p\wedge (p^{\prime }\vee q^{\prime })]}^{\prime }.$$

(1)

In this paper, we revisit QL${}^{\prime }$-implications, and we generalize them. This generalization was achieved via the possible usage of not only the same fuzzy negation function everywhere in their formula. Their formula contains three times a fuzzy negation function in it, and there is no restriction in the usage of the same fuzzy negation everywhere in the construction formula in every generation. Therefore, we are leading to a new generalized class of QL${}^{\prime }$-implications (shortly GQL${}^{\prime }$-implications). This paper follows very recent studies that have been published [5,11,34]. In other words, this is a following study, which generalizes the class of QL${}^{\prime }$-implications [4] and generates a variety of fuzzy implications, which is necessary according to many authors [9,13,14].

The paper is organized as follows. Section 2 provides the preliminaries concepts for understanding the article. Section 3 contains the main results and is divided into five subsections. In Section 3.1, we introduce GQL${}^{\prime }$-operations. We prove that they are a hyper class of QL${}^{\prime }$-operations but not always fuzzy implications. A necessary but not sufficient condition for a GQL${}^{\prime }$-operation to be a fuzzy implication is proved. A connection of GQL${}^{\prime }$-, QL-, D- and D${}^{\prime }$-operations is also presented. In Section 3.2, we present a sufficient and necessary condition for the satisfaction of the left neutrality property by a GQL${}^{\prime }$-operation. Some more results for the violation of the left neutrality property by a GQL${}^{\prime }$-operation are also presented. In Section 3.3, there is a similar study for the exchange principle property and GQL${}^{\prime }$-operations. A sufficient condition for the violation of the exchange principle by a GQL${}^{\prime }$-operation is presented. Another result for the violation of the exchange principle by a GQL${}^{\prime }$-operation is also presented. In Section 3.4, we deal with the identity principle property and GQL${}^{\prime }$-operations. Some equivalent statements under specific conditions are proved. In Section 3.5, it is proved that the identity principle and the left ordering property are equivalent for GQL${}^{\prime }$-operations. Some equivalent statements under specific conditions are also proved. The results of Section 3.3, Section 3.4, Section 3.5 also complete the study of the basic properties of QL${}^{\prime }$-implications and conditions, assuming the three fuzzy negations in the construction method as equal, i.e., as one. Section 4 contains the results of the study. A new connection of QL- and D${}^{\prime }$-operations comes out of this study. Nevertheless, this result is not the aim of this study, and it encourages the necessity of such studies. Section 5 contains the potential advantages of GQL${}^{\prime }$-implications. Section 6 contains the conclusions.

2. Preliminaries

Definition 1

([1,35,36,37]). A decreasing function $N:[0,1]\to [0,1]$ is called fuzzy negation if $N\left(0\right)=1$ and $N\left(1\right)=0$. Moreover, a fuzzy negation N is called

(i)

Strict if it is continuous and strictly decreasing;

(ii)

Strong if it is an involution, i.e.,

$$N\left(N\right(x\left)\right)=x,for all x\in [0,1];$$

(iii)

Non-filling, if

$$N\left(x\right)=1\iff x=0.$$

(2)

Example 1.

(i)

The so-called, least and greatest fuzzy negations (see [1] Example 1.4.4) are, respectively,

$$N_{D1}\left(x\right)=\left\{\begin{array}{cc}0,\hfill & if x>0\hfill \\ 1,\hfill & if x=0\hfill \end{array}\right. and N_{D2}\left(x\right)=\left\{\begin{array}{cc}0,\hfill & if x=1\hfill \\ 1,\hfill & if x<1\hfill \end{array}\right..$$

(3)

(ii)

We call $N_C\left(x\right)=1-x$ the classical fuzzy negation, which is a strong negation. Moreover, there are several types of fuzzy negations, such as $N_K\left(x\right)=1-x^2$ and $N_R\left(x\right)=1-\sqrt{x}$ (see [1] Example 1.4.4 and Table 1.6).

Lemma 1

([1] Lemma 1.4.9). If $N_1$, $N_2$ are fuzzy negations such that $N_1\circ N_2=I{d}_{[0,1]}$, then

(i)

$N_1$ is a continuous fuzzy negation;

(ii)

$N_2$ is a strictly decreasing fuzzy negation.

Definition 2

([1,36,37]). A function $T:{[0,1]}^2\to [0,1]$ is called a triangular norm (shortly t-norm) if it satisfies, for all $x,y,z\in [0,1]$, the following conditions:

$$T(x,y)=T(y,x),$$

(4)

$$T(x,T(y,z\left)\right)=T\left(T\right(x,y),z),$$

(5)

$$if y\le z,then T(x,y)\le T(x,z),i.e.,T(x,·) is increasing,$$

(6)

$$T(x,1)=x.$$

(7)

Dually, a function $S:{[0,1]}^2\to [0,1]$ is called a triangular conorm (shortly t-conorm) if it satisfies, for all $x,y,z\in [0,1]$, the above conditions (4)–(6) and additionally

$$S(x,0)=x.$$

(8)

Example 2.

(i)

There are several types of t-norms, such as the minimum t-norm $T_M(x,y)=min\{x,y\}$, the Łukasiewicz t-norm $T_L(x,y)=max\{x+y-1,0\}$ and the nilpotent minimum t-norm

$$\begin{array}{c}\hfill T_{nM}(x,y)=\left\{\begin{array}{cc}0,\hfill & if x+y\le 1\hfill \\ min\{x,y\},\hfill & otherwise\hfill \end{array}\right.\end{array}$$

(see [1] Table 2.1).

(ii)

Moreover, there are several types of t-conorms, such as the probabilistic t-conorm $S_P(x,y)=x+y-x·y$ (see [1] Table 2.2).

Definition 3

([1] Definition 2.1.2). A t-norm T is called

(i)

Idempotent, if

$$\begin{array}{c}\hfill T(x,x)=x,for all x\in [0,1],\end{array}$$

(ii)

Positive, if

$$\begin{array}{c}\hfill T(x,y)=0\iff x=0 or y=0.\end{array}$$

Definition 4

([1,37]). A t-norm T is strictly monotone if $T(x,y)<T(x,z)$ whenever $x>0$ and $y<z$.

Proposition 1

([3] Proposition 9). For all $x,y\in [0,1]$ it is

$$\begin{array}{c}\hfill T(x,y)\le x\le S(x,y) and T(x,y)\le y\le S(x,y).\end{array}$$

Remark 1.

By Proposition 1, it follows that

$$S(1,x)=S(x,1)=1,x\in [0,1]$$

(9)

and

$$T(0,x)=T(x,0)=0,x\in [0,1].$$

(10)

Definition 5

([1] Definition 2.3.14). Let T be a t-norm and N be a fuzzy negation. We say that the pair $(T,N)$ satisfies the law of contradiction if

$$T\left(N\right(x),x)=0,x\in [0,1].$$

(11)

Definition 6

([37] p. 232). Let N be a strict negation, S be a t-conorm and T be a t-norm, such that $S(x,y)=N^{-1}\left(T(N\left(x\right),N\left(y\right))\right),x,y\in [0,1]$. Then, S is said to be the N-dual of T, and we denote it by $S_{T,N}$.

Respectively, we have the following Definition.

Definition 7.

Let N be a strict negation, T be a t-norm and S be a t-conorm, such that $T(x,y)=N^{-1}\left(S(N\left(x\right),N\left(y\right))\right),x,y\in [0,1]$. Then, T is said to be the N-dual of S, and we denote it by $T_{S,N}$.

Definition 8

([1] Definition 2.3.1). Let T be a t-norm. A function $N_T:[0,1]\to [0,1]$ defined as

$$\begin{array}{c}\hfill N_T\left(x\right)=sup\{y\in [0,1]|T(x,y)=0\},x\in [0,1],\end{array}$$

is called the natural negation of T or the negation induced by T.

Remark 2.

(i)

It is easy to prove that $N_T$ is a fuzzy negation (see [1] Remark 2.3.2. (i)).

(ii)

If $T(x,y)=0$ for some $x,y\in [0,1]$, then $y\le N_T\left(x\right)$ (see [1] Remark 2.3.2. (iii)).

Corollary 1

([1] Corollary 2.3.7). Let T be any t-norm and $N_T$ be its natural negation. Then, the following statements are equivalent:

(i)

$N_T$ is strictly decreasing.

(ii)

$N_T$ is continuous.

(iii)

$N_T$ is strong.

Proposition 2

([1] Proposition 2.3.4). If a t-norm T is left continuous, then for every x, y $\in [0,1]$, it is

$$\begin{array}{c}\hfill T(x,y)=0\iff N_T\left(x\right)\ge y\end{array}$$

Definition 9

([1,38]). By Φ, we denote the family of all increasing bijections from $[0,1]$ to $[0,1]$. We say that functions $f,g:{[0,1]}^n\to [0,1]$ are Φ-conjugate if there exists a $\varphi \in \Phi$ such that $g=f_{\varphi }$, where

$$f_{\varphi }(x_1,x_2,\dots ,x_n)={\varphi }^{-1}\left(f(\varphi \left(x_1\right),\varphi \left(x_2\right),\dots ,\varphi \left(x_n\right))\right),x_1,x_2,\dots ,x_n\in [0,1].$$

Remark 3

([1] Propositions 1.4.8, Remarks 2.1.4 (vii) and 2.2.5 (vii)). It is easy to prove that if $\varphi \in \Phi$ and T is a t-norm, S is a t-conorm and N is a fuzzy negation (respectively, strict and strong), then $T_{\varphi }$ is a t-norm, $S_{\varphi }$ is a t-conorm and $N_{\varphi }$ is a fuzzy negation (respectively, strict and strong).

Definition 10

([1,35]). A function $I:{[0,1]}^2\to [0,1]$ is called a fuzzy implication if

$$I is decreasing with respect to the first variable,$$

(12)

$$I is increasing with respect to the second variable,$$

(13)

$$I(0,0)=1,$$

(14)

$$I(1,1)=1,$$

(15)

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

(16)

Remark 4.

By axioms (13) and (14), we deduce the normality condition

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

(17)

Moreover, by Definition 10, it is easy to prove the left and right boundary conditions [1]

$$I(0,y)=1,y\in [0,1],$$

(18)

$$I(x,1)=1,x\in [0,1].$$

(19)

Definition 11

([1,10]). A fuzzy implication I is said to satisfy

(i)

The left neutrality property, if

$$I(1,y)=y,y\in [0,1],$$

(20)

(ii)

The exchange principle, if

$$I(x,I(y,z\left)\right)=I(y,I(x,z\left)\right),x,y,z\in [0,1].$$

(21)

(iii)

The identity principle, if

$$I(x,x)=1,x\in [0,1],$$

(22)

(iv)

The ordering property, if

$$I(x,y)=1\iff x\le y,x,y\in [0,1].$$

(23)

(v)

The left ordering property, if

$$x\le y\Rightarrow I(x,y)=1,x,y\in [0,1].$$

(24)

(vi)

The right ordering property, if

$$x>y\Rightarrow I(x,y)\ne 1,x,y\in [0,1].$$

(25)

Remark 5.

(i)

Properties (20)–(25) are not limited to fuzzy implications, but in any function, $I:{[0,1]}^2\to [0,1]$.

(ii)

It is proved that if $\varphi \in \Phi$ and $I:{[0,1]}^2\to [0,1]$ satisfies (12) (respectively, (13)–(16)), then $I_{\varphi }$ also satisfies (12) (respectively (13)–(16)). Moreover, if $I:{[0,1]}^2\to [0,1]$ is a fuzzy implication, then $I_{\varphi }$ is also a fuzzy implication (see [1] Proposition 1.1.8).

Lemma 2

([1] Lemma 1.4.14). If a function $I:{[0,1]}^2\to [0,1]$ satisfies (12), (14) and (16), then the function $N_I:[0,1]\to [0,1]$ is a fuzzy negation, where

$$N_I\left(x\right)=I(x,0),x\in [0,1].$$

Definition 12

([1] Definition 1.4.15). Let $I:{[0,1]}^2\to [0,1]$ be a fuzzy implication. The function $N_I$ defined by Lemma 2 is called the natural negation of I.

Definition 13

([1] Definition 1.6.12). Let N be a fuzzy negation and I be a fuzzy implication. A function $I_N:{[0,1]}^2\to [0,1]$ defined by

$$I_N(x,y)=I(N\left(y\right),N\left(x\right)),x,y\in [0,1],$$

is called the N-reciprocal of I.

Theorem 1

([1] Theorem 1.6.2). If N is a fuzzy negation and I is a fuzzy implication, then $I_N$ is a fuzzy implication.

Remark 6.

The definition of the N-reciprocal of a function $I:{[0,1]}^2\to [0,1]$ is not limited to fuzzy implications, but in any function $I:{[0,1]}^2\to [0,1]$.

Definition 14

([1,13,14,35,39]). A function $I:{[0,1]}^2\to [0,1]$ is called a QL-operation if there exist a t-norm T, a t-conorm S and a fuzzy negation N such that

$$\begin{array}{c}\hfill I(x,y)=S\left(N\right(x),T(x,y\left)\right),x,y\in [0,1].\end{array}$$

If I is a QL-operation generated from the triple $(T,S,N)$, then we will often denote it by $I_{T,S,N}$.

Definition 15

([1,13,39,40]). A function $I:{[0,1]}^2\to [0,1]$ is called a D-operation if there exist a t-norm T, a t-conorm S and a fuzzy negation N such that

$$\begin{array}{c}\hfill I(x,y)=S\left(T\right(N\left(x\right),N\left(y\right)),y),x,y\in [0,1].\end{array}$$

If I is a D-operation generated from the triple $(T,S,N)$, then we will often denote it by $I^{T,S,N}$.

Definition 16

([4] Definition 12). A function $I:{[0,1]}^2\to [0,1]$ is called a D${}^{\prime }$-operation if there exist a t-conorm S, a t-norm T and a fuzzy negation N such that

$$I(x,y)=N\left(T\right(S(x,y),N\left(y\right)\left)\right),x,y\in [0,1].$$

If I is a D${}^{\prime }$-operation generated by the triple $(T,S,N)$, then we denote it by $I^{N,T,S}$.

Definition 17

([4] Definition 13). A function $I:{[0,1]}^2\to [0,1]$ is called a QL${}^{\prime }$-operation if there exist a t-norm T, a t-conorm S and a fuzzy negation N such that

$$I(x,y)=N\left(T\right(x,S\left(N\right(x),N(y\left)\right)\left)\right),x,y\in [0,1].$$

If I is a QL${}^{\prime }$-operation generated by the triple $(T,S,N)$, then we denote it by $I_{N,T,S}$.

Remark 7

([1,4,39,40]). QL-, QL${}^{\prime }$- and D-, D${}^{\prime }$-operations are not fuzzy implications in general, since (12) (for QL- and QL${}^{\prime }$-operations) or (13) (for D- and D${}^{\prime }$-operations) could not hold, respectively. Only if the QL- or QL${}^{\prime }$- or D- or D${}^{\prime }$-operation is a fuzzy implication, we will use the term QL- or QL${}^{\prime }$ or D- or D${}^{\prime }$-implication, respectively.

3. The Main Results

3.1. GQL${}^{\prime }$-Implications

Generalized QL${}^{\prime }$-implications are constructed by using not only the same fuzzy negation function to their formula. So, we are leading to the following definition.

Definition 18.

A function $I:{[0,1]}^2\to [0,1]$ is called a GQL${}^{\prime }$-operation if there exist a t-norm T, a t-conorm S and three fuzzy negations $N_1$, $N_2$ and $N_3$ such that

$$I(x,y)=N_1\left(T(x,S(N_2\left(x\right),N_3\left(y\right)))\right),x,y\in [0,1].$$

If I is a GQL${}^{\prime }$-operation generated by the quintuple $(N_1,T,S,N_2,N_3)$, then we denote it by $I_{N_1,T,S,N_2,N_3}$.

Theorem 2.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation; then, $I_{N_1,T,S,N_2,N_3}$ satisfies (13)–(18). Furthermore, $N_{I_{N_1,T,S,N_2,N_3}}=N_1$, where $N_{I_{N_1,T,S,N_2,N_3}}\left(x\right)=I_{N_1,T,S,N_2,N_3}(x,0)$, for all $x\in [0,1]$.

Proof. 

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation, then for $x,y,z\in [0,1]$,

$$\begin{array}{cc}\hfill y\le z& \Rightarrow N_3\left(y\right)\ge N_3\left(z\right)\hfill \\ \hfill & \stackrel{\left(6\right)}{\Rightarrow }S(N_2\left(x\right),N_3\left(y\right))\ge S(N_2\left(x\right),N_3\left(z\right))\hfill \\ \hfill & \stackrel{\left(6\right)}{\Rightarrow }T(x,S(N_2\left(x\right),N_3\left(y\right)))\ge T(x,S(N_2\left(x\right),N_3\left(z\right)))\hfill \\ \hfill & \Rightarrow N_1\left(T(x,S(N_2\left(x\right),N_3\left(y\right)))\right)\le N_1\left(T(x,S(N_2\left(x\right),N_3\left(z\right)))\right)\hfill \\ \hfill & \Rightarrow I_{N_1,T,S,N_2,N_3}(x,y)\le I_{N_1,T,S,N_2,N_3}(x,z),\hfill \end{array}$$

which means that $I_{N_1,T,S,N_2,N_3}$ satisfies (13).

$I_{N_1,T,S,N_2,N_3}$ satisfies (14), since

$$\begin{array}{c}\hfill I_{N_1,T,S,N_2,N_3}(0,0)=N_1\left(T(0,S(N_2\left(0\right),N_3\left(0\right)))\right)\stackrel{\left(10\right)}{=}{N}_1\left(0\right)=1.\end{array}$$

$I_{N_1,T,S,N_2,N_3}$ satisfies (15), since

$$\begin{array}{c}\hfill I_{N_1,T,S,N_2,N_3}(1,1)=N_1\left(T(1,S(N_2\left(1\right),N_3\left(1\right)))\right)=N_1\left(T(1,S(0,0))\right)\stackrel{\left(8\right)}{=}{N}_1\left(T(1,0)\right)\stackrel{\left(10\right)}{=}{N}_1\left(0\right)=1.\end{array}$$

$I_{N_1,T,S,N_2,N_3}$ satisfies (16), since

$$\begin{array}{c}\hfill I_{N_1,T,S,N_2,N_3}(1,0)=N_1\left(T(1,S(N_2\left(1\right),N_3\left(0\right)))\right)=N_1\left(T(1,S(0,1))\right)\stackrel{\left(9\right)}{=}{N}_1\left(T(1,1)\right)\stackrel{\left(7\right)}{=}{N}_1\left(1\right)=0.\end{array}$$

$I_{N_1,T,S,N_2,N_3}$ satisfies (17), since

$$\begin{array}{c}\hfill I_{N_1,T,S,N_2,N_3}(0,1)=N_1\left(T(0,S(N_2\left(0\right),N_3\left(1\right)))\right)\stackrel{\left(10\right)}{=}{N}_1\left(0\right)=1.\end{array}$$

$I_{N_1,T,S,N_2,N_3}$ satisfies (18), since $\forall y\in [0,1]$

$$\begin{array}{c}\hfill I_{N_1,T,S,N_2,N_3}(0,y)=N_1\left(T(0,S(N_2\left(0\right),N_3\left(y\right)))\right)\stackrel{\left(10\right)}{=}{N}_1\left(0\right)=1.\end{array}$$

Lastly, we have

$$\begin{array}{cc}\hfill N_{I_{N_1,T,S,N_2,N_3}}\left(x\right)& =I_{N_1,T,S,N_2,N_3}(x,0)\hfill \\ \hfill & =N_1\left(T(x,S(N_2\left(x\right),N_3\left(0\right)))\right)\hfill \\ \hfill & =N_1\left(T(x,S(N_2\left(x\right),1))\right)\hfill \\ \hfill & \stackrel{\left(9\right)}{=}{N}_1\left(T(x,1)\right)\hfill \\ \hfill & \stackrel{\left(7\right)}{=}{N}_1\left(x\right).\hfill \end{array}$$

for all $x\in [0,1]$. □

Remark 8.

Note that if $N_1=N_2=N_3=N$, the corresponding GQL${}^{\prime }$-operation is a QL${}^{\prime }$-operation. This means that GQL${}^{\prime }$-operations do not always satisfy (12). Moreover, Example 3 leads us to the same result even if we use different negations. Therefore, we called them GQL${}^{\prime }$-operations instead of GQL${}^{\prime }$-implications.

Example 3.

Consider the quintuple $(N_C,T_M,S_P,N_K,N_R)$. The corresponding GQL${}^{\prime }$-operation is

$$\begin{array}{cc}\hfill I_{N_C,T_M,S_P,N_K,N_R}(x,y)& =N_C\left(T_M(x,S_P(N_K\left(x\right),N_R\left(y\right)))\right)\hfill \\ \hfill & =1-T_M(x,S_P(1-x^2,1-\sqrt{y}))\hfill \\ \hfill & =1-T_M(x,1-x^2+1-\sqrt{y}-(1-x^2)·(1-\sqrt{y}))\hfill \\ \hfill & =1-min\{x,1-x^2·\sqrt{y}\}\hfill \\ \hfill & =max\{1-x,1-(1-x^2·\sqrt{y})\}\hfill \\ \hfill & =max\{1-x,x^2·\sqrt{y}\},\hfill \end{array}$$

which is not a fuzzy implication, since

$$\begin{array}{c}\hfill 0.1\le 1\Rightarrow I_{N_C,T_M,S_P,N_K,N_R}(0.1,1)=0.9<1=I_{N_C,T_M,S_P,N_K,N_R}(1,1).\end{array}$$

Thus, $I_{N_C,T_M,S_P,N_K,N_R}$ violates (12).

Proposition 3.

A function $I:{[0,1]}^2\to [0,1]$ is called a GQL${}^{\prime }$-implication if it is a GQL${}^{\prime }$-operation and satisfies (12).

Proof. 

The proof is obvious. □

Example 4.

Consider the quintuple $(N_{D1},T_L,S,N_C,N_{D2})$. The corresponding GQL${}^{\prime }$-operation, which is a GQL${}^{\prime }$-implication, is

$$\begin{array}{cc}\hfill I_{N_{D1},T_L,S,N_C,N_{D2}}(x,y)& =N_{D1}\left(T_L(x,S(N_C\left(x\right),N_{D2}\left(y\right)))\right)\hfill \\ \hfill & =\left\{\begin{array}{cc}{N}_{D1}\left(T_L(x,S(N_C\left(x\right),1))\right),\hfill & if y<1\hfill \\ N_{D1}\left(T_L(x,S(N_C\left(x\right),0))\right),\hfill & if y=1\hfill \end{array}\right.\hfill \\ \hfill & \stackrel{\left(9\right)}{\underset{\left(8\right)}{=}}\left\{\begin{array}{cc}{N}_{D1}\left(T_L(x,1)\right),\hfill & if y<1\hfill \\ N_{D1}\left(T_L(x,N_C\left(x\right))\right),\hfill & if y=1\hfill \end{array}\right.\hfill \\ \hfill & \stackrel{\left(7\right)}{\underset{\left(4\right)}{=}}\left\{\begin{array}{cc}{N}_{D1}\left(x\right),\hfill & if y<1\hfill \\ N_{D1}\left(max\{x+N_C\left(x\right)-1,0\}\right),\hfill & if y=1\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}{N}_{D1}\left(x\right),\hfill & if y<1\hfill \\ N_{D1}\left(max\{x+1-x-1,0\}\right),\hfill & if y=1\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}{N}_{D1}\left(x\right),\hfill & if y<1\hfill \\ N_{D1}\left(0\right),\hfill & if y=1\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}0,\hfill & if x>0 and y<1\hfill \\ 1,\hfill & if x=0 and y<1\hfill \\ 1,\hfill & if y=1\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}0,\hfill & if x>0 and y<1\hfill \\ 1,\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}1,\hfill & if x=0 or y=1\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =I_{18}(x,y).\hfill \end{array}$$

([2] Figure 5).

Remark 9.

According to the Example 4, there are GQL${}^{\prime }$-implications that are not QL${}^{\prime }$-implications, since the only QL${}^{\prime }$-implication that has $N_{D1}$ as its natural negation is

$$\begin{array}{c}\hfill I_3(x,y)=\left\{\begin{array}{cc}1,\hfill & if x=0 or y>0\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right..\end{array}$$

([4] Remark 5.14).

Since the natural negation of the GQL${}^{\prime }$-implication $I_{N_{D1},T_L,S,N_C,N_{D2}}=I_{18}$ is also $N_{D1}$, we deduce that $I_{18}$ is not a QL${}^{\prime }$-implication. Thus, the class of GQL${}^{\prime }$-implications is a hyper class of that of QL${}^{\prime }$-implications. According to Example 4, it is a non-empty set. According to Remark 8, it is a new class of fuzzy implications that contains QL${}^{\prime }$-implications.

Proposition 4.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-implication and $N_1$ be a non-filling fuzzy negation. Then, the pair $(T,N_2)$ satisfies (11).

Proof. 

If $I_{N_1,T,S,N_2,N_3}$ is a fuzzy implication, then it satisfies (19). Therefore, for all $x\in [0,1]$, it is

$$\begin{array}{cc}\hfill I_{N_1,T,S,N_2,N_3}(x,1)=1& \Rightarrow N_1\left(T(x,S(N_2\left(x\right),N_3\left(1\right)))\right)=1\hfill \\ \hfill & \Rightarrow N_1\left(T(x,S(N_2\left(x\right),0))\right)=1\hfill \\ \hfill & \stackrel{\left(8\right)}{\Rightarrow }{N}_1\left(T(x,N_2\left(x\right))\right)=1\hfill \\ \hfill & \stackrel{\left(4\right)}{\Rightarrow }{N}_1\left(T(N_2\left(x\right),x)\right)=1\hfill \\ \hfill & \stackrel{\left(2\right)}{\Rightarrow }T(N_2\left(x\right),x)=0.\hfill \end{array}$$

Therefore, the pair $(T,N_2)$ satisfies (11). □

Example 5.

Consider the quintuple $(N_K,T_{nM},S_P,N_C,N_R)$. The corresponding GQL${}^{\prime }$-operation is

$$\begin{array}{cc}\hfill I_{N_K,T_{nM},S_P,N_C,N_R}(x,y)& =N_K\left(T_{nM}(x,S_P(N_C\left(x\right),N_R\left(y\right)))\right)\hfill \\ \hfill & =N_K\left(T_{nM}(x,S_P(1-x,1-\sqrt{y}))\right)\hfill \\ \hfill & =N_K\left(T_{nM}(x,1-x+1-\sqrt{y}-(1-x)·(1-\sqrt{y}))\right)\hfill \\ \hfill & =N_K\left(T_{nM}(x,1-x·\sqrt{y})\right))\hfill \\ \hfill & =\left\{\begin{array}{cc}{N}_K\left(0\right),\hfill & if x+1-x·\sqrt{y}\le 1\hfill \\ N_K\left(min\{x,1-x·\sqrt{y}\}\right),\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}1,\hfill & if x-x·\sqrt{y}\le 0\hfill \\ 1-{\left(min\{x,1-x·\sqrt{y}\}\right)}^2,\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}1,\hfill & if x·(1-\sqrt{y})\le 0\hfill \\ 1-{\left(min\{x,1-x·\sqrt{y}\}\right)}^2,\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & \stackrel{x,y\in [0,1]}{=}\left\{\begin{array}{cc}1,\hfill & if x=0 or y=1\hfill \\ 1-min\{x^2,{(1-x·\sqrt{y})}^2\},\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}1,\hfill & if x=0 or y=1\hfill \\ max\{1-x^2,1-{(1-x·\sqrt{y})}^2\},\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}1,\hfill & if x=0 or y=1\hfill \\ max\{1-x^2,(1-(1-x·\sqrt{y}))·(1+(1-x·\sqrt{y}))\},\hfill & otherwise\hfill \end{array}\right.\hfill \\ \hfill & =\left\{\begin{array}{cc}1,\hfill & if x=0 or y=1\hfill \\ max\{1-x^2,x·\sqrt{y}·(2-x·\sqrt{y})\},\hfill & otherwise\hfill \end{array}\right.\hfill \end{array}$$

which is not a fuzzy implication, since

$$\begin{array}{c}\hfill 0.7\le 0.8\Rightarrow I_{N_K,T_{nM},S_P,N_C,N_R}(0.7,0.9)=0.88715661727<0.94189327688=I_{N_C,T_M,S_P,N_K,N_R}(0.8,0.9).\end{array}$$

Thus, $I_{N_K,T_{nM},S_P,N_C,N_R}$ violates (12).

Remark 10.

(i)

Proposition 4 gives a necessary but not sufficient condition for a GQL${}^{\prime }$-operation $I_{N_1,T,S,N_2,N_3}$ to be a fuzzy implication, when $N_1$ is a non-filling fuzzy negation. Note that $N_K$ is a non-filling fuzzy negation, $(T_{nM},N_C)$ satisfies (11), but $I_{N_K,T_{nM},S_P,N_C,N_R}$ is not a fuzzy implication (see Example 5).

(ii)

By Proposition 4, it is obvious that if $N_1$ is a non-filling negation and the pair $(T,N_2)$ does not satisfy the law of contradiction (11), i.e., $T(N_2\left(x\right),x)\ne 0$, for some $x\in (0,1)$, then the obtained $I_{N_1,T,S,N_2,N_3}$ GQL${}^{\prime }$-operation, for any t-conorm S and fuzzy negation $N_3$, is not a fuzzy implication.

Theorem 3.

By any quintuple $(N_1,T,S,N_2,N_3)$, where $N_1$ is any non-filling fuzzy negation, S is any t-conorm, $N_2$ is any continuous fuzzy negation, $N_3$ is any fuzzy negation and T is any idempotent, strict or positive t-norm, it cannot be obtained any GQL${}^{\prime }$-implication.

Proof. 

The proof is similar to the proof of Theorem 5 in [4], by using Remark 10 (ii). Therefore, it is omitted. □

Theorem 4.

If $\varphi \in \Phi$ and $I_{N_1,T,S,N_2,N_3}$ is a GQL${}^{\prime }$-operation (respectively, implication), then ${\left(I_{N_1,T,S,N_2,N_3}\right)}_{\varphi }$ is a GQL${}^{\prime }$-operation (respectively, implication) and moreover,

$${\left(I_{N_1,T,S,N_2,N_3}\right)}_{\varphi }=I_{{\left(N_1\right)}_{\varphi },T_{\varphi },S_{\varphi },{\left(N_2\right)}_{\varphi },{\left(N_3\right)}_{\varphi }}.$$

Proof. 

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation (respectively, implication), then ${\left(I_{N_1,T,S,N_2,N_3}\right)}_{\varphi }$ is a GQL${}^{\prime }$-operation (respectively, implication) according to Remark 5 (ii). Moreover, for all $x,y\in [0,1]$, we deduce that

$$\begin{array}{cc}\hfill {\left(I_{N_1,T,S,N_2,N_3}\right)}_{\varphi }(x,y)& ={\varphi }^{-1}\left(I_{N_1,T,S,N_2,N_3}(\varphi \left(x\right),\varphi \left(y\right))\right)\hfill \\ \hfill & ={\varphi }^{-1}\left(N_1\left(T(\varphi \left(x\right),S(N_2\left(\varphi \left(x\right)\right),N_3\left(\varphi \left(y\right)\right)))\right)\right)\hfill \\ \hfill & ={\varphi }^{-1}\left(N_1\left(\varphi \left({\varphi }^{-1}\left(T(\varphi \left(x\right),S(\varphi \left({\varphi }^{-1}\left(N_2\left(\varphi \left(x\right)\right)\right)\right),\varphi \left({\varphi }^{-1}\left(N_3\left(\varphi \left(y\right)\right)\right)\right)))\right)\right)\right)\right)\hfill \\ \hfill & ={\left(N_1\right)}_{\varphi }({\varphi }^{-1}(T(\varphi \left(x\right),S(\varphi ({\left(N_2\right)}_{\varphi }\left(x\right)),\varphi ({\left(N_3\right)}_{\varphi }\left(y\right))))))\hfill \\ \hfill & ={\left(N_1\right)}_{\varphi }({\varphi }^{-1}(T(\varphi \left(x\right),\varphi ({\varphi }^{-1}(S(\varphi ({\left(N_2\right)}_{\varphi }\left(x\right)),\varphi ({\left(N_3\right)}_{\varphi }\left(y\right))))))))\hfill \\ \hfill & ={\left(N_1\right)}_{\varphi }(T_{\varphi }(x,S_{\varphi }({\left(N_2\right)}_{\varphi }\left(x\right),{\left(N_3\right)}_{\varphi }\left(y\right))))\hfill \\ \hfill & =I_{{\left(N_1\right)}_{\varphi },T_{\varphi },S_{\varphi },{\left(N_2\right)}_{\varphi },{\left(N_3\right)}_{\varphi }}(x,y).\hfill \end{array}$$

□

A connection between GQL${}^{\prime }$-, QL-, D- and D${}^{\prime }$-operations is the following.

Theorem 5.

Let $I_{N^{-1},T,S,N,N}$ be a GQL${}^{\prime }$-operation generated from a t-norm T, a t-conorm S, a strict fuzzy negation N and its inverse $N^{-1}$. Then,

$$I_{N^{-1},T,S,N,N}=I_{T_{S,N},S_{T,N},N^{-1}}={\left(I^{T_{S,N},S_{T,N},N}\right)}_{N^{-1}}={\left(I^{N^{-1},T,S}\right)}_N.$$

(26)

Moreover, if one of $I_{N^{-1},T,S,N,N}$, $I_{T_{S,N},S_{T,N},N^{-1}}$, $I^{T_{S,N},S_{T,N},N}$ and $I^{N^{-1},T,S}$ is a fuzzy implication, then the other three are fuzzy implications, too.

Proof. 

Let $I_{N^{-1},T,S,N,N}$ be a GQL${}^{\prime }$-operation. Then

$$\begin{array}{cc}\hfill I_{N^{-1},T,S,N,N}(x,y)& =N^{-1}\left(T(x,S(N\left(x\right),N\left(y\right)))\right)\hfill \\ \hfill & =N^{-1}\left(T(N\left(N^{-1}\left(x\right)\right),N\left(N^{-1}\left(S(N\left(x\right),N\left(y\right))\right)\right))\right)\hfill \\ \hfill & =S_{T,N}(N^{-1}\left(x\right),T_{S,N}(x,y))\hfill \\ \hfill & =I_{T_{S,N},S_{T,N},N^{-1}}(x,y)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill I_{N^{-1},T,S,N,N}(x,y)& =I_{T_{S,N},S_{T,N},N^{-1}}(x,y)\hfill \\ \hfill & =S_{T,N}(N^{-1}\left(x\right),T_{S,N}(x,y))\hfill \\ \hfill & =S_{T,N}(N^{-1}\left(x\right),T_{S,N}(N\left(N^{-1}\left(x\right)\right),N\left(N^{-1}\left(y\right)\right)))\hfill \\ \hfill & \stackrel{\left(4\right)}{=}{S}_{T,N}(N^{-1}\left(x\right),T_{S,N}(N\left(N^{-1}\left(y\right)\right),N\left(N^{-1}\left(x\right)\right)))\hfill \\ \hfill & \stackrel{\left(4\right)}{=}{S}_{T,N}(T_{S,N}(N\left(N^{-1}\left(y\right)\right),N\left(N^{-1}\left(x\right)\right)),N^{-1}\left(x\right))\hfill \\ \hfill & =I^{T_{S,N},S_{T,N},N}(N^{-1}\left(y\right),N^{-1}\left(x\right))\hfill \\ \hfill & ={\left(I^{T_{S,N},S_{T,N},N}\right)}_{N^{-1}}(x,y).\hfill \end{array}$$

Moreover,

$$\begin{array}{cc}\hfill I_{N^{-1},T,S,N,N}(x,y)& =N^{-1}\left(T(x,S(N\left(x\right),N\left(y\right)))\right)\hfill \\ \hfill & =N^{-1}\left(T(N^{-1}\left(N\left(x\right)\right),S(N\left(x\right),N\left(y\right)))\right)\hfill \\ \hfill & \stackrel{\left(4\right)}{=}{N}^{-1}\left(T(N^{-1}\left(N\left(x\right)\right),S(N\left(y\right),N\left(x\right)))\right)\hfill \\ \hfill & \stackrel{\left(4\right)}{=}{N}^{-1}\left(T(S(N\left(y\right),N\left(x\right)),N^{-1}\left(N\left(x\right)\right))\right)\hfill \\ \hfill & =I^{N^{-1},T,S}(N\left(y\right),N\left(x\right))\hfill \\ \hfill & ={(I^{N^{-1},T,S})}_N(x,y).\hfill \end{array}$$

Therefore, it is

$$I_{N^{-1},T,S,N,N}=I_{T_{S,N},S_{T,N},N^{-1}}={\left(I^{T_{S,N},S_{T,N},N}\right)}_{N^{-1}}={\left(I^{N^{-1},T,S}\right)}_N.$$

(27)

Easily, it can be observed that

$${\left({\left(I^{T_{S,N},S_{T,N},N}\right)}_{N^{-1}}\right)}_N=I^{T_{S,N},S_{T,N},N} \mathrm{and} {({(I^{N^{-1},T,S})}_N)}_{N^{-1}}=I^{N^{-1},T,S}.$$

(28)

By virtue of Equations (27) and (28) and Theorem 1, if one of $I_{N^{-1},T,S,N,N}$, $I_{T_{S,N},S_{T,N},N^{-1}}$, $I^{T_{S,N},S_{T,N},N}$ and $I^{N^{-1},T,S}$ is a fuzzy implication, then the other three are also fuzzy implications. □

3.2. GQL${}^{\prime }$-Implications and the Left Neutrality Property (20)

Proposition 5.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation. Then, $I_{N_1,T,S,N_2,N_3}$ satisfies (20) if and only if

$$N_1\circ N_3=I{d}_{[0,1]}.$$

Proof. 

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation. If $N_1\circ N_3=I{d}_{[0,1]}$, then for all $x\in [0,1]$, it is

$$\begin{array}{cc}\hfill I_{N_1,T,S,N_2,N_3}(1,x)& =N_1\left(T(1,S(N_2\left(1\right),N_3\left(x\right)))\right)\hfill \\ \hfill & =N_1\left(T(1,S(0,N_3\left(x\right)))\right)\hfill \\ \hfill & \stackrel{\left(4\right)}{=}{N}_1\left(T(1,S(N_3\left(x\right),0))\right)\hfill \\ \hfill & \stackrel{\left(8\right)}{=}{N}_1\left(T(1,N_3\left(x\right))\right)\hfill \\ \hfill & \stackrel{\left(4\right)}{=}{N}_1\left(T(N_3\left(x\right),1)\right)\hfill \\ \hfill & \stackrel{\left(7\right)}{=}{N}_1\left(N_3\left(x\right)\right)\hfill \\ \hfill & =(N_1\circ N_3)\left(x\right)\hfill \\ \hfill & \stackrel{N_1\circ N_3=I{d}_{[0,1]}}{=}x.\hfill \end{array}$$

Thus, $I_{N_1,T,S,N_2,N_3}$ satisfies (20).

Conversely, if $I_{N_1,T,S,N_2,N_3}$ satisfies (20), then for all $x\in [0,1]$

$$\begin{array}{cc}\hfill I_{N_1,T,S,N_2,N_3}(1,x)=x& \Rightarrow N_1\left(T(1,S(N_2\left(1\right),N_3\left(x\right)))\right)=x\hfill \\ \hfill & \Rightarrow N_1\left(T(1,S(0,N_3\left(x\right)))\right)=x\hfill \\ \hfill & \stackrel{\left(4\right)}{\Rightarrow }{N}_1\left(T(1,S(N_3\left(x\right),0))\right)=x\hfill \\ \hfill & \stackrel{\left(8\right)}{\Rightarrow }{N}_1\left(T(1,N_3\left(x\right))\right)=x\hfill \\ \hfill & \stackrel{\left(4\right)}{\Rightarrow }{N}_1\left(T(N_3\left(x\right),1)\right)=x\hfill \\ \hfill & \stackrel{\left(7\right)}{\Rightarrow }{N}_1\left(N_3\left(x\right)\right)=x\hfill \\ \hfill & \Rightarrow (N_1\circ N_3)\left(x\right)=x\hfill \\ \hfill & \Rightarrow N_1\circ N_3=I{d}_{[0,1]}.\hfill \end{array}$$

□

Corollary 2.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation, where at least one of $N_1$ and $N_3$ is a two-valued fuzzy negation, i.e., $Ran\left(N_1\right)\in \{0,1\}$ or $Ran\left(N_2\right)\in \{0,1\}$. Then, $I_{N_1,T,S,N_2,N_3}$ violates (20).

Proof. 

If at least one of $N_1$ and $N_3$ is a two-valued fuzzy negation, then $N_1\circ N_3\ne I{d}_{[0,1]}$. So, if we assume that $I_{N_1,T,S,N_2,N_3}$ satisfies (20), that is a contradiction according to Proposition 5. □

According to Lemma 1 and Proposition 5, we obtain the following corollaries. Their proofs are omitted, since they are obvious.

Corollary 3.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation, where $N_1$ is not a continuous fuzzy negation. Then, $I_{N_1,T,S,N_2,N_3}$ violates (20).

Corollary 4.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation, where $N_3$ is not a strictly decreasing fuzzy negation. Then, $I_{N_1,T,S,N_2,N_3}$ violates (20).

3.3. GQL${}^{\prime }$-Implications and the Exchange Principle (21)

Proposition 6.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation. If $I_{N_1,T,S,N_2,N_3}$ satisfies (21), then $N_1\circ N_3\circ N_1=N_1$.

Proof. 

We assume that $I_{N_1,T,S,N_2,N_3}$ satisfies (21). Then, for all $x\in [0,1]$

$$\begin{array}{cc}\hfill & I_{N_1,T,S,N_2,N_3}(1,I_{N_1,T,S,N_2,N_3}(x,0))=I_{N_1,T,S,N_2,N_3}(x,I_{N_1,T,S,N_2,N_3}(1,0))\hfill \\ \hfill & \stackrel{\left(16\right)}{\Rightarrow }{I}_{N_1,T,S,N_2,N_3}(1,N_1\left(T(x,S(N_2\left(x\right),N_3\left(0\right)))\right))=I_{N_1,T,S,N_2,N_3}(x,0)\hfill \\ \hfill & \Rightarrow I_{N_1,T,S,N_2,N_3}(1,N_1\left(T(x,S(N_2\left(x\right),1))\right))=N_1\left(T(x,S(N_2\left(x\right),1))\right)\hfill \\ \hfill & \stackrel{\left(9\right)}{\Rightarrow }{I}_{N_1,T,S,N_2,N_3}(1,N_1\left(T(x,1)\right))=N_1\left(T(x,1)\right)\hfill \\ \hfill & \stackrel{\left(7\right)}{\Rightarrow }{I}_{N_1,T,S,N_2,N_3}(1,N_1\left(x\right))=N_1\left(x\right)\hfill \\ \hfill & \Rightarrow N_1\left(T(1,S(N_2\left(1\right),N_3\left(N_1\left(x\right)\right)))\right)=N_1\left(x\right)\hfill \\ \hfill & \Rightarrow N_1\left(T(1,S(0,N_3\left(N_1\left(x\right)\right)))\right)=N_1\left(x\right)\hfill \\ \hfill & \stackrel{\left(4\right)}{\Rightarrow }{N}_1\left(T(1,S(N_3\left(N_1\left(x\right)\right),0))\right)=N_1\left(x\right)\hfill \\ \hfill & \stackrel{\left(8\right)}{\Rightarrow }{N}_1\left(T(1,N_3\left(N_1\left(x\right)\right))\right)=N_1\left(x\right)\hfill \\ \hfill & \stackrel{\left(4\right)}{\Rightarrow }{N}_1\left(T(N_3\left(N_1\left(x\right)\right),1)\right)=N_1\left(x\right)\hfill \\ \hfill & \stackrel{\left(7\right)}{\Rightarrow }{N}_1\left(N_3\left(N_1\left(x\right)\right)\right)=N_1\left(x\right).\hfill \end{array}$$

□

Example 6.

Let $S_{SS}^{\lambda }$ be the Schweizer–Sklar t-conorm ([1] Example 2.6.15), then for $\lambda =2$,

$S_{SS}^2(x,y)=1-\sqrt{max\{{(1-x)}^2+{(1-y)}^2-1,0\}}$.

The obtained GQL${}^{\prime }$-implication by the quintuple $(N_C,T_P,S_{SS}^2,N_C,N_C)$ is

$$\begin{array}{cc}\hfill I_{N_C,T_P,S_{SS}^2,N_C,N_C}(x,y)& =I_{T_P,S_{SS}^2,N_C}(x,y)=I_{PC}(x,y)=1-\sqrt{max\{x·(x+x·y^2-2·y),0\}},\hfill \end{array}$$

which does not satisfy (21) ([1] p. 135).

Remark 11.

(i)

Proposition 6 gives us the sufficient condition that if there is an $x_0\in (0,1)$ such that $N_1\left(N_3\left(N_1\left(x_0\right)\right)\right)\ne N_1\left(x_0\right)$, then $I_{N_1,T,S,N_2,N_3}$ violates (21).

(ii)

Since $N_C$ is a strong negation, we obtain that $N_C\left(N_C\left(N_C\left(x\right)\right)\right)=N_C\left(x\right)$, but according to Example 6, $I_{N_C,T_P,S_{SS}^2,N_C,N_C}$ violates (21). Thus, Proposition 6 gives us a necessary but not sufficient condition for the satisfaction of (21) when we have a GQL${}^{\prime }$-operation.

Proposition 7.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation, where $N_1$ is strictly decreasing with a fixed point. If $N_3$ does not have any fixed point, or $N_1$ and $N_3$ have different fixed points, then $I_{N_1,T,S,N_2,N_3}$ violates (21).

Proof. 

The proof is similar with the proofs of Proposition 3.13 in [5] and Proposition 7 in [34]. Therefore, it is omitted. □

3.4. GQL${}^{\prime }$-Implications and the Identity Principle (22)

Proposition 8.

Let the quintuple be $(N_1,T,S,N_2,N_3)$, where $N_1$ is any non-filling fuzzy negation. If $N_T$ is strong and the corresponding GQL${}^{\prime }$-operation, $I_{N_1,T,S,N_2,N_3}$ satisfies (22), then

$$\begin{array}{c}\hfill S(N_2\left(x\right),N_3\left(x\right))\le N_T\left(x\right),for all x\in [0,1].\end{array}$$

Proof. 

Consider that $I_{N_1,T,S,N_2,N_3}$ satisfies (22). Therefore, for all x$\in [0,1]$

$$\begin{array}{cc}\hfill I_{N_1,T,S,N_2,N_3}(x,x)=1& \Rightarrow N_1\left(T(x,S(N_2\left(x\right),N_3\left(x\right)))\right)=1\hfill \\ \hfill & \stackrel{\left(2\right)}{\Rightarrow }T(x,S(N_2\left(x\right),N_3\left(x\right)))=0\hfill \\ \hfill & \stackrel{\mathrm{Remark}2\left(\mathrm{ii}\right)}{\Rightarrow }S(N_2\left(x\right),N_3\left(x\right))\le N_T\left(x\right).\hfill \end{array}$$

□

Theorem 6.

Let the quintuple $(N_1,T,S,N_2,N_3)$ be given, where $N_1$ is any non-filling fuzzy negation. Let $N_T$ be a strong fuzzy negation and T be a left continuous t-norm. Then, the following statements are equivalent:

(i)

$I_{N_1,T,S,N_2,N_3}$ satisfies (22).

(ii)

$S(N_2\left(x\right),N_3\left(x\right))\le N_T\left(x\right)$, for any $x\in [0,1]$.

Proof. 

(i) ⇒ (ii) The proof is obvious. Just apply Proposition 8.

(ii) ⇒ (i) For any x$\in [0,1]$

$$\begin{array}{cc}\hfill S(N_2\left(x\right),N_3\left(x\right))\le N_T\left(x\right)& \stackrel{\mathrm{Proposition}2}{\Rightarrow }T(x,S(N_2\left(x\right),N_3\left(x\right)))=0\hfill \\ \hfill & \Rightarrow N_1\left(T(x,S(N_2\left(x\right),N_3\left(x\right)))\right)=N_1\left(0\right)\hfill \\ \hfill & \Rightarrow I_{N_1,T,S,N_2,N_3}(x,x)=1.\hfill \end{array}$$

□

Remark 12.

Note that Proposition 8 and Theorem 6 hold when $N_T$ is strictly decreasing, or continuous, since it is also strong, according to Corollary 1.

3.5. GQL${}^{\prime }$-Implications and the Left Ordering Property (24)

It is obvious that the satisfaction of the ordering property (23) implies the satisfaction of the identity principle (22). Moreover, the ordering property (23) is divided in two sub-properties, the so-called left ordering property (24) and right ordering property (25). It is easy to observe that if a fuzzy implication I satisfies both (24) and (25), it satisfies the ordering property (23). In the following, we prove a useful theorem.

Theorem 7.

Let $I:{[0,1]}^2\to [0,1]$ be a function that satisfies (13). Then, I satisfies the identity principle (22) if and only if it satisfies the left ordering property (24).

Proof. 

We assume that I satisfies the identity principle (22). For $x,y\in [0,1]$ and $x\le y$, we have

$$\begin{array}{c}\hfill x\le y\stackrel{\left(13\right)}{\Rightarrow }I(x,x)\le I(x,y)\stackrel{\left(22\right)}{\Rightarrow }1\le I(x,y)\Rightarrow I(x,y)=1.\end{array}$$

Thus, I satisfies the left ordering property (24).

Conversely, we assume that I satisfies the left ordering property (24). Therefore, for any $x\in [0,1]$

$$\begin{array}{c}\hfill x\le x\stackrel{\left(24\right)}{\Rightarrow }I(x,x)=1.\end{array}$$

Thus, I satisfies the identity principle (22). □

Corollary 5.

Let $I_{N_1,T,S,N_2,N_3}$ be a GQL${}^{\prime }$-operation. Then, $I_{N_1,T,S,N_2,N_3}$ satisfies the identity principle (22) if and only if it satisfies the left ordering property (24).

Proof. 

By virtue of Theorem 2, $I_{N_1,T,S,N_2,N_3}$ satisfies (13). Therefore, the proof is deduced by applying Theorem 7. □

Corollary 6.

Let the quintuple $(N_1,T,S,N_2,N_3)$ be given, where $N_1$ is any non-filling fuzzy negation. Let $N_T$ be a strong fuzzy negation and T be a left continuous t-norm. Then, the following statements are equivalent:

(i)

$I_{N_1,T,S,N_2,N_3}$ satisfies (22).

(ii)

$I_{N_1,T,S,N_2,N_3}$ satisfies (24).

(iii)

$S(N_2\left(x\right),N_3\left(x\right))\le N_T\left(x\right)$, for any $x\in [0,1]$.

Proof. 

By virtue of Theorem 6 and Corollary 5, the proof is obvious. □

4. Results

In this paper, we have introduced a hyper class of the known class of QL${}^{\prime }$-implications. This hyper class has been named GQL${}^{\prime }$-implications. Similar to QL${}^{\prime }$-implications, they are not always fuzzy implications, since they sometimes violate (12). Therefore, we use the term GQL${}^{\prime }$-operations instead of implications. The sufficient and necessary conditions under a GQL${}^{\prime }$-operation that is a fuzzy implication have not been discovered, and the same open problem holds for QL-, QL${}^{\prime }$-, D- and D${}^{\prime }$-operations, respectively.

Moreover, it has been proved that the class of GQL${}^{\prime }$-operations is a hyper class of that of QL${}^{\prime }$-operations. A necessary but not sufficient condition for a GQL${}^{\prime }$-operation to be a fuzzy implication has been proved (see Proposition 4 and Remark 10). A relation between GQL${}^{\prime }$-, QL-, D- and D${}^{\prime }$-operations has been presented and proved (see Theorem 5). In Theorem 3, quintuples that do not generate GQL${}^{\prime }$-implications have been excluded. In Theorem 4, the relation of $\Phi$-conjugation in GQL${}^{\prime }$-operations was studied.

In the following subsections, the basic properties (20)–(22) have been studied extensively. The sufficient and necessary condition under a GQL${}^{\prime }$-operation satisfies (20), which has been presented and proved (see Proposition 5). Moreover, a sufficient condition under a GQL${}^{\prime }$-implication does not satisfy (21), which has been presented and proved. A study for the satisfaction of (22) has also been made, and some more general results for the satisfaction of (22) and (24) have been presented. The equivalence of (22) and (24) in GQL${}^{\prime }$-operations has also been proved.

Lastly, we have to remark a result that was not the aim of this study, but it came out. In [4], Theorem 4 has been proved that a D${}^{\prime }$-operation is the reciprocal of a QL-operation when they are generated from a strong fuzzy negation. In Theorem 5, Equation (26) expands this result by substituting strong fuzzy negations with strict fuzzy negations. Thus, a D${}^{\prime }$-operation is the reciprocal of a QL-operation when they are generated from a strict fuzzy negation, as it is described in Equation (26).

5. Discussion

There is a great repertoire of fuzzy implication functions that is also uncountable. Do we need more? Fuzzy logic is mostly generated from classical logic, but it is not classical logic. Although in classical logic everything is unique and uniquely determined, this uniqueness fortunately does not hold in fuzzy logic. Moreover, we have not stopped with the definition of a fuzzy implication via the five axioms (12)–(16); we expand to many other properties that restrict the number of fuzzy implications in order to be useful to specific applications. Therefore, it is ensured that we need more fuzzy implications.

What is the benefit of this study? We believe that the main benefit of this hyper class is the variety of the generated fuzzy implications. We obtain this variety from the possible usage of up to three different fuzzy negations in the generation formula. Moreover, we can generate many fuzzy implications which have the same fuzzy negation by stabilizing $N_1$ in their formula.

6. Conclusions

In this paper, a hyper class of fuzzy implications, the so-called GQL${}^{\prime }$-implications, was introduced and studied. The main idea was to use not only the same fuzzy negation in their formula, which is a fact that increases the variety of fuzzy implications we can obtain. Unfortunately, they are not always fuzzy implications. It remains an open problem whether a GQL${}^{\prime }$-operation is a fuzzy implication. Moreover, some properties of fuzzy implications, such as (20)–(22) and (24) for GQL${}^{\prime }$-operations, have been studied. Surprisingly, a new connection of QL- and D${}^{\prime }$-operations has also been discovered. This connection was discovered accidentally and is another reason for the necessity of this study. Lastly, this study completes the study of some basic properties of QL${}^{\prime }$-operations.