A Study of (T, N)– and (N^′, T, N)– Implications

Abstract

In this paper, we revisit and study the basic properties of two families of fuzzy implications, the so-called $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications. More specific, we study when these fuzzy implications satisfy, or not, the neutrality property $\left(\mathrm{N}\mathrm{P}\right)$, the exchange principle $\left(\mathrm{E}\mathrm{P}\right)$, the identity principle $\left(\mathrm{I}\mathrm{P}\right)$ and the ordering property $\left(\mathrm{O}\mathrm{P}\right)$. Moreover, a study is presented for the law of importation $\left(\mathrm{L}\mathrm{I}\right)$ with respect to a t- norm. Also, we study the relation of $\mathrm{\Phi }-$conjugation in $(N^{\mathrm{\prime }},T,N)-$ implications.

Keywords:

• $(T,N)-$ Implication $(N^{\mathrm{\prime }},T,N)-$ Implication
• Exchange Principle
• Ordering Property
• Law of Importation

1. Introduction

The generalization of the notion of implication from classical to fuzzy logic is a known process with its difficulties [1,2]. In this paper, we will study the properties of such generalizations, the so-called $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications. These implications are the generalization of the following classical tautology (1) $(p\Rightarrow q)\equiv (p\wedge q^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$(1) Firstly, $(T,N)-$ implications mentioned by many authors, such as Dubois and Prade [3] in 1984, Fodor [4,5] in 1991, Fodor and Roubens [2] in 1994 and Klement et al. [6] in 2000. These authors addressed their correlation with another family, the so called $(S,N)-$ implications, when we use a strong negation N. Bedregal [7] in 2007 defined them for any t-norm and any fuzzy negation. Baczyński and Jayaram [1] in 2008 related them with R-implications, when the t-norm T is left continuous and N is a strong negation. Pradera et al. [8] in 2016 mentioned the formula of $(T,N)-$ implications, using aggregation functions in general. Despite these references, they remained anonymous and unstudied until 2017.

In 2017, Pinheiro et al. [9] in their homonymous paper named them $(T,N)-$ implications. They studied them in 2017 [9] and 2018 [10] for strong and not, negations. Some more results on functional equations and $(T,N)-$ implications are presented by Pinheiro et al. in 2018 [11]. In 2020 [12], the intersection between $(S,N)-$ and $(T,N)-$ implications presented and moreover their close relation in a way that they simulate the same (or a similar) way of classical thinking. Additionally, since a fuzzy negation is not a uniquely determined function in 2018 Pinheiro at al. [13] studied $(N^{\mathrm{\prime }},T,N)-$ implications, which are $(T,N)-$ implications generated from not necessary the same negation.

The aforementioned recent interest was the motivation of our study. In this paper, we revisit $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications and we study whether or not they satisfy basic properties of fuzzy implications and the law of importation with respect to a t-norm [1].

2. Preliminaries

Definition 2.1

[1,2,6,14]

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

1. strict, if it is continuous and strictly decreasing,

2. strong, if it is an involution, i.e. $N\left(N\right(x\left)\right)=x,\text{for all}x\in [0,1],$

3. and non-filling, if $N\left(x\right)=1\iff x=0.$

Remark 2.1

The so-called crisp fuzzy negations ([15, Remark 2.1]) are (2) $\begin{array}{rl}{N}^{\alpha }\left(x\right)& =\left\{\begin{array}{ll}0,& \text{if}x\ge \alpha \\ 1,& \text{if}x<\alpha \end{array}\right.,\text{where}\alpha \in (0,1]\text{and}\end{array}$(2) (3) $\begin{array}{rl}{N}_{\alpha }\left(x\right)& =\left\{\begin{array}{ll}0,& \text{if}x>\alpha \\ 1,& \text{if}x\le \alpha \end{array}\right.,\text{ where }\alpha \in [0,1).\end{array}$(3)

Lemma 2.1

[1, Lemma 1.4.9].

If $N^{\mathrm{\prime }}$, N are fuzzy negations such that $N^{\mathrm{\prime }}\circ N=I{d}_{[0,1]}$, then

(i)	$N^{\mathrm{\prime }}$ is a continuous fuzzy negation,
(ii)	N is a strictly decreasing fuzzy negation.

Definition 2.2

[1,6,14]

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 (T1) $T(x,y)=T(y,x),$(T1) (T2) $T(x,T(y,z\left)\right)=T\left(T\right(x,y),z),$(T2) (T3) $\text{if}y\le z,\text{then}T(x,y)\le T(x,z)\text{, i.e., }T(x,\cdot )\text{is increasing},$(T3) (T4) $T(x,1)=x.$(T4) 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 (T1), (T2), (T3), which are denoted by (S1), (S2), (S3) and additionally (S4) $S(x,0)=x.$(S4)

Definition 2.3

[1,6]

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

Definition 2.4

[1,6]

A t-norm T is called continuous if it is continuous in both the arguments.

Definition 2.5

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

Definition 2.6

[1,16]

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 $\varphi \in \mathrm{\Phi }$ such that $g=f_{\varphi }$, where $f_{\varphi }(x_1,x_2,\cdots ,x_n)={\varphi }^{-1}\left(f\right(\varphi \left(x_1\right),\varphi \left(x_2\right),\cdots ,\varphi \left(x_n\right)\left)\right),x_1,x_2,\cdots ,x_n\in [0,1].$

Remark 2.2

[1, Propositions 1.4.8, Remarks 2.1.4(vii) and 2.2.5(vii)].

It is easy to prove that if $\varphi \in \mathrm{\Phi }$ and T is a t-norm, S is a t-conorm and N is a fuzzy negation (respectively strict, strong), then $T_{\varphi }$ is a t-norm, $S_{\varphi }$ is a t-conorm and $N_{\varphi }$ is a fuzzy negation (respectively strict, strong).

Definition 2.7

[1,2]

A function $I:[0,1{]}^2\to [0,1]$ is called a fuzzy implication if (I1) $I\text{is decreasing with respect to the first variable,}$(I1) (I2) $I\text{is increasing with respect to the second variable,}$(I2) (I3) $I(0,0)=1,$(I3) (I4) $I(1,1)=1,$(I4) (I5) $I(1,0)=0.$(I5)

Definition 2.8

[1,10]

A fuzzy implication I is said to satisfy

1. the left neutrality property, if (NP) $I(1,y)=y,y\in [0,1],$(NP)

2. the exchange principle, if (EP) $I(x,I(y,z\left)\right)=I(y,I(x,z\left)\right),x,y,z\in [0,1].$(EP)

3. the identity principle, if (IP) $I(x,x)=1,x\in [0,1],$(IP)

4. the ordering property, if (OP) $I(x,y)=1\iff x\le y,x,y\in [0,1].$(OP)

5. the left ordering property, if (LOP) $x\le y\Rightarrow I(x,y)=1,x,y\in [0,1].$(LOP)

6. the right ordering property, if (ROP) $x>y\Rightarrow I(x,y)\ne 1,x,y\in [0,1].$(ROP)

Remark 2.3

[1, Proposition 1.1.8].

It is proved that, if $\varphi \in \mathrm{\Phi }$ and $I:[0,1{]}^2\to [0,1]$ is a fuzzy implication, then $I_{\varphi }$ is also a fuzzy implication.

Definition 2.9

[1, page 223].

A fuzzy implication I is said to satisfy the law of importation with respect to a t-norm T, if (LI) $I\left(T\right(x,y),z)=I(x,I(y,z\left)\right),x,y,z\in [0,1].$(LI)

Remark 2.4

[1, Remark 7.3.1].

It can be immediately seen that if a fuzzy implication I satisfies (LI) with respect to any t-norm T, by the commutativity of the t-norm T, we have that I satisfies (EP).

Definition 2.10

[1, Lemma 1.4.14 and Definition 1.4.15].

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

Definition 2.11

[1,6]

A function $I:[0,1{]}^2\to [0,1]$ is called an $(S,N)-$ implication if there exist a t-conorm S and a fuzzy negation N such that (4) $I(x,y)=S\left(N\right(x),y),x,y\in [0,1].$(4) Moreover, if I is an $(S,N)-$ implication generated from S and N, then we will often denote it by $I_{S,N}$.

Proposition 2.2

[1, Definition 2.4.3(i)]

If $I_{S,N}$ is an $(S,N)-$ implication, then it satisfies (NP) and (EP).

3. $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ Implications

In this section we investigate the properties of $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications.

Definition 3.1

[10, Proposition 3.1 and Definition 3.1].

A function $I:[0,1{]}^2\to [0,1]$ is called a $(T,N)-$ implication if there exist a t-norm T and a fuzzy negation N such that (5) $I(x,y)=N\left(T\right(x,N\left(y\right)),x,y\in [0,1].$(5) Moreover, if I is a $(T,N)-$ implication generated from T and N, then we will often denote it by $I_T^N$.

Definition 3.2

[13, Proposition 3.1 and Definition 3.2].

A function $I:[0,1{]}^2\to [0,1]$ is called an $(N^{\mathrm{\prime }},T,N)-$ implication if there exist a t-norm T and two fuzzy negations $N^{\mathrm{\prime }}$, N such that (6) $I(x,y)=N^{\mathrm{\prime }}\left(T\right(x,N\left(y\right)),x,y\in [0,1].$(6) Moreover, if I is an $(N^{\mathrm{\prime }},T,N)-$ implication generated from T, $N^{\mathrm{\prime }}$ and N, then we will often denote it by $I_{T,N}^{N^{\mathrm{\prime }}}$.

Definition 3.3

[6, page 232].

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

The above Definition 3.3 addresses that a $(T,N)-$ implication generated from a strong negation N is always an $(S,N)-$ implication and more specific it is $I_T^N=I_{S_{T,N},N}$. Thus, the properties of $(T,N)-$ implications generated from strong negations N are the same with them of $(S,N)-$ implications, as they are studied in [1, Section 2.4]. So, a $(T,N)-$ implication generated from a strong negation N always satisfies (NP) and (EP) according to Proposition 2.2. We get the same results for the $(N^{\mathrm{\prime }},T,N)-$ implication $I_{T,(N^{\mathrm{\prime }}{)}^{-1}}^{N^{\mathrm{\prime }}}$ (where $N^{\mathrm{\prime }}$ is strict) according to the following Proposition 3.1.

Proposition 3.1

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication generated from t-norm T and two strict fuzzy negations $N^{\mathrm{\prime }}$, N, such that $(N^{\mathrm{\prime }}{)}^{-1}=N$. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ is an $(S,N)-$ implication. Moreover, it is $I_{T,N}^{N^{\mathrm{\prime }}}=I_{S_{T,(N^{\mathrm{\prime }}{)}^{-1}},N^{\mathrm{\prime }}}$.

Proof.

Since $N^{\mathrm{\prime }}$ is a strict fuzzy negation, then it is a bijection. So there exists the fuzzy negation $(N^{\mathrm{\prime }}{)}^{-1}$. Moreover, if $(N^{\mathrm{\prime }}{)}^{-1}=N$ then for all $x,y\in [0,1]$ it is $\begin{array}{rl}& I_{T,N}^{N^{\mathrm{\prime }}}(x,y)\stackrel{(N^{\mathrm{\prime }}{)}^{-1}=N}{=}{I}_{T,(N^{\mathrm{\prime }}{)}^{-1}}^{N^{\mathrm{\prime }}}(x,y)\\ & =N^{\mathrm{\prime }}\left(T\right(x,\left(N^{\mathrm{\prime }}{)}^{-1}\right(y\left)\right))\\ & =\left(\right(N^{\mathrm{\prime }}{)}^{-1}{)}^{-1}\left(T\right(\left(N^{\mathrm{\prime }}{)}^{-1}\right(N^{\mathrm{\prime }}\left(x\right)),(N^{\mathrm{\prime }}{)}^{-1}\left(y\right)\left)\right)\\ & =S_{T,(N^{\mathrm{\prime }}{)}^{-1}}\left(N^{\mathrm{\prime }}\right(x),y)\\ & =I_{S_{T,(N^{\mathrm{\prime }}{)}^{-1}},N^{\mathrm{\prime }}}(x,y).\end{array}$ Thus $I_{T,N}^{N^{\mathrm{\prime }}}$ is an $(S,N)-$ implication. Moreover, it is $I_{T,N}^{N^{\mathrm{\prime }}}=I_{T,(N^{\mathrm{\prime }}{)}^{-1}}^{N^{\mathrm{\prime }}}=I_{S_{T,(N^{\mathrm{\prime }}{)}^{-1}},N^{\mathrm{\prime }}}.$

Proposition 3.2

[13, Proposition 3.4(i)]

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. Then $N_{I_{T,N}^{N^{\mathrm{\prime }}}}=N^{\mathrm{\prime }}$.

Remark 3.1

Note that if $N^{\mathrm{\prime }}=N$ the corresponding $(N^{\mathrm{\prime }},T,N)-$ implication is a $(T,N)-$ implication. More specific $I_{T,N}^{N^{\mathrm{\prime }}}=I_T^N$ and $N_{I_T^N}=N$.

3.1. $(T,N)-$, $(N^{\mathrm{\prime }},T,N)-$ Implications and the Neutrality Property (NP)

In this section, we investigate whether or not, $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications satisfy (NP).

Proposition 3.3

Let $I_T^N$ be a $(T,N)-$ implication.

(i)	If N is strong, then $I_T^N$ satisfies (NP).
(ii)	If N is not strong, then $I_T^N$ violates (NP).

Proof.

(i) Since N is a strong negation then $I_T^N=I_{S_{T,N},N}$. Thus $I_T^N$ is an $(S,N)-$ implication and it satisfies (NP) according to Proposition 2.2.

(ii) It is proved in Proposition 3.3(i) in [10].

The above Proposition 3.3 is very important, because it fully characterizes the intersection between the sets of $(S,N)-$ and $(T,N)-$ implications. This characterization is addressed in [12] Figure 1.

Figure 1. Intersection between $(S,N)$- and $(T,N)$-implications.

Proposition 3.4

[13, Proposition 3.4(i)]

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (NP), if and only if $N^{\mathrm{\prime }}\circ N=I{d}_{[0,1]}$.

The above Proposition 3.4 is very important. According to the Lemma 2.1 if $N^{\mathrm{\prime }}\circ N=I{d}_{[0,1]}$, then

1. $N^{\mathrm{\prime }}$ is a continuous fuzzy negation,

2. N is a strictly decreasing fuzzy negation.

Therefore, the following Corollaries are presented without proofs, since they are obvious.

Corollary 3.5

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is not a continuous fuzzy negation. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (NP).

Corollary 3.6

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where N is not strictly decreasing. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (NP).

Corollary 3.7

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where at least one of $N^{\mathrm{\prime }}$ and N is a crisp fuzzy negation. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (NP).

Proof.

If N is a crisp fuzzy negation, then it is not strictly decreasing and according to the Corollary 3.6, the corresponding $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (NP).

If $N^{\mathrm{\prime }}$ is a crisp fuzzy negation, then it is not continuous and according to the Corollary 3.5, the corresponding $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (NP).

Remark 3.2

1. Corollaries 3.5 and 3.6 are very important. Note that, they prove that if N is not a strictly decreasing fuzzy negation, or $N^{\mathrm{\prime }}$ is not a continuous fuzzy negation, then the corresponding $I_{T,N}^{N^{\mathrm{\prime }}}$ is not an $(S,N)-$ implication according to Proposition 2.2.

2. Corollary 3.7 is also very important. Note that, it proves that if at least one of $N^{\mathrm{\prime }}$ and N is a crisp fuzzy negation, then the corresponding $I_{T,N}^{N^{\mathrm{\prime }}}$ is not an $(S,N)-$ implication according to Proposition 2.2.

3. Proposition 3.3 can be deduced by Proposition 3.4.

3.2. $(T,N)-$, $(N^{\mathrm{\prime }},T,N)-$ Implications and the Exchange Principle (EP)

In this section, we investigate whether or not, $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications satisfy (EP).

Proposition 3.8

Let $I_T^N$ be a $(T,N)-$ implication.

(i)	If N is strong, then $I_T^N$ satisfies (EP).
(ii)	If N is a crisp fuzzy negation, then $I_T^N$ satisfies (EP).
(iii)	If N is strict, but not strong, then $I_T^N$ violates (EP).

Proof.

(i) Since N is a strong negation then $I_T^N=I_{S_{T,N},N}$. Thus $I_T^N$ is an $(S,N)-$ implication and it satisfies (EP) according to Proposition 2.2.

(ii) It is proved in Theorem 3.2(i) in [10].

(iii) It is proved in Proposition 3.3(ii) in [10].

Proposition 3.9

[13, Proposition 3.5]

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. If $N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$, then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP).

Proposition 3.10

[13, Proposition 3.9(i)]

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ and N are crisp fuzzy negations. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP).

Proposition 3.11

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. If $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP), then $N^{\mathrm{\prime }}\circ N\circ N^{\mathrm{\prime }}=N^{\mathrm{\prime }}$.

Proof.

Since $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP), then for all $x\in [0,1]$ it is $\begin{array}{rl}& I_{T,N}^{N^{\mathrm{\prime }}}(1,I_{T,N}^{N^{\mathrm{\prime }}}(x,0\left)\right)=I_{T,N}^{N^{\mathrm{\prime }}}(x,I_{T,N}^{N^{\mathrm{\prime }}}(1,0\left)\right)\\ & \stackrel{\left(I5\right)}{\Rightarrow }{I}_{T,N}^{N^{\mathrm{\prime }}}(1,N^{\mathrm{\prime }}(T(x,N(0\left)\right)\left)\right)=I_{T,N}^{N^{\mathrm{\prime }}}(x,0)\\ & \Rightarrow I_{T,N}^{N^{\mathrm{\prime }}}(1,N^{\mathrm{\prime }}(T(x,1)\left)\right)=N^{\mathrm{\prime }}\left(T\right(x,N\left(0\right)\left)\right)\\ & \Rightarrow I_{T,N}^{N^{\mathrm{\prime }}}(1,N^{\mathrm{\prime }}(T(x,1)\left)\right)=N^{\mathrm{\prime }}\left(T\right(x,1\left)\right)\\ & \stackrel{\left(T4\right)}{\Rightarrow }{I}_{T,N}^{N^{\mathrm{\prime }}}(1,N^{\mathrm{\prime }}(x\left)\right)=N^{\mathrm{\prime }}\left(x\right)\\ & \Rightarrow N^{\mathrm{\prime }}\left(T\right(1,N\left(N^{\mathrm{\prime }}\right(x\left)\right))=N^{\mathrm{\prime }}(x)\\ & \stackrel{\left(T4\right)}{\Rightarrow }{N}^{\mathrm{\prime }}\left(N\right(N^{\mathrm{\prime }}\left(x\right)\left)\right)=N^{\mathrm{\prime }}\left(x\right).\text{■}\end{array}$

Corollary 3.12

Let $I_T^N$ be a $(T,N)-$ implication generated from a non- strong negation N. If $I_T^N$ satisfies (EP), then $N\left(N\right(N\left(x\right)\left)\right)=N\left(x\right)$, for all $x\in [0,1]$.

Proof.

It is deduced by Proposition 3.11, for $N^{\mathrm{\prime }}=N$.

Proposition 3.13

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is strictly decreasing with a fixed point. If N does not have any fixed point, or $N^{\mathrm{\prime }}$ and N have different fixed points, then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (EP).

Proof.

We assume that $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP). Therefore, from Proposition 3.11, we deduce that for all $x\in [0,1]$ it is $(N^{\mathrm{\prime }}\circ N\circ N^{\mathrm{\prime }})\left(x\right)=N^{\mathrm{\prime }}\left(x\right)$. Since $N^{\mathrm{\prime }}$ has a fixed point (this is unique since $N^{\mathrm{\prime }}$ is strictly decreasing, therefore it is an injection), there is an $e\in (0,1)$ such that $N^{\mathrm{\prime }}\left(e\right)=e$. So, $\begin{array}{rl}(N^{\mathrm{\prime }}\circ N\circ N^{\mathrm{\prime }})\left(e\right)=N^{\mathrm{\prime }}\left(e\right)& \Rightarrow N^{\mathrm{\prime }}\left(N\right(N^{\mathrm{\prime }}\left(e\right)\left)\right)=N^{\mathrm{\prime }}\left(e\right)\\ & \stackrel{N^{\mathrm{\prime }}\left(e\right)=e}{\Rightarrow }{N}^{\mathrm{\prime }}\left(N\right(e\left)\right)=N^{\mathrm{\prime }}\left(e\right)\\ & \stackrel{N^{\mathrm{\prime }}}{\underset{injection}{\Rightarrow }}N\left(e\right)=e.\end{array}$ So, $N^{\mathrm{\prime }}$ and N have the same fixed point. That is a contradiction, therefore, $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (EP).

Remark 3.3

1. Proposition 3.11 gives us the sufficient condition, that if there is an $x_0\in (0,1)$ such that $N^{\mathrm{\prime }}\left(N\right(N^{\mathrm{\prime }}\left(x_0\right)\left)\right)\ne N^{\mathrm{\prime }}\left(x_0\right)$, then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (EP). This result in the case of $(T,N)-$ implications, where $N^{\mathrm{\prime }}=N$ is transformed to the sufficient condition, that if there is an $x_0\in (0,1)$ such that $N\left(N\right(N\left(x_0\right)\left)\right)\ne N\left(x_0\right)$, then $I_T^N$ violates (EP).

2. Note that, if there is an $x_0\in (0,1)$ such that $N\left(N\right(N\left(x_0\right)\left)\right)\ne N\left(x_0\right)$, then N is not a strong fuzzy negation.

3. According to [1] Theorem 1.4.7, every continuous fuzzy negation N has a unique fixed point. Therefore, Proposition 3.13 holds if $N^{\mathrm{\prime }}$ is strict, N is continuous and they have different fixed points.

Proposition 3.14

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is a strictly decreasing fuzzy negation. The following statements are equivalent:

(i)	$I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP).
(ii)	$N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$.

Proof.

(i)⇒(ii)

Since $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP), then from Proposition 3.11, we deduce that for all $x\in [0,1]$ it is $N^{\mathrm{\prime }}\left(N\right(N^{\mathrm{\prime }}\left(x\right)\left)\right)=N^{\mathrm{\prime }}\left(x\right)$. Moreover, $N^{\mathrm{\prime }}$ is an injection, since it is a strictly decreasing function. So, for any $x\in [0,1]$ it is $N^{\mathrm{\prime }}\left(N\right(N^{\mathrm{\prime }}\left(x\right)\left)\right)=N^{\mathrm{\prime }}\left(x\right)\Rightarrow N\left(N^{\mathrm{\prime }}\right(x\left)\right)=x\Rightarrow N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}.$ (ii)⇒(i)

It is deduced by Proposition 3.9.

According to the Lemma 2.1, if $N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$, then N is a continuous fuzzy negation. Therefore, by Proposition 3.14, we deduce the following Corollary without proof, since is is obvious.

Corollary 3.15

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where N is not a continuous fuzzy negation and $N^{\mathrm{\prime }}$ is a strictly decreasing fuzzy negation. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (EP).

A very helpful Theorem is the following.

Theorem 3.16

[1, Theorem 2.4.10]

For a function $I:[0,1{]}^2\to [0,1]$ the following statements are equivalent:

(i)	I is an $(S,N)-$ implication with a continuous fuzzy negation N.
(ii)	I satisfies (I1), (EP) and $N_I$ is a continuous fuzzy negation.

The above Theorem 3.16 helps to the proof of the following Propositions.

Proposition 3.17

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is a continuous fuzzy negation and N is not a strictly decreasing fuzzy negation. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (EP).

Proof.

We assume that $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP). By Proposition 3.2 it is deduced that $N^{\mathrm{\prime }}=N_{I_{T,N}^{N^{\mathrm{\prime }}}}$. Also, $N^{\mathrm{\prime }}$ is a continuous fuzzy negation, by the hypothesis. Moreover, since $I_{T,N}^{N^{\mathrm{\prime }}}$ is a fuzzy implication, it satisfies (I1). So, $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies the statement (ii) in Theorem 3.16. By virtue of Theorem 3.16, we deduce that $I_{T,N}^{N^{\mathrm{\prime }}}$ is an $(S,N)-$ implication. Thus, $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (NP), according to Proposition 2.2. So, by Proposition 3.4 it must be $N^{\mathrm{\prime }}\circ N=I{d}_{[0,1]}$. This means that N is strictly decreasing according to Lemma 2.1. That is a contradiction, since N is not strictly decreasing. Thus, $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (EP).

Proposition 3.18

Let $I_T^N$ be a $(T,N)-$ implication generated from a continuous non- strong negation N. Then $I_T^N$ violates (EP).

Proof.

We assume that $I_T^N$ satisfies (EP). By Remark 3.1, it is deduced that $N=N_{I_T^N}$. Also, N is a continuous fuzzy negation, by the hypothesis. Moreover, since $I_T^N$ is a fuzzy implication, it satisfies (I1). So, $I_T^N$ satisfies the statement (ii) in Theorem 3.16. By virtue of Theorem 3.16, we deduce that $I_T^N$ is an $(S,N)-$ implication. Thus, $I_T^N$ satisfies (NP), according to Proposition 2.2. That is a contradiction according to Proposition 3.3(ii), since N is not strong. Thus, $I_T^N$ violates (EP).

Corollary 3.19

Let $I_T^N$ be a $(T,N)-$ implication generated from a strictly decreasing non- strong negation N. Then $I_T^N$ violates (EP).

Proof.

Since N is a non- strong negation, there is an $x_0\in (0,1)$, such that $N\left(N\right(x_0\left)\right)\ne x_0$. We assume that $I_T^N$ satisfies (EP), than by virtue of Proposition 3.14 it is $N\circ N=I{d}_{[0,1]}$, i.e. N is a strong negation. That is a contradiction. Thus, $I_T^N$ violates (EP).

Remark 3.4

Proposition 3.8(iii) can be deduced by Proposition 3.18 or Corollary 3.19.

3.3. $(T,N)-$, $(N^{\mathrm{\prime }},T,N)-$ Implications and the Identity Principle (IP)

In this section, we investigate whether $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications satisfy (IP).

Proposition 3.20

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. Moreover, let the pair $(T,N)$ satisfies (LC). Then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (IP).

Proof.

For all $x\in [0,1]$ it is $I_{T,N}^{N^{\mathrm{\prime }}}(x,x)=N^{\mathrm{\prime }}\left(T\right(x,N\left(x\right)\left)\right)\stackrel{\left(T1\right)}{=}{N}^{\mathrm{\prime }}\left(T\right(N\left(x\right),x\left)\right)\stackrel{\left(\mathrm{L}\mathrm{C}\right)}{=}{N}^{\mathrm{\prime }}\left(0\right)=1.\text{■}$

Corollary 3.21

If $I_T^N$ is a $(T,N)-$ implication and the pair $(T,N)$ satisfies (LC), then $I_T^N$ satisfies (IP).

Proof.

Note that it is Proposition 3.20, where $N^{\mathrm{\prime }}=N$.

Proposition 3.22

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is a non- filling fuzzy negation. Then the following statements are equivalent:

(i)	The pair $(T,N)$ satisfies (LC)).
(ii)	$I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (IP).

Proof.

(i) ⇒ (ii)

It is deduced by Proposition 3.20.

(ii)⇒ (i)

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (IP). Then for all $x\in [0,1]$ it is $\begin{array}{rl}{I}_{T,N}^{N^{\mathrm{\prime }}}(x,x)=1& \Rightarrow N^{\mathrm{\prime }}\left(T\right(x,N\left(x\right)\left)\right)=1\\ & \stackrel{N^{\mathrm{\prime }}}{\underset{non-filling}{\Rightarrow }}T(x,N(x\left)\right)=0\\ & \stackrel{\left(T1\right)}{\Rightarrow }T\left(N\right(x),x)=0.\end{array}$ Thus the pair $(T,N)$ satisfies (LC).

Corollary 3.23

Let $I_T^N$ be a $(T,N)-$ implication generated from a non- filling fuzzy negation N. Then the following statements are equivalent:

(i)	The pair $(T,N)$ satisfies (LC).
(ii)	$I_T^N$ satisfies (IP).

Proof.

Note that it is Proposition 3.22, where $N^{\mathrm{\prime }}=N$.

In the case, we use a crisp fuzzy negation we get the following $(N^{\mathrm{\prime }},T,N)-$ implications. (7) $\begin{array}{rl}{I}_{T,N}^{N^{\alpha }}(x,y)& =\left\{\begin{array}{ll}0,& \text{if}T(x,N(y\left)\right)\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.,\end{array}$(7) (8) $\begin{array}{rl}{I}_{T,N}^{N_{\alpha }}(x,y)& =\left\{\begin{array}{ll}0,& \text{if}T(x,N(y\left)\right)>\alpha \\ 1,& \text{otherwise}\end{array}\right.,\end{array}$(8) (9) $\begin{array}{rl}{I}_{T,N^{\alpha }}^N(x,y)& =\left\{\begin{array}{ll}1,& \text{if}y\ge \alpha \\ N\left(x\right),& \text{otherwise }\end{array}\right.,\end{array}$(9) and (10) $I_{T,N_{\alpha }}^N(x,y)=\left\{\begin{array}{ll}1,& \text{if}y>\alpha \\ N\left(x\right),& \text{otherwise }\end{array}\right..$(10)

Proposition 3.24

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is a crisp fuzzy negation. If the pair $(T,N)$ satisfies (LC), then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (IP).

Proof.

Let $N^{\mathrm{\prime }}=N^{\alpha }$. For all $x\in [0,1]$, it is $I_{T,N}^{N^{\alpha }}(x,y)\stackrel{\left(7\right)}{=}\left\{\begin{array}{ll}0,& \text{if}T(x,N(y\left)\right)\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.,\alpha \in (0,1].$ Moreover, the pair $(T,N)$ satisfies (LC). So, for all $x\in [0,1]$, it is $T(x,N(x\left)\right)=0<\alpha \in (0,1]\Rightarrow I_{T,N}^{N^{\alpha }}(x,x)=1.$ Thus, $I_{T,N}^{N^{\alpha }}$ satisfies (IP).

Let $N^{\mathrm{\prime }}=N_{\alpha }$. For all $x\in [0,1]$, it is $I_{T,N}^{N_{\alpha }}(x,y)\stackrel{\left(8\right)}{=}\left\{\begin{array}{ll}0,& \text{if}T(x,N(y\left)\right)>\alpha \\ 1,& \text{otherwise }\end{array}\right.,\alpha \in [0,1).$ Moreover, the pair $(T,N)$ satisfies (LC). So, for all $x\in [0,1]$, it is $T(x,N(x\left)\right)=0\le \alpha \in [0,1)\Rightarrow I_{T,N}^{N_{\alpha }}(x,x)=1.$ Thus, $I_{T,N}^{N_{\alpha }}$ satisfies (IP).

Remark 3.5

(i) It is easy to prove that $I_{T,N_{\text{β}}}^{N_{\alpha }}$, $I_{T,N^{\text{β}}}^{N_{\alpha }}$ and $I_{T,N^{\text{β}}}^{N^{\alpha }}$ satisfy (IP), if and only if $\alpha \ge \text{β}$, since (11) $I_{T,N_{\text{β}}}^{N_{\alpha }}(x,y)\stackrel{\left(10\right)}{=}\left\{\begin{array}{ll}1,& \text{if}y>\text{β}\\ N_{\alpha }\left(x\right),& \text{otherwise }\end{array}\right.=\left\{\begin{array}{ll}0,& \text{if}y\le \text{β}\text{and}x>\alpha \\ 1,& \text{otherwise }\end{array}\right.,$(11) where $\alpha ,\text{β}\in [0,1)$, (12) $I_{T,N^{\text{β}}}^{N_{\alpha }}(x,y)\stackrel{\left(9\right)}{=}\left\{\begin{array}{ll}1,& \text{if}y\ge \text{β}\\ N_{\alpha }\left(x\right),& \text{otherwise }\end{array}\right.=\left\{\begin{array}{ll}0,& \text{if}y<\text{β}\text{and}x>\alpha \\ 1,& \text{otherwise }\end{array}\right.,$(12) where $\alpha \in [0,1)$ and $\text{β}\in (0,1]$, (13) $I_{T,N^{\text{β}}}^{N^{\alpha }}(x,y)\stackrel{\left(9\right)}{=}\left\{\begin{array}{ll}1,& \text{if}y\ge \text{β}\\ N^{\alpha }\left(x\right),& \text{otherwise }\end{array}\right.=\left\{\begin{array}{ll}0,& \text{if}y<\text{β}\text{and}x\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.,$(13) where $\alpha ,\text{β}\in (0,1]$.

(ii) From (11(I2) $I\text{is increasing with respect to the second variable,}$(I2) ) and (13(I4) $I(1,1)=1,$(I4) ), for $\alpha =\text{β}$ we deduce that a $(T,N)-$ implication generated from a crisp fuzzy negation satisfies (IP).

(iii) It is easy to prove that $I_{T,N_{\text{β}}}^{N^{\alpha }}$ satisfies (IP), if and only if $\alpha >\text{β}$, since (14) $I_{T,N_{\text{β}}}^{N^{\alpha }}(x,y)\stackrel{\left(10\right)}{=}\left\{\begin{array}{ll}1,& \text{if}y>\text{β}\\ N^{\alpha }\left(x\right),& \text{otherwise }\end{array}\right.=\left\{\begin{array}{ll}0,& \text{if}y\le \text{β}\text{and}x\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.,$(14) where $\alpha \in (0,1]$ and $\text{β}\in [0,1)$.

3.4. $(T,N)-$, $(N^{\mathrm{\prime }},T,N)-$ Implications and the Ordering Property (OP)

In this section, we investigate whether or not, $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications satisfy (OP). Sometimes (OP) is divided in two sub- properties, the so-called (LOP) and (ROP). It is easy to observe that, if a fuzzy implication I satisfies both (LOP) and (ROP), it satisfies (OP).

Proposition 3.25

[10, Theorem 3.2 and Remark 3.2]

Let $I_T^N$ be a $(T,N)-$ implication generated from a crisp fuzzy negation. Then $I_T^N$ satisfies (LOP) and violates (ROP) and (OP).

Proposition 3.26

[13, Proposition 3.9(iii),(v)]

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$, N are crisp fuzzy negations.

(i)	$I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (LOP) if and only if $N\le N^{\mathrm{\prime }}$.
(ii)	$I_{T,N}^{N^{\mathrm{\prime }}}$ does not satisfy (ROP).

A direct conclusion from Proposition 3.26 is the following.

Corollary 3.27

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$, N are crisp fuzzy negations. Then $I_{T,N}^{N^{\mathrm{\prime }}}$ violates (OP).

Proposition 3.28

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. Moreover, let the pair $(T,N)$ satisfies (LC). Then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (LOP).

Proof.

For all $x,y\in [0,1]$, such that $x\le y$ it is $N\left(x\right)\ge N\left(y\right)$, since N is decreasing. So, $\begin{array}{rl}x\le y& \Rightarrow N\left(x\right)\ge N\left(y\right)\\ & \stackrel{\left(T3\right)}{\Rightarrow }T(x,N(x\left)\right)\ge T(x,N(y\left)\right)\\ & \stackrel{\left(T1\right)}{\Rightarrow }T\left(N\right(x),x)\ge T(x,N(y\left)\right)\\ & \stackrel{\left(\mathrm{L}\mathrm{C}\right)}{\Rightarrow }0\ge T(x,N(y\left)\right)\\ & \stackrel{\text{(Definition~{2})}}{\Rightarrow }T(x,N(y\left)\right)=0\\ & \Rightarrow N^{\mathrm{\prime }}\left(T\right(x,N\left(y\right)\left)\right)=N^{\mathrm{\prime }}\left(0\right)\\ & \Rightarrow I_{T,N}^{N^{\mathrm{\prime }}}(x,y)=1.\end{array}$ Thus, $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (LOP).

Proposition 3.29

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication generated from strictly decreasing fuzzy negations $N^{\mathrm{\prime }}$, N and a strictly monotone t- norm T such that, the pair $(T,N)$ satisfies (LC). Then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (ROP).

Proof.

For all $x,y\in [0,1]$, such that x>y it is x>0 and $N\left(x\right)<N\left(y\right)$, since N is strictly decreasing. So, $x>y\Rightarrow N\left(x\right)<N\left(y\right)\Rightarrow T(x,N(x\left)\right)<T(x,N(y\left)\right),$ since T is strictly monotone. By virtue of (T1) we have $\begin{array}{rl}T\left(N\right(x),x)<T(x,N(y\left)\right)& \stackrel{\left(\mathrm{L}\mathrm{C}\right)}{\Rightarrow }0<T(x,N(y\left)\right)\\ & \Rightarrow N^{\mathrm{\prime }}\left(0\right)>N^{\mathrm{\prime }}\left(T\right(x,N\left(y\right)\left)\right)\\ & \Rightarrow 1>I_{T,N}^{N^{\mathrm{\prime }}}(x,y)\\ & \Rightarrow I_{T,N}^{N^{\mathrm{\prime }}}(x,y)\ne 1,\end{array}$ since $N^{\mathrm{\prime }}$ is strictly decreasing. Thus, $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (ROP).

Corollary 3.30

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication generated from strictly decreasing fuzzy negations $N^{\mathrm{\prime }}$, N and a strictly monotone t-norm T such that, the pair $(T,N)$ satisfies (LC). Then $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (OP).

Proof.

It is deduced by virtue of Propositions 3.28 and 3.29.

Corollary 3.31

Let $I_T^N$ be a $(T,N)-$ implication.

(i)	If the pair $(T,N)$ satisfies (LC), then $I_T^N$ satisfies (LOP).
(ii)	If N is strictly decreasing fuzzy negation and the t-norm T is strictly monotone such that, the pair $(T,N)$ satisfies (LC), then $I_T^N$ satisfies (ROP) and (OP).

Proof.

(i) Note that it is Proposition 3.28 for $N^{\mathrm{\prime }}=N$.

(ii) Note that it is an application of Proposition 3.29 and Corollary 3.30 for $N^{\mathrm{\prime }}=N$.

3.5. $(T,N)-$, $(N^{\mathrm{\prime }},T,N)-$ Implications and the Law of Importation (LI)

In this section, we study whether or not, $(T,N)-$ and $(N^{\mathrm{\prime }},T,N)-$ implications satisfy the law of importation (LI) with respect to any, or which, t-norm $T^{\mathrm{\prime }}$.

Proposition 3.32

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. If $N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$, then the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and T satisfies (LI). Moreover, this couple of functions is unique, i.e. for any other t-norm $T^{\mathrm{\prime }}$ the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ violates (LI).

Proof.

For all $x,y,z\in [0,1]$, it is $\begin{array}{rl}{I}_{T,N}^{N^{\mathrm{\prime }}}\left(T\right(x,y),z)& =N^{\mathrm{\prime }}\left(T\right(T(x,y),N\left(z\right)\left)\right)\\ & \stackrel{\left(T2\right)}{=}{N}^{\mathrm{\prime }}\left(T\right(x,T(y,N(z\left)\right)\left)\right)\\ & \stackrel{N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}}{=}{N}^{\mathrm{\prime }}\left(T\right(x,N\left(N^{\mathrm{\prime }}\right(T(y,N(z\left)\right)\left)\right)\left)\right)\\ & =I_{T,N}^{N^{\mathrm{\prime }}}(x,I_{T,N}^{N^{\mathrm{\prime }}}(y,z\left)\right).\end{array}$ Thus, the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and T satisfies (LI).

We assume that, there is an other t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI). Then, for all $x,y,z\in [0,1]$, it is $I_{T,N}^{N^{\mathrm{\prime }}}\left(T\right(x,y),z)=I_{T,N}^{N^{\mathrm{\prime }}}(x,I_{T,N}^{N^{\mathrm{\prime }}}(y,z\left)\right)=I_{T,N}^{N^{\mathrm{\prime }}}\left(T^{\mathrm{\prime }}\right(x,y),z).$ So, $\begin{array}{rl}& N^{\mathrm{\prime }}\left(T\right(T(x,y),N\left(z\right)\left)\right)=N^{\mathrm{\prime }}\left(T\right(T^{\mathrm{\prime }}(x,y),N\left(z\right)\left)\right)\\ & \Rightarrow N\left(N^{\mathrm{\prime }}\right(T\left(T\right(x,y),N(z\left)\right)\left)\right)=N\left(N^{\mathrm{\prime }}\right(T\left(T^{\mathrm{\prime }}\right(x,y),N(z\left)\right)\left)\right)\\ & \stackrel{N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}}{\Rightarrow }T\left(T\right(x,y),N(z\left)\right)=T\left(T^{\mathrm{\prime }}\right(x,y),N(z\left)\right).\end{array}$ Putting z = 0 we have $\begin{array}{rl}T\left(T\right(x,y),N(0\left)\right)=T\left(T^{\mathrm{\prime }}\right(x,y),N(0\left)\right)& \Rightarrow T\left(T\right(x,y),1)=T\left(T^{\mathrm{\prime }}\right(x,y),1)\\ & \stackrel{\left(T4\right)}{\Rightarrow }T(x,y)=T^{\mathrm{\prime }}(x,y),\end{array}$ that is a contradiction, since $T\ne T^{\mathrm{\prime }}$.

Corollary 3.33

Let $I_T^N$ be a $(T,N)-$ implication generated from a t-norm T and a strong negation N. The couple of functions $I_T^N$ and T satisfies (LI). Moreover, this couple of functions is unique, i.e. for any other t-norm $T^{\mathrm{\prime }}$ the couple of functions $I_T^N$ and $T^{\mathrm{\prime }}$ violates (LI).

Proof.

Note that it is Proposition 3.32 since N is strong, therefore $N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$.

Corollary 3.34

Let $I_{T,(N^{\mathrm{\prime }}{)}^{-1}}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is a strict fuzzy negation. Then the couple of functions $I_{T,(N^{\mathrm{\prime }}{)}^{-1}}^{N^{\mathrm{\prime }}}$ and T satisfies (LI). Moreover, this couple of functions is unique.

Proof.

The proof is deduced by Proposition 3.32 since $(N^{\mathrm{\prime }}{)}^{-1}\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$.

Proposition 3.35

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication. Let there is an $x_0\in (0,1)$ such that $N^{\mathrm{\prime }}\left(N\right(N^{\mathrm{\prime }}\left(x_0\right)\left)\right)\ne N^{\mathrm{\prime }}\left(x_0\right)$. Then there is not any t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI).

Proof.

Consider that there is a t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI). Then, by Remark 2.4 follows that $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP), a contradiction by virtue of Remark 3.3(i).

Corollary 3.36

Let $I_T^N$ be a $(T,N)-$ implication generated from a non- strong negation N. Moreover, there is an $x_0\in (0,1)$ such that $N\left(N\right(N\left(x_0\right)\left)\right)\ne N\left(x_0\right)$. Then there is not any t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_T^N$ and $T^{\mathrm{\prime }}$ satisfies (LI).

Proof.

Note that $N^{\mathrm{\prime }}=N$ and apply Proposition 3.35.

Proposition 3.37

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is strictly decreasing with a fixed point. If N does not have any fixed point, or $N^{\mathrm{\prime }}$ and N have different fixed points, then there is not any t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI).

Proof.

Consider that there is a t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI). Then, by Remark 2.4 follows that $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP), a contradiction by virtue of Proposition 3.13.

Proposition 3.38

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where $N^{\mathrm{\prime }}$ is a strictly decreasing fuzzy negation. The following statements are equivalent:

(i)	$I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP).
(ii)	The couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and T satisfies (LI) and this couple of functions is unique.
(iii)	$N\circ N^{\mathrm{\prime }}=I{d}_{[0,1]}$.

Proof.

(i)⇔(iii)

See Proposition 3.14.

(ii)⇒(i)

See Remark 2.4.

(iii)⇒(ii)

See Proposition 3.32.

Proposition 3.39

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, where N is not a continuous fuzzy negation and $N^{\mathrm{\prime }}$ is a strictly decreasing fuzzy negation. Then there is not any t- norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI).

Proof.

Consider that there is a t- norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_{T,N}^{N^{\mathrm{\prime }}}$ and $T^{\mathrm{\prime }}$ satisfies (LI). Then, by Remark 2.4 follows that $I_{T,N}^{N^{\mathrm{\prime }}}$ satisfies (EP), a contradiction by virtue of Corollary 3.15.

Proposition 3.40

Let $I_T^N$ be a $(T,N)-$ implication generated from a continuous non- strong negation N. Then there is not any t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_T^N$ and $T^{\mathrm{\prime }}$ satisfies (LI).

Proof.

Consider that there is a t- norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_T^N$ and $T^{\mathrm{\prime }}$ satisfies (LI). Then, by Remark 2.4 follows that $I_T^N$ satisfies (EP), a contradiction by virtue of Proposition 3.18.

Proposition 3.41

Let $I_T^N$ be a $(T,N)-$ implication generated from a strictly decreasing non- strong negation N. Then there is not any t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_T^N$ and $T^{\mathrm{\prime }}$ satisfies (LI).

Proof.

Consider that there is a t-norm $T^{\mathrm{\prime }}$, such that the couple of functions $I_T^N$ and $T^{\mathrm{\prime }}$ satisfies (LI). Then, by Remark 2.4 follows that $I_T^N$ satisfies (EP), a contradiction by virtue of Corollary 3.19.

Proposition 3.42

Let $I_{T,N}^{N^{\alpha }}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, generated from a t-norm T and two crisp fuzzy negations $N^{\alpha }$ and N. The couple of functions $I_{T,N}^{N^{\alpha }}$ and $T^{\mathrm{\prime }}$ satisfies (LI) if and only if $min\{x,y\}\ge \alpha \Rightarrow T^{\mathrm{\prime }}(x,y)\ge \alpha .$

Proof.

Let $N=N^{\text{β}}$. It is $I_{T,N^{\text{β}}}^{N^{\alpha }}(x,y)\stackrel{\left(13\right)}{=}\left\{\begin{array}{ll}0,& \text{if}y<\text{β}\text{and}x\ge \alpha \\ 1,& \text{otherwise }\end{array}\right..$ Let $T^{\mathrm{\prime }}$ be any t-norm. For all $x,y,z\in [0,1]$, it is $I_{T,N^{\text{β}}}^{N^{\alpha }}\left(T^{\mathrm{\prime }}\right(x,y),z)=\left\{\begin{array}{ll}0,& \text{if}z<\text{β}\text{and}{T}^{\mathrm{\prime }}(x,y)\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.$ and $\begin{array}{rl}& I_{T,N^{\text{β}}}^{N^{\alpha }}(x,I_{T,N^{\text{β}}}^{N^{\alpha }}(y,z\left)\right)=\left\{\begin{array}{ll}{I}_{T,N^{\text{β}}}^{N^{\alpha }}(x,0),& \text{if}z<\text{β}\text{and}y\ge \alpha \\ I_{T,N^{\text{β}}}^{N^{\alpha }}(x,1),& \text{otherwise }\end{array}\right.\\ & \stackrel{\text{β}\in (0,1]}{=}\left\{\begin{array}{ll}0,& \text{if}z<\text{β}\text{and}x\ge \alpha \text{and}y\ge \alpha \\ 1,& \text{otherwise }\end{array}\right..\end{array}$ It is easy to prove that for any t-norm $T^{\mathrm{\prime }}$ and $x,y\in [0,1]$, it is $T^{\mathrm{\prime }}(x,y)\le x$ and $T^{\mathrm{\prime }}(x,y)\le y$ (see [17, Proposition 9]). So we deduce that (15) $T^{\mathrm{\prime }}(x,y)\ge \alpha \Rightarrow x\ge \alpha \text{and}y\ge \alpha \Rightarrow min\{x,y\}\ge \alpha .$(15) “⇒”

We assume that the couple of functions $I_{T,N^{\text{β}}}^{N^{\alpha }}$ and $T^{\mathrm{\prime }}$ satisfies (LI). It must be for any $x,y,z\in [0,1]$, $\begin{array}{rl}& I_{T,N^{\text{β}}}^{N^{\alpha }}\left(T^{\mathrm{\prime }}\right(x,y),z)=I_{T,N^{\text{β}}}^{N^{\alpha }}(x,I_{T,N^{\text{β}}}^{N^{\alpha }}(y,z\left)\right)\\ & \iff \left\{\begin{array}{ll}0,& \text{if}z<\text{β}\text{and}{T}^{\mathrm{\prime }}(x,y)\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.=\left\{\begin{array}{ll}0,& \text{if}z<\text{β}\text{and}x\ge \alpha \text{and}y\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.\\ & \Rightarrow \left(T^{\mathrm{\prime }}\right(x,y)\ge \alpha \iff x\ge \alpha \text{and}y\ge \alpha )\\ & \Rightarrow \left(T^{\mathrm{\prime }}\right(x,y)\ge \alpha \iff min\{x,y\}\ge \alpha )\\ & \stackrel{\left(15\right)}{\Rightarrow }\left(min\right\{x,y\}\ge \alpha \Rightarrow T^{\mathrm{\prime }}(x,y)\ge \alpha ).\end{array}$ “⇐”

We assume that $(min\{x,y\}\ge \alpha \Rightarrow T^{\mathrm{\prime }}(x,y)\ge \alpha )\Rightarrow (x\ge \alpha \text{and}y\ge \alpha \Rightarrow T^{\mathrm{\prime }}(x,y)\ge \alpha )$ this means, that $T^{\mathrm{\prime }}(x,y)\ge \alpha \iff x\ge \alpha \text{and}y\ge \alpha .$ By the above equivalence, we conclude that $\begin{array}{rl}{I}_{T,N^{\text{β}}}^{N^{\alpha }}\left(T^{\mathrm{\prime }}\right(x,y),z)& =\left\{\begin{array}{ll}0,& \text{if}z<\text{β}\text{and}{T}^{\mathrm{\prime }}(x,y)\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.\\ & =\left\{\begin{array}{ll}0,& \text{if}z<\text{β}\text{and}x\ge \alpha \text{and}y\ge \alpha \\ 1,& \text{otherwise }\end{array}\right.\\ & =I_{T,N^{\text{β}}}^{N^{\alpha }}(x,I_{T,N^{\text{β}}}^{N^{\alpha }}(y,z\left)\right).\end{array}$ Thus, the couple of functions $I_{T,N^{\text{β}}}^{N^{\alpha }}$ and $T^{\mathrm{\prime }}$ satisfies (LI). If we assume that $N=N_{\text{β}}$. the proof is similar. So it is omitted.

Proposition 3.43

Let $I_{T,N}^{N_{\alpha }}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, generated from a t-norm T and two crisp fuzzy negations $N_{\alpha }$ and N. The couple of functions $I_{T,N}^{N_{\alpha }}$ and $T^{\mathrm{\prime }}$ satisfies (LI) if and only if $min\{x,y\}>\alpha \Rightarrow T^{\mathrm{\prime }}(x,y)>\alpha .$

Proof.

The proof is similar with the proof of Proposition 3.42. So it is omitted.

Remark 3.6

1. It is obvious that Proposition 3.42 (respectively Proposition 3.43) holds for $(T,N)-$ implications generated from a t-norm T and a crisp fuzzy negation $N^{\alpha }$ (respectively, $N_{\alpha }$).

2. From Propositions 3.42 and 3.43 we deduce that an $(N^{\mathrm{\prime }},T,N)-$ implication (respectively, a $(T,N)-$ implication) generated from crisp fuzzy negations (respectively negation) always satisfies (LI) with respect to the minimum t- norm $T_M(x,y)=min\{x,y\}$.

3. From Proposition 3.42 we deduce that $I_{T,N}^{N^1}$, where N is a crisp fuzzy negation satisfies (LI) with respect to any t-norm $T^{\mathrm{\prime }}$.

4. From Proposition 3.43 we deduce that $I_{T,N}^{N_0}$, where N is a crisp fuzzy negation satisfies (LI) with respect to any strictly monotone t-norm $T^{\mathrm{\prime }}$.

5. Proposition 3.10 can be deduced by Remarks 3.6(ii) and 2.4.

3.6. $(N^{\mathrm{\prime }},T,N)-$ Implications and Φ- Conjugacy Classes

Theorem 3.44

If $\varphi \in \mathrm{\Phi }$ and $I_{T,N}^{N^{\mathrm{\prime }}}$ is an $(N^{\mathrm{\prime }},T,N)-$ implication, then $(I_{T,N}^{N^{\mathrm{\prime }}}{)}_{\varphi }$ is an $(N^{\mathrm{\prime }},T,N)-$ implication and moreover $(I_{T,N}^{N^{\mathrm{\prime }}}{)}_{\varphi }=I_{T_{\varphi },N_{\varphi }}^{N_{\varphi }^{\mathrm{\prime }}}.$

Proof.

Let $I_{T,N}^{N^{\mathrm{\prime }}}$ be an $(N^{\mathrm{\prime }},T,N)-$ implication, then $(I_{T,N}^{N^{\mathrm{\prime }}}{)}_{\varphi }$ is an $(N^{\mathrm{\prime }},T,N)-$ implication according to the Remark 2.3. Moreover, for all $x,y\in [0,1]$, we deduce that $\begin{array}{rl}\left(I_{T,N}^{N^{\mathrm{\prime }}}{)}_{\varphi }\right(x,y)& ={\varphi }^{-1}\left(I_{T,N}^{N^{\mathrm{\prime }}}\right(\varphi \left(x\right),\varphi \left(y\right)\left)\right)\\ & ={\varphi }^{-1}\left(N^{\mathrm{\prime }}\right(T\left(\varphi \right(x),N(\varphi \left(y\right)\left)\right)\left)\right)\\ & ={\varphi }^{-1}\left(N^{\mathrm{\prime }}\right(\varphi \left({\varphi }^{-1}\right(T\left(\varphi \right(x),\varphi ({\varphi }^{-1}\left(N\right(\varphi \left(y\right)\left)\right)\left)\right)\left)\right)\left)\right)\\ & =N_{\varphi }^{\mathrm{\prime }}\left(T_{\varphi }\right(x,N_{\varphi }\left(y\right))\\ & =I_{T_{\varphi },N_{\varphi }}^{N_{\varphi }^{\mathrm{\prime }}}(x,y).\text{■}\end{array}$

4. Final Remarks

In this study, we dealt with $(T,N)-$ implications and $(N^{\mathrm{\prime }},T,N)-$ implications, which are a generalization of $(T,N)-$ implications. Firstly, we connected a form of $(N^{\mathrm{\prime }},T,N)-$ implications with $(S,N)-$ implications and we investigated when an $(N^{\mathrm{\prime }},T,N)-$ implication satisfies, or not, (NP). In the following, the conditions under an $(N^{\mathrm{\prime }},T,N)-$ implication (respectively a $(T,N)-$ implication) satisfies or violates (EP) have been studied. A study for the satisfaction of (IP) and the satisfaction or violation of (OP) has also been made. Furthermore, except the basic properties of fuzzy implications we expanded our study to the law of importation (LI) with respect to a t-norm $T^{\mathrm{\prime }}$. Our study focused, not only to the satisfaction of violation of (LI) with respect to a t-norm $T^{\mathrm{\prime }}$, but also to the uniqueness, or not of the t-norm $T^{\mathrm{\prime }}$. Moreover, the sufficient and necessary conditions under an $(N^{\mathrm{\prime }},T,N)-$ implication (respectively a $(T,N)-$ implication), generated from crisp fuzzy negations (respectively a crisp fuzzy negation), satisfies (LI) with respect to a t-norm $T^{\mathrm{\prime }}$ were presented and proved. Also, the relation of $\mathrm{\Phi }-$conjugation in $(N^{\mathrm{\prime }},T,N)-$ implications was studied.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

No data were used to support this study.

Additional information

Notes on contributors

Dimitrios S. Grammatikopoulos

Dimitrios S. Grammatikopoulos is with the Department of Civil Engineering, School of Engineering at Democritus University of Thrace, Greece. He received his Bachelor degree in Mathematics in 2005 and his MSc in Statistics and Operational Research in 2008, both from the Department of Mathematics at Aristotle University of Thessaloniki, Greece. He received his second MSc in Applied Mathematics in 2018, from the School of Engineering at Democritus University of Thrace, Greece. His research interests include fuzzy theory and systems with applications. He is currently working towards his PhD degree.

Basil K. Papadopoulos

Basil K. Papadopoulos is currently a Professor at the Department of Civil Engineering, School of Engineering at Democritus University of Thrace, Greece. He is also the Director of the MSc in Applied Mathematics at the School of Engineering at Democritus University of Thrace, Greece. For more information, please visit his webpage: http://utopia.duth.gr/papadob/.