# A Method of Generating Fuzzy Implications with Specific Properties

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**.**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

**Definition**

**3**

**.**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 following conditions

**Definition**

**4**

- (i)
- Idempotent, if$T(x,x)=x$, for all $x\in [0,1]$,(respectively $S(x,x)=x$, for all $x\in [0,1]$),
- (ii)
- Positive, if$T(x,y)=0\iff x=0$or$y=0$,(respectively $S(x,y)=1\iff x=1$ or $y=1$).

**Definition**

**5**

**.**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 $\varphi \in \mathsf{\Phi}$ such that $g={f}_{\varphi}$, where

**Remark**

**1**

**.**It is easy to prove that if $\varphi \in \mathsf{\Phi}$ and T is a t-norm, S is a t-conorm, and N is a fuzzy negation, then ${T}_{\varphi}$ is a t-norm, ${S}_{\varphi}$ is a t-conorm, and ${N}_{\varphi}$ is a fuzzy negation.

**Definition**

**6**

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

**Definition**

**7**

- (i)
- The left neutrality property, if$$I(1,y)=y,y\in [0,1];$$
- (ii)
- The identity principle, if$$I(x,x)=1,x\in [0,1];$$
- (iii)
- The exchange principle, if$$I(x,I(y,z))=I(y,I(x,z)),x,y,z\in [0,1];$$
- (iv)
- The ordering property, if$$I(x,y)=1\iff x\le y,x,y\in [0,1].$$

**Remark**

**2**

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

**Definition**

**8**

**.**Let I be a fuzzy implication and N be a fuzzy negation. I is said to satisfy the

- (i)
- Law of contraposition with respect to N, if$$I(x,y)=I(N(y),N(x)),x,y\in [0,1];$$
- (ii)
- Law of left contraposition with respect to N, if$$I(N(x),y)=I(N(y),x),x,y\in [0,1];$$
- (iii)
- Law of right contraposition with respect to N, if$$I(x,N(y))=I(y,N(x)),x,y\in [0,1].$$

**Lemma**

**1**

**.**If a function $I:{[0,1]}^{2}\to [0,1]$ satisfies (9), (11) and (13), then the function ${N}_{I}:[0,1]\to [0,1]$ is a fuzzy negation, where

**Definition**

**9**

**.**Let $I:{[0,1]}^{2}\to [0,1]$ be a fuzzy implication. The function ${N}_{I}$ defined by Lemma 1 is called the natural negation of I.

**Definition**

**10**

**.**Let N be a fuzzy negation and I be a fuzzy implication. A function ${I}_{N}:{[0,1]}^{2}\to [0,1]$ defined by

## 3. The Main Results

#### 3.1. Fuzzy Implications Generated by Known Fuzzy Implications

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Example**

**1.**

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

#### 3.2. Fuzzy Implications Generated by Fuzzy Connectives and Fuzzy Implications

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Example**

**3.**

**Corollary 5.**(i) If $\varphi \in \mathsf{\Phi}$ and ${I}_{T,{I}_{(1)},{I}_{(2)}}$ is a fuzzy implication, then ${({I}_{T,{I}_{(1)},{I}_{(2)}})}_{\varphi}$ is a fuzzy implication, and moreover,

**Proof.**

**Proposition**

**9.**

**Proof.**

#### 3.3. Fuzzy Connectives’ Classes of Fuzzy Implications

**Corollary**

**6.**

**Corollary**

**7.**

- (i)
- (ii)
- (iii)
- (iv)

**Corollary**

**8.**

- (i)
- (ii)
- (iii)
- (iv)

**Example**

**4.**

## 4. Conclusions

- If we want a fuzzy implication that satisfies (15), its construction is completed by two fuzzy implications ${I}_{(1)}$, ${I}_{(2)}$ that satisfy (15) and any t-norm or t-conorm. On the other hand, if we want to construct a fuzzy implication that violates (15), we can construct it by two ways. The first way is to consider ${I}_{T,{I}_{(1)},{I}_{(2)}}$, where T is any t-norm and at least one of ${I}_{(1)}$, ${I}_{(2)}$ violate (15), according to Proposition 4. The second way is to consider ${I}_{T,{I}_{(1)},{I}_{(2)}}$, where S is any positive t-conorm and the choice of the fuzzy implications ${I}_{(1)}$, ${I}_{(2)}$ is made, such that there exists at least one $x\in (0,1)$, such that$$\begin{array}{c}\hfill {I}_{(1)}(x,x)\ne 1\text{}\mathrm{and}\text{}{I}_{(2)}(x,x)\ne 1,\end{array}$$
- If we want a fuzzy implication that satisfies (17), its construction is achieved similar to the previous ones, if we consider two fuzzy implications ${I}_{(1)}$, ${I}_{(2)}$ that satisfy (17) and any t-norm or any positive t- conorm. On the other hand, the construction of a fuzzy implication that does not satisfy (17) is achieved by two ways. The first way is to consider ${I}_{T,{I}_{(1)},{I}_{(2)}}$, where T is any t-norm and at least one of ${I}_{(1)}$, ${I}_{(2)}$ violate (17), according to Proposition 4. The second way is to consider ${I}_{S,{I}_{(1)},{I}_{(2)}}$, where S is any positive t- conorm and the choice of the fuzzy implications ${I}_{(1)},{I}_{(2)}$ will be done, such that there exist at least one $(x,y)\in {[0,1]}^{2}$, such$$\begin{array}{c}\hfill x\le y\iff {I}_{(1)}(x,y)\ne 1\text{}\mathrm{and}\text{}{I}_{(2)}(x,y)\ne 1,\end{array}$$
- If we want a fuzzy implication that satisfies (14) its construction is given in Proposition 8. On the other hand, the construction of a fuzzy implication that does not satisfy (14) is given by the same Proposition. This construction is achieved by using two implications ${I}_{(i)},i=1,2$ that satisfy (14) and any t-norm or t-conorm, except ${T}_{M}$ and ${S}_{M}$.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Grammatikopoulos, D.S.; Papadopoulos, B.K.
A Method of Generating Fuzzy Implications with Specific Properties. *Symmetry* **2020**, *12*, 155.
https://doi.org/10.3390/sym12010155

**AMA Style**

Grammatikopoulos DS, Papadopoulos BK.
A Method of Generating Fuzzy Implications with Specific Properties. *Symmetry*. 2020; 12(1):155.
https://doi.org/10.3390/sym12010155

**Chicago/Turabian Style**

Grammatikopoulos, Dimitrios S., and Basil K. Papadopoulos.
2020. "A Method of Generating Fuzzy Implications with Specific Properties" *Symmetry* 12, no. 1: 155.
https://doi.org/10.3390/sym12010155