# Three-Way Fuzzy Sets and Their Applications (III)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Let U be a non-empty set, L, M, N be three lattices. The whole composed $\langle f,g,h\rangle $ of three partial mappings, which are defined as follows,

**Definition**

**2**

**.**Suppose that $(L,\vee ,\wedge ,0,1)$ is a bounded lattice, where the order relation on L is ≤. The binary relation ${\le}_{t}$ on ${D}_{*}\left(L\right)$ is defined as follows: $\forall x,y\in {D}_{*}\left(L\right)$,

**Remark**

**1.**

**Theorem**

**1**

**.**Suppose that $(L,\vee ,\wedge ,0,1)$ is a bounded lattice, where the order relation on L is ≤. Then, $({D}_{*}\left(L\right),{\le}_{t})$ is a bounded partially ordered set, where ${0}_{t}=(0,0,1)$, ${1}_{t}=(1,1,0)$ are the minimal element and maximal element, respectively. Furthermore, $({D}_{*}\left(L\right),{\vee}_{t},{\wedge}_{t},{0}_{t},{1}_{t})$ is a bounded lattice, $\forall x,y\in {D}_{*}\left(L\right)$,

**Definition**

**3**

**.**Suppose U is a universe, $TFS1\left(U\right)$ is the ordinary three-way fuzzy sets on U, $A\in TFS1\left(U\right)$. The complement of A (denote as ${A}^{ct}$) is defined as follows: $\forall x\in U$,

**Definition**

**4**

**.**Let $(L,{\le}_{L})$ be a complete lattice. A t-norm on $(L,{\le}_{L})$ is a commutative, associative, increasing mapping $T:{L}^{2}\to L$, which satisfies $T({1}_{L},u)=u$, for all $u\in L$.

**Definition**

**5**

**.**A three-way fuzzy implication operator is a mapping "→": $D\left(\right[0,1\left]\right)\times D\left(\right[0,1\left]\right)\to D\left(\right[0,1\left]\right)$ that satisfies the following conditions:

**Example**

**1**

**.**The definitions of common three-way fuzzy implication operators are as follows: $\forall x,y\in D\left(\right[0,1\left]\right)$,

**Remark**

**2.**

## 3. Three-Way t-Norm and Its Residual Implication

**Proof.**

**Theorem**

**2.**

**Proof.**

**Definition**

**6.**

**Definition**

**7.**

- (1)
- ${R}_{0}$ adjoint pairs $({\otimes}_{{r}_{0}},{\to}_{{r}_{0}})$$$x{\to}_{{r}_{0}}y=\left(\right)open="\{"\; close>\begin{array}{cc}{1}_{t},\hfill & if\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x{\le}_{t}y;\\ {x}^{ct}{\vee}_{t}y,& otherwise.\end{array}$$$$x{\otimes}_{{r}_{0}}y=\left(\right)open="\{"\; close>\begin{array}{cc}{0}_{t},\hfill & if\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x{\le}_{t}{y}^{ct};\\ x{\wedge}_{t}y,& otherwise.\end{array}$$
- (2)
- G$\ddot{o}$del adjoint pairs $({\otimes}_{g},{\to}_{g})$$$x{\to}_{g}y=\left(\right)open="\{"\; close>\begin{array}{cc}{1}_{t},& if\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}x{\le}_{t}y;\\ y,& otherwise.\end{array}$$$$x{\otimes}_{g}y=x{\wedge}_{t}y$$

**Remark**

**3.**

**Theorem**

**3.**

**Proof.**

## 4. The Triple I Algorithm of Three-Way Fuzzy Inference

#### 4.1. The Triple I Algorithm Based on Three-Way Residual Implication

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Remark**

**4.**

#### 4.2. The Triple I Algorithm Based on Three-Way Fuzzy Implication Operator

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Remark**

**5.**

#### 4.3. Three-Way Fuzzy Inference Based on Triple I Algorithm

**Example**

**2.**

- (1)
- ${B}^{*}$ is calculated using the TCRI method.$$\begin{array}{cc}\hfill {B}^{*}& ={A}^{*}\circ \left(A{\to}_{z}B\right)\hfill \\ \hfill & ={A}^{*}\circ R\hfill \\ \hfill & =\langle (0.2,0.4,0.4),(0.6,0.0,0.4),(0.7,1.0,0.2)\rangle .\hfill \end{array}$$
- (2)
- ${B}^{*}$ is calculated using the triple I algorithm. From ${\left({A}^{*}\right)}^{ct}{<}_{t}R$ and ${R}^{ct}{<}_{t}R$, we have$${B}^{*}=\langle (0.2,0.7,0.8),(0.6,0.5,0.5),(0.7,1.0,0.2)\rangle .$$

**Example**

**3.**

- (1)
- ${B}^{*}$ is calculated using the TCRI method.$$\begin{array}{cc}\hfill {B}^{*}& ={A}^{*}\circ \left(A{\to}_{{r}_{0}}B\right)\hfill \\ \hfill & =\langle (0.2,0.4,0.4),(0.6,0.0,0.4),(0.7,0.4,0.1)\rangle .\hfill \end{array}$$
- (2)
- ${B}^{*}$ is calculated using the triple I algorithm. From ${\left({A}^{*}\right)}^{ct}{<}_{t}R$, we have$${B}^{*}=\langle (0.2,0.7,0.8),(0.6,0.5,0.5),(0.7,0.4,0.1)\rangle .$$

- (1)
- For the same sets of given: A, B and ${A}^{*}$, the results are different because of the different R, which are generated by different three-way fuzzy implication operators.
- (2)
- For the same sets of given: A, B and ${A}^{*}$, and the same three-way fuzzy implication operator, the results are different due to different inference methods, and the triple I algorithm is better.

**Example**

**4.**

**Example**

**5.**

## 5. Application of Three-Way Fuzzy Inference

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The truth-membership functions (

**a**), indeterminacy-membership functions (

**b**) and falsity-membership functions (

**c**) of three-way fuzzy subsets of e, $ec$ and u.

**Figure 4.**Inference results of the triple I algorithm based on Zadeh’s three-way fuzzy implication operator (or ${R}_{0}$ three-way residual implication).

${\mathit{C}}_{\mathbf{i}\mathbf{j}}$ | ${\mathit{B}}_{\mathit{j}}$ | NB | NM | NS | ZE | PS | PM | PB | |
---|---|---|---|---|---|---|---|---|---|

${\mathit{A}}_{\mathit{i}}$ | |||||||||

NB | PB | PB | PM | PM | PS | PS | ZE | ||

NM | PB | PM | PM | PS | PS | ZE | NS | ||

NS | PM | PM | PS | PS | ZE | NS | NS | ||

ZE | PM | PS | PS | ZE | NS | NS | NM | ||

PS | PS | PS | ZE | NS | NS | NM | NM | ||

PM | PS | ZE | NS | NS | NM | NM | NB | ||

PB | ZE | NS | NS | NM | NM | NB | NB |

_{i}, B

_{j}, C

_{ij}(i, j = 1, 2, … , 7) are three-way fuzzy subsets of e, ec, u, respectively.

u | IP | Mamdani | G$\ddot{\mathit{o}}$del | Zadeh | ${\mathit{R}}_{0}$ | |
---|---|---|---|---|---|---|

DF | ||||||

WA | Figure 4A [26] | Figure 5b | Figure 5b | Figure 5b | ||

MOM | Figure 5a | Figure 4a | Figure 4a | Figure 4a |

_{0}, G$\ddot{o}$del three-way residual implications and Mamdani’s and Zadeh’s three-way fuzzy implication operator are considered.

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**MDPI and ACS Style**

Hu, Q.; Zhang, X.
Three-Way Fuzzy Sets and Their Applications (III). *Axioms* **2023**, *12*, 57.
https://doi.org/10.3390/axioms12010057

**AMA Style**

Hu Q, Zhang X.
Three-Way Fuzzy Sets and Their Applications (III). *Axioms*. 2023; 12(1):57.
https://doi.org/10.3390/axioms12010057

**Chicago/Turabian Style**

Hu, Qingqing, and Xiaohong Zhang.
2023. "Three-Way Fuzzy Sets and Their Applications (III)" *Axioms* 12, no. 1: 57.
https://doi.org/10.3390/axioms12010057