# Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- $A({a}_{1},{a}_{2},\dots ,{a}_{n})\le A({a}_{1}^{\prime},{a}_{2}^{\prime},\dots ,{a}_{n}^{\prime})$ whenever ${a}_{i}\le {a}_{i}^{\prime}$ for $1\le i\le n$.
- (ii)
- $A({0}_{L},{0}_{L},\dots ,{0}_{L})={0}_{L}$ and $A({1}_{L},{1}_{L},\dots ,{1}_{L})={1}_{L}$.

**Definition**

**2.**

- (T1)
- $a\otimes b=b\otimes a$.
- (T2)
- ${a}_{1}\otimes b\le {a}_{2}\otimes b$ if ${a}_{1}\le {a}_{2}$.
- (T3)
- $a\otimes (b\otimes c)=(a\otimes b)\otimes c$.
- (T4)
- $a\otimes {1}_{L}=a$.

**Example**

**1.**

**Example**

**2.**

## 3. Sufficient Conditions for an Aggregation Operator Preserving $\otimes -$Convexity

**Theorem**

**1.**

**Proof.**

**Example**

**3.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Example**

**4.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Example**

**5.**

## 4. Sufficient Conditions for Triangular Norm Preserving $\otimes -$Convexity

**Theorem**

**6.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Example**

**6.**

**Theorem**

**9.**

**Proof.**

**Example**

**7.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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A | ${0}_{\mathit{L}}$ | a | b | ${1}_{\mathit{L}}$ |
---|---|---|---|---|

${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ |

a | ${0}_{L}$ | 0 | b | b |

b | ${0}_{L}$ | a | b | b |

${1}_{L}$ | ${0}_{L}$ | a | b | ${1}_{L}$ |

A | ${0}_{\mathit{L}}$ | a | b | ${1}_{\mathit{L}}$ |
---|---|---|---|---|

${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ |

a | ${0}_{L}$ | a | a | a |

b | ${0}_{L}$ | a | a | b |

${1}_{L}$ | ${0}_{L}$ | a | b | ${1}_{L}$ |

${\mathit{T}}_{\mathit{b}}$ | ${0}_{\mathit{L}}$ | a | b | c | d | ${1}_{\mathit{L}}$ |
---|---|---|---|---|---|---|

${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ |

a | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | a |

b | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | b |

c | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | c | ${0}_{L}$ | c |

d | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | ${0}_{L}$ | d | d |

${1}_{L}$ | ${0}_{L}$ | a | b | c | d | ${1}_{L}$ |

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**MDPI and ACS Style**

Li, L.; Luo, Q.
Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity. *Symmetry* **2018**, *10*, 729.
https://doi.org/10.3390/sym10120729

**AMA Style**

Li L, Luo Q.
Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity. *Symmetry*. 2018; 10(12):729.
https://doi.org/10.3390/sym10120729

**Chicago/Turabian Style**

Li, Lifeng, and Qinjun Luo.
2018. "Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity" *Symmetry* 10, no. 12: 729.
https://doi.org/10.3390/sym10120729