# Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

**Definition**

**2.**

- (i)
- Increasing, if, for all $x,y\in X$,$$x\precsim y\Rightarrow f\left(x\right)\le f\left(y\right);$$
- (ii)
- Order-preserving, if f is increasing and, for all $x,y\in X$,$$x\prec y\Rightarrow f\left(x\right)<f\left(y\right).$$

**Definition**

**3.**

**Definition**

**4.**

**Proposition**

**1.**

- (i)
- f is ≾-C-compatible;
- (ii)
- ${s}_{x}^{C}\left(f\right)<{i}_{y}^{C}\left(f\right)$ for every pair $(x,y)\in {E}_{\prec}^{C}$;
- (iii)
- For every pair $(x,y)\in {E}_{\prec}^{C}$, the following implication holds:$${f}^{-1}\left(\right]-\infty ,{s}_{x}^{C}\left(f\right)\left]\right)\cup {f}^{-1}\left(\right[{i}_{y}^{C}\left(f\right),\infty \left[\right)=C\Rightarrow {s}_{x}^{C}\left(f\right)<{i}_{y}^{C}\left(f\right).$$

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**1**

**Definition**

**8.**

## 3. The Lifting Theorems

**Theorem**

**2**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**2.**

**Theorem**

**3.**

- (i)
- ≾ is representable by a continuous multi-utility;
- (ii)
- If ≾ is any closed preorder on $(X,t)$, then the following property is verified:“If C is any closed subset of X, and $f:(C,{\precsim}_{\mid C},{t}_{\mid C})\to (R,\le ,{t}_{nat})$ is any bounded, continuous, order-preserving and ≾-C-compatible function, then ${F}_{\mid C}=f$ for some continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,{t}_{nat})$";
- (iii)
- If ≾ is any closed preorder on $(X,t)$, then the following property is verified:“If C is any compact subset of X, and $f:(C,{\precsim}_{\mid C},{t}_{\mid C})\to (R,\le ,{t}_{nat})$ is any continuous order-preserving function, then ${F}_{\mid C}=f$ for some continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,{t}_{nat})$”.

**Proof.**

**Case****1:**- $C\cap d\left(x\right)=\varnothing $ and $C\cap i\left(y\right)=\varnothing $.
**Case****2:**- $C\cap d\left(x\right)\ne \varnothing $ and $C\cap i\left(y\right)=\varnothing $.
**Case****3:**- $C\cap d\left(x\right)=\varnothing $ and $C\cap i\left(y\right)\ne \varnothing $.
**Case****4:**- $C\cap d\left(x\right)\ne \varnothing $ and $C\cap i\left(y\right)\ne \varnothing $.

**Theorem**

**4.**

- (i)
- $(X,t)$ is σ-compact;
- (ii)
- If ≾ is any closed preorder on $(X,t)$, then the following property is verified:“If C is any compact subset of X, and $f:(C,{\precsim}_{\mid C},{t}_{\mid C})\to (R,\le ,{t}_{nat})$ is any continuous order-preserving function, then ${F}_{\mid C}=f$ for some continuous order-preserving function $F:(X,\precsim ,t)\to (R,\le ,{t}_{nat})$".

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Bosi, G.; Zuanon, M.
Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility. *Axioms* **2023**, *12*, 123.
https://doi.org/10.3390/axioms12020123

**AMA Style**

Bosi G, Zuanon M.
Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility. *Axioms*. 2023; 12(2):123.
https://doi.org/10.3390/axioms12020123

**Chicago/Turabian Style**

Bosi, Gianni, and Magalì Zuanon.
2023. "Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility" *Axioms* 12, no. 2: 123.
https://doi.org/10.3390/axioms12020123