# Continuous Multi-Utility Representations of Preorders and the Chipman Approach

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Chipman Approach

**LR**is based upon the validity of condition

**lSB**.

**SB:**- In order for ${t}_{\mid \sim}^{\precsim}$ to be Hausdorff, it is necessary that the sets $l\left(x\right)$ and $r\left(x\right)$, where x runs through X, constitute a sub-basis of ${t}^{\precsim}$.

**LR:**- In order for ${t}_{\mid \sim}^{\precsim}$ to be Hausdorff, it is necessary that, for all points $x\in X$ and $y\in X$, the validity of the following implication holds for all $z\in l\left(x\right)$ and for all $u\in r\left(x\right)$:$$(y\in r(z\left)\right)\wedge (y\in d(u\left)\right)\Rightarrow x\sim y.$$

**SB**and

**LR**already implies that ${t}_{\mid \sim}^{\precsim}$ is Hausdorff. The general case, however, is difficult. No simple solution can be expected.

#### 1.2. The Continuous Multi-Utility Representation Theorem

## 2. On an Order-Embedding Theorem of Chipman-Type

**Theorem**

**1.**

- (i)
- There exists some cardinal number κ for which there exists an order-embedding $\psi :(X,\precsim )\u27f6({\{0,1\}}^{\kappa},{\le}_{lex})$.
- (ii)
- There exists some cardinal number κ and a (complete) preorder ≲ on ${\{0,1\}}^{\kappa}$, that is coarser than ${\le}_{lex}$, for which there exists a continuous order-embedding $\vartheta :(X,\precsim ,t)\u27f6({\{0,1\}}^{\kappa},\lesssim ,{t}^{\lesssim})$.

**Proof.**

**SB**and the definition of $\psi $ applied to the equations $\underset{i\in I}{lim}\phantom{\rule{0.166667em}{0ex}}{x}_{i}=x=\underset{j\in J}{lim}\phantom{\rule{0.166667em}{0ex}}{y}_{j}$. Since this argument is routine and obvious in nature, it may be omitted for the sake of brevity. The independence of (crucial) gaps from particularly chosen nets that converge to x allows us to now proceed by considering the collection $\mathbb{H}$ of all half-closed half-open and all half-open half-closed gaps of $\psi \left(X\right)$. In accordance with Beardon [1], we define as follows an equivalence relation ∼ on ${\{0,1\}}^{\kappa}$ with respect to $\mathbb{H}$. If $[r,s[$ and $[s,t[$ are adjacent gaps of $X\setminus \bigcup \mathbb{H}$, then $[r,t]$ is an equivalence class of ∼. In addition, if $[u,v[$ and $]w,z]$, respectively, are gaps that do not belong to pairs of adjacent gaps of $X\setminus \bigcup \mathbb{H}$, then $[u,v]$ and $[w,z]$, respectively, define the corresponding equivalence classes of ∼. All the other equivalence classes of ∼ are defined to be singletons. Since the equivalence classes of ∼ are closed intervals of $({\{0,1\}}^{\kappa},{\le}_{lex})$, we may define the desired preorder ≲ on ${\{0,1\}}^{\kappa}$ that is coarser than ${\le}_{lex}$ by setting

**Example**

**1.**

**ORD**be the class of ordinal numbers. As it already has been outlined to some degree in the last paragraph of Section 1.1, we now prove a theorem that underlines the universal character of the class $\{({\{0,1\}}^{\lambda},{\le}_{lex})\mid \lambda \in \mathbf{ORD}\}$ of chains.

**Theorem**

**2.**

- (i)
- There exist some ordinal number λ and order-embeddings$$\psi :(X,\precsim )\u27f6({\{0,1\}}^{\lambda},{\le}_{lex})$$and$$\eta :\u27f6({\{0,1\}}^{\lambda},{\le}_{lex})$$such that the following diagram commutes$$\begin{array}{ccc}({\{0,1\}}^{\lambda},{\le}_{lex})& \stackrel{\eta}{\leftarrow}& (D,\le )\\ \uparrow \psi & \nearrow \varphi & \\ (X,\precsim )& & \end{array}.$$
- (ii)
- In addition to Assertion(i), (total) preorders $\le \subset \u22b4$ on D and ${\le}_{lex}\subset \lesssim $ and order-embeddings$$\nu :\left(\varphi \right(X),\le )\u27f6(D,\u22b4)$$and$$\rho :(\psi \left(X\right),{\le}_{lex})\u27f6({\{0,1\}}^{\lambda},\lesssim )$$can be chosen in such a way that the compositions$$\theta :=\nu \circ \varphi :(X\precsim ,t)\u27f6(D,\u22b4,{t}^{\u22b4}),$$$$\vartheta :=\rho \circ \psi :(X\precsim ,t)\u27f6({\{0,1\}}^{\lambda},\lesssim ,{t}^{\lesssim})$$and$$\chi :=\rho \circ \eta :(D,\u22b4,{t}^{\u22b4})\u27f6({\{0,1\}}^{\lambda},\lesssim ,{t}^{\lesssim})$$are continuous and the following diagram commutes$$\begin{array}{ccc}({\{0,1\}}^{\lambda},\lesssim ,{t}^{\lesssim})& \stackrel{\chi}{\leftarrow}& (D,\u22b4,{t}^{\u22b4})\\ \uparrow \vartheta & \nearrow \theta & \\ (X,\precsim ,t)& & \end{array}.$$

**Proof.**

**LR**implies the existence of an order-embedding $\eta :(D,\le )\u27f6({\{0,1\}}^{\lambda},{\le}_{lex})$ that is defined by identifying $\eta \left(a\right)$, for every $a\in D$, with the tuple $\left(\underset{\lambda -\mathrm{times}}{\underbrace{1,1,\dots ,1,\dots}}\right)\in {\{0,1\}}^{\lambda}$, if $a\notin \varphi \left(X\right)$. Let, therefore, $a\in \varphi \left(X\right)$. Then, we must at first verify that the subsequent definition is independent of the particular chosen point $x\in X$ such that $\varphi \left(x\right)=a$. This means that we must prove that the equation $\varphi \left(x\right)=\varphi \left(y\right)$ implies that $\left[x\right]=\left[y\right]$. Indeed, if $\varphi \left(x\right)=\varphi \left(y\right)$, then we have that $l\left(x\right)=l\left(y\right)$ and $r\left(x\right)=r\left(y\right)$. Hence, it follows that assumptions of condition

**LR**are satisfied, which implies that $\left[x\right]=\left[y\right]$. Let, consequently, some $x\in X$ such that $a=\varphi \left(x\right)$ be arbitrarily chosen. Then, we may identify $\eta \left(a\right)$ with the tuple ${\left({a}_{\alpha}\right)}_{\alpha <\lambda}$ that is defined by setting

## 3. On a Relation of the Chipman Approach with the Continuous Multi-Utility Representation Problem of Preorders

**Lemma**

**1.**

**HD:**- Let ≾ have a continuous multi-utility representation. Then, $({X}_{\mid \sim},{\precsim}_{\mid \sim})$ is a Hausdorff space.
**OC:**- Let $({X}_{\mid \sim},{\precsim}_{\mid \sim})$ be a Hausdorff space and let ≾ be closed. Then, $d\left(y\right)$ is open (and closed) for every point $y\in {\mathbf{N}}_{\precsim}\left(x\right)$ that is maximal with respect to $(X,\precsim )$.

**Proof.**

**HD**:

**OC:**Let $y\in {\mathbf{N}}_{\precsim}\left(x\right)$ be a maximal element of $(X,\precsim )$, which means that $r\left(y\right)=\varnothing $. Then, the assumption according to which ${t}_{\mid \sim}^{\precsim}$ is Hausdorff implies, with help of condition

**SB**, that $l\left(y\right)\ne \varnothing $. Hence, we may distinguish between the cases when $\left(l\right(y),\precsim )$ has a maximal element, and, respectively $\left(l\right(y),\precsim )$ has no maximal element. Let us, therefore, assume at first that $\left(l\right(y),\precsim )$ has a maximal element m. Then, the interval $]m,y[$ is empty. This means, in particular, that there exists no net ${\left(m\left(s\right)\right)}_{s\in S}$ of points $m\left(s\right)\in r\left(m\right)$ that converges to y. Hence, the set $\mathbf{U}\left(y\right):=\{t\in r(m)\mid t\in r(m)\setminus [y\left]\right\}$ must be closed, and we may conclude that $\left[y\right]=r\left(m\right)\setminus \mathbf{U}\left(y\right)$ is open and closed. We, thus, proceed by showing that both sets $l\left(y\right)$ as well as $d\left(y\right)$ are open and closed. In order to verify these properties of $l\left(y\right)$ and $d\left(y\right)$, respectively, it suffices to prove that $l\left(y\right)$ is closed and that $d\left(y\right)$ is open. Let, therefore, in a first step, some point $p\in \overline{l\left(y\right)}$ be arbitrarily chosen. Then, we have to show that $p\in l\left(y\right)$. We, thus, consider some net ${\left({p}_{o}\right)}_{o\in O}$ of points ${p}_{o}\in l\left(y\right)$ that converges to p. Since ≾ is closed and ${p}_{o}\prec y$ for all $o\in O$, it follows that $p\precsim y$, and it remains to verify that the equivalence $p\sim y$ can be excluded. Indeed, if $p\sim y$, then the just proved property that $\left[y\right]$ is open (and closed) implies that there exists some index ${o}_{y}$ such that ${p}_{o}\sim y$ for all points $o\in O$ which are at least as great as ${o}_{y}$. This contradiction implies that $l\left(y\right)$ must be closed. For later use, in particular in the proof of Theorem 3, we abbreviate this conclusion by (*). Since $l\left(y\right)$ is open and $\left[y\right]$ is open, it follows, in a second step, that $d\left(y\right)=l\left(y\right)\cup \left[y\right]$ is open, which completes the discussion of the case $\left(l\right(y),\precsim )$ to have a maximal element. We now still must think of the situation $\left[{\displaystyle \underset{q\in l\left(y\right)}{sup}q}\right]$ to coincide with $\left[y\right]$. Let, in this situation, $(C,\precsim )$ be some sub-chain of $\left(l\right(y),\precsim )$ such that $\left[{\displaystyle \underset{c\in C}{sup}c}\right]=\left[y\right]$ Because of property (*), we may assume without loss of generality that $\left[y\right]$ is not open (and closed). We, thus, may arbitrarily choose some point $c\in C$ in order to then consider some net ${\left({y}_{i}\right)}_{i\in I}$ of points ${y}_{i}\in r\left(c\right)$ which converges to y. Because of the maximality of y with respect to $(X,\precsim )$, it follows that, for every $i\in I$, there exist points ${c}^{\prime}\in C$ and ${c}^{\prime \prime}\in C$ such that $c\precsim {c}^{\prime}\precsim {y}_{i}\precsim {c}^{\prime \prime}$. Indeed, otherwise the definition of ${t}^{\precsim}$ implies that $\left[y\right]$ is the meet of two open intervals and, thus, it is open (and closed), which contradicts our assumption according to which $\left[y\right]$ is not open (and closed). This argument will be abbreviated by (

**M**). But this consideration allows us to conclude that, for every point $l\in l\left(y\right)$, the set $r\left(l\right)\cap d\left(y\right)$ is an open neighborhood of y. Hence, it follows that $d\left(y\right)=l\left(y\right)\cup \left(r\right(l)\cap d(y\left)\right)$ is open (and closed) for every point $l\in l\left(y\right)$, which still was to be shown. □

**Lemma**

**2.**

**SB**and

**LR**.

**Proof.**

**SB**and

**LR**is necessary in order to guarantee that ${t}_{\mid \sim}^{\precsim}$ is Hausdorff. Hence, we may concentrate on the sufficient part of the lemma. In order to verify that the assumption according to which ≾ is semi-closed implies, in combination with validity of the conditions

**SB**and

**LR**, that ${t}_{\mid \sim}^{\precsim}$ is a Hausdorff topology on ${X}_{\mid \sim}$ we notice at first that condition

**SB**is equivalent to condition

**LU**, which states that, for every point $x\in X$, at least one of the sets $l\left(x\right)$ or $r\left(x\right)$ is not empty. Let now points $x\in X$ and $y\in X$ such that $not(x\precsim y)$ be arbitrarily chosen. Then, the cases $y\prec x$ and $not(y\precsim x)$ are possible. Therefore, we have to distinguish between these possible cases.

**Case 1:**$y\prec x$. In this situation we distinguish between two more cases.

**Case 1.1:**There exist points $u\in X$ and $v\in X$ such that the interval $]u,v[$ is empty and $y\precsim u\prec v\precsim x$. In this case, $l\left(v\right)$ is an open set that contains y, and $r\left(u\right)$ is an open set that contains x. Therefore, the equation $l\left(v\right)\cap r\left(u\right)=\varnothing $ settles 1.1.

**Case 1.2:**The closed interval $[y,x]$ does not contain any jump. In this situation there exists some point $z\in X$ such that $y\prec z\prec x$. Hence $l\left(z\right)$ and $r\left(z\right)$, respectively, are disjoint open sets, which contain y and x, respectively.

**Case 2:**$not(y\precsim x)$. In this situation condition

**LU**implies that the lemma will be proven if the cases $l\left(x\right)\ne \varnothing $ or $r\left(x\right)\ne \varnothing $ successfully have been handled. Since both cases can be settled by completely analogous arguments, it suffices to concentrate on the case when $l\left(x\right)$ is not empty. The inequality $l\left(x\right)\ne \varnothing $ implies, with help of condition

**LR**, that there exists some $z\in l\left(x\right)$ such that $y\notin i\left(z\right)$, in case that $r\left(x\right)=\varnothing $, or that there exist points $v\in l\left(x\right)$ and $z\in r\left(x\right)$ such that $y\notin [v,z]$, in case that $r\left(x\right)\ne \varnothing $. Since ≾ is semi-closed it, thus, follows that $r\left(z\right)$ and $X\setminus i\left(z\right)$, respectively, or $]v,z[$ and $X\setminus [v,z]$, respectively, are disjoint open sets which contain the point x and the point y, respectively, which still was to be shown. □

**Proposition**

**1.**

**Proof.**

**Theorem**

**3.**

- (i)
- ≾ admits a continuous multi-utility representation.
- (ii)
- ${t}_{\mid \sim}^{\precsim}$ is Hausdorff and ≾ is closed.
- (iii)
- ${t}_{\mid \sim}^{\precsim}$ is Hausdorff and ≾ is semi-closed.

**Proof.**

**OC**in order to set

**Theorem**

**4.**

- (i)
- There exists some cardinal number κ and a preorder ≲ on ${\{0,1\}}^{\kappa}$, which is coarser than ${\le}_{lex}$, for which there exists a continuous order-embedding $\vartheta :(X\precsim ,t)\u27f6({\{0,1\}}^{\kappa},\lesssim ,{t}^{\lesssim})$;
- (ii)
- ≾ admits a continuous multi-utility representation.

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Bosi, G.; Daris, R.; Zuanon, M.
Continuous Multi-Utility Representations of Preorders and the Chipman Approach. *Axioms* **2024**, *13*, 148.
https://doi.org/10.3390/axioms13030148

**AMA Style**

Bosi G, Daris R, Zuanon M.
Continuous Multi-Utility Representations of Preorders and the Chipman Approach. *Axioms*. 2024; 13(3):148.
https://doi.org/10.3390/axioms13030148

**Chicago/Turabian Style**

Bosi, Gianni, Roberto Daris, and Magalì Zuanon.
2024. "Continuous Multi-Utility Representations of Preorders and the Chipman Approach" *Axioms* 13, no. 3: 148.
https://doi.org/10.3390/axioms13030148