\documentclass[11pt]{amsart} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \begin{document} \setlength{\unitlength}{0.01in} \linethickness{0.01in} \begin{center} \begin{picture}(474,66)(0,0) \multiput(0,66)(1,0){40}{\line(0,-1){24}} \multiput(43,65)(1,-1){24}{\line(0,-1){40}} \multiput(1,39)(1,-1){40}{\line(1,0){24}} \multiput(70,2)(1,1){24}{\line(0,1){40}} \multiput(72,0)(1,1){24}{\line(1,0){40}} \multiput(97,66)(1,0){40}{\line(0,-1){40}} \put(143,66){\makebox(0,0)[tl]{\footnotesize Proceedings of the Ninth Prague Topological Symposium}} \put(143,50){\makebox(0,0)[tl]{\footnotesize Contributed papers from the symposium held in}} \put(143,34){\makebox(0,0)[tl]{\footnotesize Prague, Czech Republic, August 19--25, 2001}} \end{picture} \end{center} \vspace{0.25in} \title[sublinear continuous order-preserving function]{Sublinear and continuous order-preserving functions for noncomplete preorders} \author{Gianni Bosi} \address{Dipartimento di Matematica\\ Applicata {\em ``Bruno de Finetti''}\\ Universit\`a di Trieste, Piazzale\\ Europa 1, 34127, Trieste, Italy\\ Phone: +39 040 6767115. Fax: +39 040 54209.} \email{giannibo@econ.univ.trieste.it} \author{Magal\`i E. Zuanon} \address{Istituto di Econometria e Matematica per le Decisioni Economiche\\ Universit\`a Cattolica del Sacro Cuore\\ Largo A. Gemelli 1, 20123, Milano, Italy} \email{mzuanon@mi.unicatt.it} \thanks{\textit{JEL classification.} C60; D80} \thanks{\textit{Corresponding author.} Gianni Bosi} \thanks{Gianni Bosi and Magal\`i E. Zuanon, {\em Sublinear and continuous order-preserving functions for noncomplete preorders}, Proceedings of the Ninth Prague Topological Symposium (Prague, 2001), pp.~1--8, Topology Atlas, Toronto, 2002; {\tt arXiv:math.GN/0204101}} \begin{abstract} We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder. \end{abstract} \keywords{Topological vector space; preordered real cone; sublinear order-preserving function; decreasing scale} \subjclass[2000]{28B15, 06A06, 91B16} \maketitle \section{Introduction} Necessary and sufficient conditions for the existence of a continuous linear order-preserving function for a complete preorder on a topological real vector space are already found in the literature (see e.g.\ Candeal and Indur\'ain \cite{ci1995} and Neuefeind and Trockel \cite{nt1995}). It is well known that there are important applications of such results in expected utility theory and collective decision making. A characterization of the existence of a continuous linear utility function for a complete preorder on a convex set in a normed real vector space was presented by B\"{u}ltel \cite{b2001}. Further, some authors were concerned with the existence of a homogeneous of degree one and continuous order-preserving function for a complete preorder on a real cone in a topological real vector space (see e.g.\ Bosi \cite{b1998}, Bosi, Candeal and Indur\'ain \cite{bci2000}, and Dow and Werlang \cite{dw1992}). More recently, Bosi and Zuanon \cite{bz2001a} presented a characterization of the existence of a nonnegative, homogeneous of degree one and continuous order-preserving function for a noncomplete preorder on a real cone in a topological real vector space. In a different context, other authors were concerned with the existence of an additive order-preserving function on a completely preordered semigroup (see e.g.\ Allevi and Zuanon \cite{az2000}, and Candeal, de Miguel and Indur\'ain \cite{cdi1999}). Bosi and Zuanon \cite{bz2001b} presented a characterization of the existence of a Choquet integral representation for a complete preorder on the space of all continuous real-valued functions on a compact topological space. In this paper we provide an axiomatization of the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in a topological real vector space. All the results we present are based upon the notion of a {\em decreasing scale} (or {\em linear separable system}) which was introduced by Herden \cite{h1989a, h1989b} in order to characterize the existence of a continuous order-preserving function for a preorder on a topological space (see also Burgess and Fitzpatrick \cite{bf1977}, and Mehta \cite{m1998}). It should be noted that such a topic can be of some interest in the applications to economics. Consider the following example concerning decision theory under uncertainty. Let ${\bf M}=\{\mu_{n}: n \in \{1,\ldots,n^{*}\}\}$ be a finite family of {\em concave capacities} on a measurable space $(\Omega , \mathcal{A})$, with $\Omega$ the {\em state space}, and $\mathcal{A}$ a {\em $\sigma$-algebra} of subsets of $\Omega$. We recall that a {\em capacity} $\mu$ on $\mathcal{A}$ (i.e., a function from $\mathcal{A}$ into $[0,1]$ such that $\mu ( \emptyset ) = 0$, $\mu ( \Omega ) = 1$, and $\mu (A) \leq \mu ( B)$ for all $A \subseteq B$, $A,B \in \mathcal{A}$) is said to be concave if for all sets $A,B \in \mathcal{A}$, $$ \mu (A \cup B ) + \mu (A \cap B ) \leq \mu (A) + \mu (B) $$ (see e.g.\ Chateauneuf \cite{c1996}). Let $X$ be a {\em real convex cone} of nonnegative {\em real random variables} (i.e., measurable real functions) on $(\Omega , \mathcal{A})$, and assume that $X$ is contained in $L^{1}(\Omega , \mathcal{A}, \mu_{n})$ for every $n \in \{1,\ldots,n^{*}\}$, where $L^{1}(\Omega, \mathcal{A}, \mu)$ stands for the normed space of all the real random variables $x$ such that the {\em Choquet integral} $$ \int_{\Omega}x d\mu = \int_{0}^{\infty}\mu(\{x \geq t\})dt + \int_{- \infty }^{0}(\mu(\{x \geq t\})-1)dt $$ is finite (see e.g.\ Denneberg \cite{d1994}). Define a binary relation $\preceq$ on $X$ as follows: $$ x \preceq y \mbox{ if and only if } \int_{\Omega}x d\mu_{n} \leq \int_{\Omega}y d\mu_{n} \mbox{ for all } n \in \{1,\ldots,n^{*}\}. $$ It is clear that $\preceq$ is a preorder on $X$, and that $\preceq$ is not complete in general. For every $n \in \{1,\ldots,n^{*}\}$, denote by $\tau_{n}$ the norm topology on $X$ which is associated to $\mu_{n}$, and let $\tau$ be any (vector) topology on $X$ which is stronger than $\tau_{n}$ for all $n \in \{1,\ldots,n^{*}\}$ (i.e., $\tau_{n} \subseteq \tau$ for all $n \in \{1,\ldots,n^{*}\}$). Then the real-valued function $u$ on $X$ defined by $$ u(x) = \sum_{n=1}^{n^{*}} \int_{\Omega}x d\mu_{n} \quad(x \in X) $$ is a nonnegative, sublinear and $\tau$-continuous {\em order-preserving function} for $\preceq$. Indeed, it is clear that $x \preceq y$ implies $u(x) \leq u(y)$ for all $x,y \in X$. If $x \prec y$, then we have $u(x) < u(y)$, since $\int_{\Omega}x d\mu_{n} \leq \int_{\Omega}y d\mu_{n}$ for all $n \in \{1,\ldots,n^{*}\}$, and there exists at least one index $\bar n \in \{1,\ldots,n^{*}\}$ such that $\int_{\Omega}x d\mu_{\bar n} < \int_{\Omega}y d\mu_{\bar n}$. Further, $u$ is sublinear since the functional $x \rightarrow \int_{\Omega}x d\mu_{n}$ is sublinear for all $n \in \{1,\ldots,n^{*}\}$. Finally, $u$ is $\tau$-continuous since the functional $x \rightarrow \int_{\Omega}x d\mu_{n}$ is $\tau_{n}$-continuous, and therefore $\tau$-continuous for all $n \in \{1,\ldots,n^{*}\}$ (see Denneberg \cite[Proposition 9.4]{d1994}). \section{Notation and preliminaries} A {\em preorder} $\preceq$ on an arbitrary set $X$ is a reflexive and transitive binary relation on $X$. The {\em strict part} and the {\em symmetric part} of a given preorder $\preceq$ will be denoted by $\prec$, and respectively $\sim$. A preorder $\preceq$ on a set $X$ is said to be {\em complete} if for any two elements $x,y \in X$ either $x \preceq y$ or $y \preceq x$. If $\preceq$ is a preorder on a set $X$, then the pair $(X,\preceq )$ will be referred to as a {\em preordered set}. Define, for every $x \in X$, $L_{\prec}(x)=\{z \in X: z \prec x\}$, $U_{\prec}(x)=\{z \in X: x \prec z\}$. Given a preordered set $(X, \preceq)$, a real-valued function $u$ on $X$ is said to be \begin{enumerate} \item {\em increasing} if $u(x) \leq u(y)$ for every $x,y \in X$ such that $x \preceq y$; \item {\em order-preserving} if it is increasing and $u(x) < u(y)$ for every $x,y \in X$ such that $x \prec y$. \end{enumerate} If $(X, \preceq )$ is a preordered set, and $\tau$ is a topology on $X$, then the triple $(X, \tau , \preceq )$ will be referred to as a {\em topological preordered space}. If $(X, \tau , \preceq )$ is a topological completely preordered space, then the complete preorder $\preceq$ is said to be {\em continuous} if $L_{\prec}(x)$ and $U_{\prec}(x)$ are open subsets of $X$ for every $x \in X$. Given a preordered set $(X, \preceq )$, a subset $A$ of $X$ is said to be {\em decreasing} if $y\in A$ whenever $y \preceq x$ and $x \in A$. In the sequel, the symbol ${\mathbb Q}^{++}$ (${\mathbb R}^{++}$) will stand for the set of all positive rational (real) numbers. If $(X, \tau )$ is a topological space, then denote by $\overline A$ the topological closure of any subset $A$ of $X$. We say that a family $\mathcal{G}=\{G_{r}: r \in {\mathbb Q}^{++}\}$ is a {\em countable decreasing scale} ({\em countable linear separable system}) in a topological preordered space $(X,\tau , \preceq )$ if \begin{enumerate} \item $G_{r}$ is an open decreasing subset of $X$ for every $r \in {\mathbb Q}^{++}$; \item $\overline{G_{r_{1}}} \subseteq G_{r_{2}}$ for every $r_{1},r_{2} \in {\mathbb Q}^{++}$ such that $r_{1} < r_{2}$; \item $\displaystyle{\bigcup_{r \in {\mathbb Q}^{++}}G_{r}=X}$. \end{enumerate} If $E$ is a real vector space, then define, for every subset $A$ of $E$ and any real number $t$, $tA=\{ta:a \in A\}$. Further, if $A$ and $B$ are any two subsets of a (real) vector space $E$, then define $A+B=\{a+b:a \in A,b\in B\}$. A subset $X$ of a {\em real vector space} $E$ is said to be \begin{enumerate} \item a {\em real cone} if $tx \in X$ for every $x \in X$ and $t \in {\mathbb R}^{++}$; \item a {\em real convex cone} if it is a real cone and $x+y \in X$ for every $x,y \in X$. \end{enumerate} A real-valued function $u$ on a real cone $X$ in a real vector space $E$ is said to be {\em homogeneous of degree one} if $u(tx) = t u(x)$ for every $x \in X$ and $t \in {\mathbb R}^{++}$. A real-valued function $u$ on a real convex cone $X$ in a real vector space $E$ is said to be {\em sublinear} if it is homogeneous of degree one and {\em subadditive} (i.e., $u(x+y) \leq u(x) + u(y)$ for every $x,y \in X$). Given a {\em topological real vector space} $E$, denote by $\tau$ the vector topology for $E$ (i.e., the topology on $E$ which makes the vector operations continuous). If $X$ is any subset of a topological real vector space $E$, denote by $\tau_{X}$ the topology induced on $X$ by the vector topology $\tau$ on $E$. If $(X, \preceq )$ is a preordered real cone in a topological real vector space $E$, then we say that a countable decreasing scale $\mathcal{G}=\{G_{r}: r \in {\mathbb Q}^{++}\}$ in $(X, \tau_{X},\preceq )$ is {\em homogeneous} if $qG_{r}=G_{qr}$ for every $q,r \in {\mathbb Q}^{++}$. If $(X, \preceq )$ is a preordered real convex cone in a topological real vector space $E$, then we say that a countable decreasing scale $\mathcal{G}=\{G_{r}: r \in {\mathbb Q}^{++}\}$ in $(X, \tau_{X},\preceq )$ is {\em subadditive} if $G_{q}+G_{r}\subseteq G_{q+r}$ for every $q,r \in {\mathbb Q}^{++}$. \section{Existence of a sublinear continuous order-preserving function} In the following theorem we characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in a topological real vector space. \begin{theorem}\label{thm} Let $\preceq$ be a preorder on a real convex cone $X$ in a topological real vector space $E$. Then the following conditions are equivalent: \begin{enumerate} \item\label{theorem1} There exists a nonnegative, sublinear and continuous order-preserving function $u$ for $\preceq$. \item\label{theorem2} There exists a homogeneous and subadditive countable decreasing scale $\mathcal{G}=\{G_{r}: r \in {\mathbb Q}^{++}\}$ in $(X, \tau_{X},{\preceq})$ such that for every $x,y \in X$ with $x \prec y$ there exist $r_{1},r_{2} \in {\mathbb Q}^{++}$ with $$ r_{1} < r_{2}, x \in G_{r_{1}}\ \mbox{and}\ y \not \in G_{r_{2}}. $$ \end{enumerate} \end{theorem} \begin{proof} (\ref{theorem1}) $\Rightarrow$ (\ref{theorem2}). Assume that there exists a nonnegative, sublinear and continuous order-preserving function $u$ for $\preceq$. Define $G_{r}=u^{-1}([0,r[)$ for every $r \in {\mathbb Q}^{++}$. Let us show that $\mathcal{G}=\{G_{r}: r \in {\mathbb Q}^{++}\}$ is a homogeneous and subadditive countable decreasing scale satisfying condition (\ref{theorem2}). Using the fact that $u$ is nonnegative and order-preserving, we have that for every $x,y \in X$ such that $x \prec y$, there exist $r_{1},r_{2} \in {\mathbb Q}^{++}$ such that $u(x) < r_{1} < r_{2} < u(y)$, and therefore $x \in G_{r_{1}}$, $y \not \in G_{r_{2}}$. Further, since $u$ is homogeneous of degree one, $$ qG_{r}= qu^{-1}([0,r[) = u^{-1}([0,qr[) =G_{qr}\ \mbox{for every $q,r \in {\mathbb Q}^{++}$.} $$ Hence, $\mathcal{G}$ is homogeneous. It only remains to show that $\mathcal{G}$ is subadditive. To this aim, consider any two rational numbers $q,r \in {\mathbb Q}^{++}$, and let $z \in G_{q} + G_{r}$. Then there exist two elements $x,y \in X$ such that $z = x+y$, $u(x) < q$, $u(y) < r$. Hence, using the fact that $u$ is subadditive, we have $u(z) = u(x+y) \leq u(x) + u(y) < q+r$, and therefore $z \in G_{q+r}$. (\ref{theorem2}) $\Rightarrow$ (\ref{theorem1}). Define, for every $x \in X$, \[u(x)= \inf \{r \in {\mathbb Q}^{++}: x \in G_{r}\}.\] Then $u$ is a nonnegative continuous order-preserving function for $\preceq$. Indeed, by using the fact that $G_{r}$ is a decreasing set for every $r \in {\mathbb Q}^{++}$, it is easily seen that $u$ is increasing. Further, $u$ is order-preserving by condition (\ref{theorem2}) above, and $u$ is continuous since $u(x)= \inf \{r \in {\mathbb Q}^{++}: x \in \overline{G_{r}}\}$ (see e.g.\ Theorem 1 in Bosi and Mehta \cite{bm2001} for details). In order to show that $u$ is homogeneous of degree one, it suffices to prove that for no $r \in {\mathbb Q}^{++}$, and $x \in X$ it is $u(rx) \neq ru(x)$. Then the thesis follows by a standard continuity argument. This part of the proof is already found in Bosi and Zuanon \cite[Theorem 1]{bz2001a}. Nevertheless, we present all the details here for reader's convenience. By contradiction, assume that there exist $r \in {\mathbb Q}^{++}$, and $x \in X$ such that $u(rx) < ru(x)$. Then, from the definition of $u$, there exists $r' \in {\mathbb Q}^{++}$ such that $u(rx) < r' < ru(x)$, $rx \in G_{r'}$. Since $u(x) > \frac{r'}{r}$, it follows that $x \not \in G_{\frac{r'}{r}}=\frac{1}{r}G_{r'}$, and therefore we arrive at the contradiction $rx \not \in G_{r'}$. Analogously it can be shown that for no $r \in {\mathbb Q}^{++}$, and $x \in X$ it is $ru(x) < u(rx)$. It remains to prove that $u$ is subadditive. By contradiction, assume that there exist two elements $x,y \in X$ such that $u(x) + u(y) < u(x+y)$. Then, from the definition of $u$, there exist two rational numbers $q,r \in {\mathbb Q}^{++}$ such that $u(x) + u(y) < q + r < u(x+y)$, $x \in G_{q}$, $y \in G_{r}$. Since the countable decreasing scale $\mathcal{G}= \{G_{r}: r \in {\mathbb Q}^{++}\}$ is subadditive, we have $x + y \in G_{q+r}$, and therefore $q + r < u(x+y)$ is contradictory from the definition of $u$. This consideration completes the proof. \end{proof} If $\preceq$ is a {\em homothetic} complete preorder on a real cone $X$ in a real vector space $E$ (i.e., $x \preceq y$ entails $tx \preceq ty$ for every $x,y \in X$, and $t \in {\mathbb R}^{++}$), then consider the following subcones of $X$: $$X_{0} = \{x \in X: x \sim tx \mbox{ for some positive real number } t \neq 1\};$$ $$X_{+} = \{x \in X: x \prec tx \mbox{ for some real number } t > 1\};$$ $$X_{-} = \{x \in X: tx \prec x \mbox{ for some real number } t > 1\}.$$ In the following corollary, we present a characterization of the existence of a nonnegative, sublinear and continuous order-preserving function for a complete preorder on a real convex cone in a topological real vector space. We recall that, given a preordered set $(X, \preceq)$, a subset $A$ of $X$ is said to be an {\em order-dense subset of} $(X, \preceq)$ if for every $x,y \in X$ such that $x \prec y$ there exists $a \in A$ such that $x \prec a \prec y$. \begin{corollary}\label{cor} Let $\preceq$ be a complete preorder on a real convex cone $X$ in a topological real vector space $E$, and assume that $X_{0}$ and $X_{+}$ are both nonempty, while $X_{-}$ is empty. Then the following conditions are equivalent: \begin{enumerate} \item\label{cor1} There exists a nonnegative, sublinear and continuous order-preserving function $u$ for $\preceq$. \item\label{cor2} The following conditions are satisfied: \begin{enumerate} \item\label{cora} $\preceq$ is homothetic; \item\label{corb} $\preceq$ is continuous; \item\label{corc} The set $\{qx_{+}: q \in {\mathbb Q}^{++}\}$ is an order-dense subset of $(X_{+}, {\preceq})$ for every element $x_{+} \in X_{+}$; \item\label{cord} $x \sim y$ for every $x,y \in X_{0}$; \item\label{core} $x \prec x_{+}$ for every $x \in X_{0}$, $x_{+} \in X_{+}$; \item\label{corf} $x + y \prec (q+r)x_{+}$ for every $x,y \in X$, $x_{+} \in X_{+}$, $q,r \in {\mathbb Q}^{++}$ such that $x \prec qx_{+}$, $y \prec rx_{+}$. \end{enumerate} \end{enumerate} \end{corollary} \begin{proof} (\ref{cor1}) $\Rightarrow$ (\ref{cor2}). Assume that there exists a nonnegative, sublinear and continuous order-preserving function $u$ for $\preceq$. Then it is clear that $\preceq$ is homothetic and continuous. If we consider any element $x_{+} \in X$ such that $x_{+} \prec tx_{+}$ for some real number $t>1$, then it is necessarily $u(x_{+}) > 0$, and therefore, using the fact that $u$ is homogeneous of degree one, condition (\ref{corc}) easily follows. Finally, it is easily seen that conditions (\ref{cord}), (\ref{core}) and (\ref{corf}) are verified. (\ref{cor2}) $\Rightarrow$ (\ref{cor1}). Consider any element $x_{+} \in X$ such that $x_{+} \prec tx_{+}$ for some real number $t>1$, and let $G_{r}=L_{\prec}(rx_{+})$ for every $r \in {\mathbb Q}^{++}$. Then it is easy to check that the family $\mathcal{G}= \{G_{r}: r \in {\mathbb Q}^{++}\}$ is a countable decreasing scale satisfying condition (\ref{theorem2}) of Theorem \ref{thm}. Indeed, by homotheticity of $\preceq$, $x \prec rx_{+}$ is equivalent to $qx \prec qrx_{+}$ ($q \in {\mathbb Q}^{++}$), and therefore $\mathcal{G}$ is homogeneous (see Bosi and Zuanon \cite[Corollary 2]{bz2001a}). Further, $\mathcal{G}$ is subadditive by condition (\ref{corf}) since, for every $q,r \in {\mathbb Q}^{++}$, and $x,y \in X$, $x \in G_{q}$ and $y \in G_{r}$ is equivalent to $x \prec qx_{+}$ and $y \prec rx_{+}$, which implies $x+y \prec (q+r)x_{+}$ or equivalently $x+y \in G_{q+r}$. Finally, by condition (\ref{corc}) above, for every $x,y \in X$ such that $x \prec y$ there exist $r_{1},r_{2} \in {\mathbb Q}^{++}$ such that $$ r_{1} < r_{2}, \quad x \prec r_{1}x_{+} \prec r_{2}x_{+} \prec y, $$ or equivalently $$ x \in L_{\prec}(r_{1}x_{+}), \quad y \not \in L_{\prec}(r_{2}x_{+}). $$ So the proof is complete. \end{proof} Denote by $\bar 0$ the {\em zero vector} in a real vector space $E$. In the following corollary we are concerned with a sublinear representation of a complete preorder on a real convex cone containing the zero vector. \begin{corollary} Let $\preceq$ be a complete preorder on a real convex cone $X$ in a topological real vector space $E$, and assume that $X_{+}$ is nonempty, while $X_{-}$ is empty. If in addition $\bar 0$ belongs to $X$, then there exists a nonnegative, sublinear and continuous order-preserving function $u$ for $\preceq$ if and only if $\preceq$ is homothetic and continuous, and it satisfies condition (\ref{corf}) of Corollary \ref{cor}. \end{corollary} \begin{proof} From the corollary in Bosi, Candeal and Indur\'ain \cite{bci2000}, there exists a nonnegative, homogeneous of degree one and continuous order-preserving function $u$ for $\preceq$. Indeed, the complete preorder $\preceq$ on the real (convex) cone $X$ is homothetic and continuous, and we have in addition $\bar 0 \in X$. Let us show that $u$ must be subadditive as a consequence of condition (\ref{corf}) of Corollary \ref{cor}. Assume by contraposition that there exist $x,y \in X$ such that $u(x) + u(y) < u(x+y)$, and consider any element $x_{+} \in X_{+}$. Then it must be $u(x_{+}) > 0$ since $u$ is a homogeneous of degree one utility function for $\preceq$, and there exist two positive rational numbers $q$ and $r$ such that $$ u(x) + u(y) < (q+r)u(x_{+}) < u(x+y),\quad x \prec qx_{+}, \quad y \prec rx_{+}. $$ But here we have a contradiction since it should be $u(x+y) < (q+r)u(x_{+})$ by condition (\ref{corf}) of Corollary \ref{cor}. So the proof is complete. \end{proof} %\bibliographystyle{amsplain} %\bibliography{01} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{10} \bibitem{az2000} E.~Allevi and Magal{\`\i}~E. Zuanon, \emph{Representation of preference orderings on totally ordered semigroups}, Pure Math. Appl. \textbf{11} (2000), no.~1, 13--21. \MR{2001m:91121} \bibitem{b1998} Gianni Bosi, \emph{A note on the existence of continuous representations of homothetic preferences on a topological vector space}, Ann. Oper. 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