Representing preorders with injective monotones

We introduce a new class of real-valued monotones in preordered spaces, injective monotones. We show that the class of preorders for which they exist lies in between the class of preorders with strict monotones and preorders with countable multi-utilities, improving upon the known classification of preordered spaces through real-valued monotones. We extend several well-known results for strict monotones (Richter–Peleg functions) to injective monotones, we provide a construction of injective monotones from countable multi-utilities, and relate injective monotones to classic results concerning Debreu denseness and order separability. Along the way, we connect our results to Shannon entropy and the uncertainty preorder, obtaining new insights into how they are related. In particular, we show how injective monotones can be used to generalize some appealing properties of Jaynes’ maximum entropy principle, which is considered a basis for statistical inference and serves as a justification for many regularization techniques that appear throughout machine learning and decision theory.


Introduction
The set of all preordered spaces ðX; "Þ is structured according to how well their preorder can be represented by real-valued monotones, that is, functions u : X ! R such that x " y implies f ðxÞ f ðyÞ 8x; y 2 X (Evren & Ok, 2011;Ok, 2002). Two major classification methods can be distinguished depending on whether one considers a single monotone (Alcantud et al., 2016) or a whole family U of monotones encapsulating all the information in ", called a multi-utility (Evren & Ok, 2011). More precisely, if U is a multi-utility for ðX; "Þ then 8x; y 2 X we have x " y if and only if uðxÞ uðyÞ 8u 2 U. Without further constraints, monotones and multi-utilities are, however, not very useful from a classification perspective as they exist for any preordered space. They become more useful when adding constraints. For example, there are preordered spaces without strict monotones, that is, without monotones u such that, uðxÞ\uðyÞ whenever x 0 y. 1 Strict monotones, also known as Richter-Peleg functions, have been extensively studied (Alcantud et al., 2013(Alcantud et al., , 2016Peleg, 1970;Richter, 1966) and are related to other features of the preorder such as its maximal elements. In the case of multi-utilities, the cardinality is an important property for the classification of preordered spaces, with countable multi-utilities playing a central role (Bevilacqua et al., 2018c). Of particular importance are utility functions (Debreu, 1954(Debreu, , 1964, that is, multi-utilities consisting of a single function. 2 Here, we introduce injective monotones, which are monotones u such that uðxÞ ¼ uðyÞ implies both x " y and y " x. Preorders for which they exist form a category between preorders with strict monotones and preorders with countable multiutilities, as we show in Propositions 1, 5 and 8. Hence, we improve on the existing classification of preorders by adding a new distinct class. More precisely, in Sect. 3, we define injective monotones and prove some simple properties. After discussing their relation to optimization in Sect. 4, we take a look at the role of multi-utilities in Sect. 5, in particular, we construct injective monotones from countable multiutilities and show that the converse does not hold. Finally, in Sect. 6, we consider separability properties of preorders that are sufficient for the existence of strict and injective monotones, introducing a new notion of Debreu separability, that allows to extend previous results on strict monotones to corresponding analogues for injective monotones. In the following section, we introduce our running example to which we come back several times throughout the development of the general theory. In particular, we discuss the relation between the uncertainty preorder from majorization theory (Arnold, 2018), which has Shannon entropy as a strict monotone, and the maximum entropy principle that appears in many different parts of science.

Example: the uncertainty preorder and Shannon entropy
The outcome of a random variable with a narrow probability distribution is easier to predict than the outcome of a random variable with a less concentrated distribution. For example, the result of throwing an unbalanced coin is easier to predict than the one of a balanced coin. In other words, a wider distribution contains more uncertainty than a narrower distribution. This idea is captured by a binary relation on the space P X of probability distributions on a set X: the uncertainty preorder " U , defined for finite X by p " U q () u i ðpÞ u i ðqÞ 8i 2 f1; ::; jXj À 1g ; where u i ðpÞ:¼ À P i n¼1 p # n and p # denotes the decreasing rearrangement of p (same components as p but ordered decreasingly). Notice, " U is known in mathematics, economics, and quantum physics as majorization (Arnold, 2018;Brandao et al., 2015;Hardy et al., 1952;Marshall et al., 1979), originally developed by Lorenz (1905) and Dalton (1920) among others, to measure wealth and income inequality. An intuitive way to think of p " U q is that q is the result of finitely many transfers of pieces of probability from a more likely to a less likely option in p (Gottwald & Braun, 2019). In other words, q is more spread out or less biased, and thus, contains more uncertainty than p. For instance, a Dirac distribution is the smallest, and the uniform distribution is the largest, with respect to " U , among all distributions on X.
There is, however, a downside to this intuitive notion of uncertainty: what if p and q do not have this relationship? For example, if p ¼ ð0:6; 0:2; 0:2; 0; ::; 0Þ and q ¼ ð0:5; 0:4; 0:1; 0; ::; 0Þ, then p and q cannot be related by " U . Instead, the most common way to measure uncertainty is to use an entropy functional, such as the Shannon entropy, HðpÞ:¼ À E p ½log p, or one of various alternative entropy proposals, including Renyi entropy (Rényi et al., 1961), Tsallis entropy (Tsallis, 1988), and many more (Csiszár, 2008). Even though, in general, " U cannot be fully represented by any of these so-called generalized entropies F, it is noteworthy that all of them are monotones with respect to " U . 3 While the converse is not true for any single F, there are collections F which constitute a multi-utility, e.g., in the case of finite X, F ¼ f P jXj n¼1 f ðp n Þ j f concaveg (Schur, 1923), or even F ¼ fu i g jXjÀ1 i¼1 by the definition of " U (1).
The preference towards unbiased distributions, that is represented by any monotone of " U , is of particular relevance in the maximum entropy principle, where (Shannon) entropy serves as a counter-acting force against the bias towards the maximal elements of a given ''energy'' function E. Going back to the principle of insufficient reason (Bernoulli, 1713), today the maximum entropy principle appears in virtually all branches of science. For example, it is often used as a general principle to explain the raison d'etre behind all kinds of ''soft'' versions of known machine learning methods, especially in reinforcement learning (Fox et al., 2016;Williams & Peng, 1991), but also in models of robust and resource-aware decision making (Maccheroni et al., 2006;Ortega & Braun, 2013;Still, 2009;Tishby & Polani, 2011). Basically, whenever there appears a trade-off between precision and uncertainty, there is a good chance that the maximum entropy principle is applied (Gottwald & Braun, 2020).
The underlying goal of the maximum entropy principle is to select a typical distribution among a set of candidate distributions satisfying a given constraint, usually of the form hEi p ¼ c, where hEi p denotes the expectation of a random variable E with respect to the probability measure p. In Wallis' derivation of the maximum entropy principle, typicality is measured by the number of possibilities of assigning n elements among N groups, under the limit of infinitely many elements (n ! 1) such that the statistical probabilities p i ¼ n i N of belonging to a specific group i remain finite (n i denotes the number of elements in group i) Jaynes (2003). However, we can also think of typicality as containing the least amount of bias, or in other words, the maximal amount of uncertainty. Thus, when considering the uncertainty preorder " U as the most basic way to decide about the difference in uncertainty between two distributions, then the ultimate goal of the maximum entropy principle becomes to obtain the maximal elements of " U , inside the given constraint set.
Even though, generally, we are not guaranteed to find all maximal elements of " U when maximizing entropy, maximum entropy solutions are in fact maximal elements of " U , as entropy is a strict monotone. Furthermore, since the maximum entropy principle maximizes a strictly concave functional H over a convex subset, it yields a unique maximal element of " U . In contrast, injective monotones, which exist for " U (see Proposition 5), preserve this uniqueness property up to equivalence (see Proposition 3), without asking for the additional structural requirements of concavity.

Injective monotones
A preorder " on a set X is a reflexive (x " x 8x 2 X) and transitive (x " y and y " z implies x " z 8x; y; z 2 X) binary relation. A tuple ðX; "Þ is called a preordered space. An antisymmetric (x " y and y " x imply x ¼ y 8x; y 2 X) preorder " is called a partial order. The relation x $ y, defined by x " y and y " x, forms an equivalence relation on X, that is, it fulfills the reflexive, transitive and symmetric (x $ y if and only if y $ x 8x; y 2 X) properties. Notice, a preorder " is a partial order on the quotient set X= $ ¼ f½xjx 2 Xg, consisting of all equivalence classes ½x ¼ fy 2 Xjy $ xg. In case x " y and :ðx $ yÞ for some x; y 2 X we say y is strictly preferred to x, denoted by x 0 y. If :ðx " yÞ and :ðy " xÞ, we say x and y are incomparable, denoted by x ffl y. Whenever there are no incomparable elements, a preordered space is called total. By the Szpilrajn extension theorem (Szpilrajn, 1930), every partial order can be extended to a total order, that is, to a partial order that is total. Notice, the set P X of probability distributions on X equipped with the uncertainty preorder " U forms a non-antisymmetric preordered space, because equivalent elements are only equal up to permutations (Arnold, 2018).
A real-valued function f : X ! R is called a monotone if x " y implies f ðxÞ f ðyÞ. If also the converse is true, then f is called a utility function. Furthermore, if f is a monotone and x 0 y implies f ðxÞ\f ðyÞ, then f is called a strict monotone (or a Richter-Peleg function Alcantud et al., 2016).
Definition 1 (Injective monotones) A monotone f : X ! R on a preordered space ðX; "Þ is called an injective monotone if f ðxÞ ¼ f ðyÞ implies x $ y, that is, if f is injective considered as a function on the quotient set X= $ .
Clearly, an injective monotone is also a strict monotone, since x 0 y and f ðxÞ ¼ f ðyÞ contradicts injectivity. The converse is not true, for example, Shannon entropy is a strict monotone for the uncertainty preorder " U (Appendix A.1) but not an injective monotone, nor a utility. In fact, preorders that have an injective monotone form a class in between preorders that have a strict monotone and preorders that have a utility function.
Proposition 1 (i) There are preorders with strict monotones but without injective monotones.
(ii) There are preorders with injective monotones and without utility functions.

Proof
(i) Consider ðPðRÞ; "Þ, the power set PðRÞ of the reals equipped with the preorder " defined by U " V if and only if U ¼ V, or U ¼ f0g and V ¼ f1g.
Then v : PðRÞ ! R, given by vðf1gÞ ¼ 1 and vðUÞ ¼ 0 8U 6 ¼ f1g, is a strict monotone. However, there cannot be injective monotones, because here jPðRÞ= $ j ¼ jPðRÞj and by Cantor's theorem the cardinality of R is strictly smaller than the cardinality of PðRÞ. (ii) Consider ðR; "Þ, where x " y if and only if x y and x; y 6 ¼ 0 or The identity I : R ! R is an injective monotone. However, ðR; "Þ is nontotal since 0 ffl x 8x 2 R=f0g and, thus, has no utility function. h Since every preorder has a monotone (constant functions) and there are preorders without strict monotones (see Appendix A.2.1 for an example), we arrive at the picture shown in Fig. 1. Notice, the closer we are to the center, the better a monotone represents the underlying preorder. In particular, injective monotones contain more information about the preorder than strict monotones.
Nevertheless, in well-behaved cases, it is possible to construct an injective monotone out of a strict monotone. A negative example is the strict monotone that appears in the proof of (i) in Proposition 1, which maps uncountably many incomparable elements to a single number (zero). If a strict monotone fails to be an injective monotone because of only countably many points, however, then it can easily be turned into an injective monotone by consecutive elimination.
Proposition 2 A preordered space ðX; "Þ has an injective monotone if and only if it has a strict monotone f whose non-injective set I f :¼ fx 2 Xj 9y 2 X s:t: f ðxÞ ¼ f ðyÞ and x ffl yg is countable.
Proof By definition, for an injective monotone f, we have I f ¼ ;. Conversely, consider a strict monotone f with a countable non-injective set. Given a numeration fx n g n ! 0 of I f , define f 0 : X ! R by & Notice, by definition, 8x; y 2 X, f ðxÞ f ðyÞ implies f 0 ðxÞ f 0 ðyÞ, I f 0 & I f , and f 0 is injective up to equivalence at x 0 , in particular x 0 6 2 I f 0 . Therefore, we can consecutively eliminate the elements in I f by defining for all n 2 N, f n ðxÞ:¼f nÀ1 ðxÞ þ 2 Àn if f nÀ1 ðxÞ ! f nÀ1 ðx n Þ and :ðx $ x n Þ, and f n ðxÞ:¼f nÀ1 ðxÞ otherwise, analogously to f 0 . It is then straightforward to see that the pointwise limit cðxÞ:¼ lim n!1 f n ðxÞ exists for all x 2 X and that c is an injective monotone. h Notice, the technique in the proof of Proposition 2 does not work if I f is uncountable. In particular, it cannot be used to construct an injective monotone from Shannon entropy f ¼ H for the uncertainty preorder " U , because if N:¼jXj ! 3 then for all c 2 ð0; log NÞ there are p; q 2 P X with c ¼ HðpÞ ¼ HðqÞ but p ffl q (see Appendix A.1). In other words, we can construct an injective map g : ð0; log NÞ ! I H and thus I H has the same cardinality as R, in particular I H is not countable.

Relating monotones to optimization
An element x 2 X is called a maximal element of " if there exists no y 2 X such that x 0 y. For any B X, an element x 2 B is called a maximal element of " in B if there exists no y 2 B such that x 0 y.
Definition 2 (Representing maximal elements) We say, a function f : X ! R is effective for B X if argmax B f 6 ¼ ;, where argmax B f :¼fx 2 Bj 6 9y 2 B such that f ðxÞ\f ðyÞg. We say, a function f : X ! R represents maximal elements of ", if for any B X where B " M denotes the set of maximal elements of " in B. Similarly, we say, a function f injectively represents maximal elements of ", if for any B X for which f is effective, there exists x 0 2 B " M such that where ½x 0 j B is the equivalence class of x 0 restricted to B. Moreover, we say, ðX; "Þ has an (injective) optimization principle if there exists a function f : X ! R which (injectively) represents maximal elements of ".
Even though Shannon entropy does not represent " U as a utility, its property as a strict monotone guarantees that its maxima are in fact maximal elements of " U , i.e., H represents maximal elements of the uncertainty preorder according to Definition 2. Indeed, any p 2 argmax B H is a maximal element of " U for any B P X on which H is effective, as p 0 q for some q 2 B would lead to the contradiction HðqÞ [ HðpÞ. In fact, representing maximal elements is closely related to being a monotone for preorders in general.
Proposition 3 Given a preordered space ðX; "Þ and a monotone u : X ! R, then (i) u is a strict monotone if and only if u represents maximal elements of ". (ii) u is an injective monotone if and only if u injectively represents maximal elements of ".

Proof
(i) If u is a strict monotone, then argmax B u B " M (by the same argument as for entropy). Conversely, consider x; y 2 X with x 0 y. For B:¼fx; yg, we have B " M ¼ fyg and thus fyg ¼ argmax B u, i.e., uðxÞ\uðyÞ.
(ii) For any B X on which u is effective, if x, y 2 argmax B u, we have uðxÞ ¼ uðyÞ and, since u is an injective monotone, x $ y. Conversely, consider x; y 2 X and B :¼ fx; yg. If uðxÞ ¼ uðyÞ then by hypothesis fx; yg ¼ argmax x2B fuðxÞg ¼ ½x 0 j B for some x 0 2 B. In particular, x $ y. h Notice, for the ''if'' part in (ii), we do not have to assume that u is a monotone, that is, if the maxima of some real-valued function u form an equivalence class in the set of maximal elements, then it already follows that u is a monotone.
For any preordered space ðX; "Þ, thus, the existence of a strict monotone implies the existence of an optimization principle and the existence of an injective monotone is equivalent to the existence of an injective optimization principle. One can contrast the global injective representation of maximal elements which characterizes injective monotones in Proposition 3 with local approaches, for some specific B X, present in the literature (Bevilacqua et al., 2018b;White, 1980). Choosing a particular strict monotone u and optimizing it in a set B might, however, not yield all the maximal elements in B " M . For example, take p; q 2 P X with p ffl q and HðpÞ\HðqÞ, then B ¼ fp; qg has the two maximal elements p and q, but argmax B H ¼ fpg. Notice, this is not only an issue for trivial examples like this, but also happens for the maximum entropy principle with linear constraint sets. In particular, if B ¼ fp j hEi ¼ cg, for a given random variable E and some c 2 R, crosses two incomparable elements that turn out to be maximal (see Fig. 2), then only part of the actual maximal elements of " U can be found by maximizing entropy.
Similarly, while optimizing an injective monotone in a set B results in equivalent elements, in general, we only find a slice of the set of all maximal elements in B. In fact, for every maximal element x in B " M , we can construct an injective monotone c such that x 2 argmax c (e.g., in the proof of Proposition 4 below, take c x if x 2 A c and c otherwise). This means that the problem of selecting a maximal equivalence class can be replaced by the problem of selecting an injective monotone.
In the following section, we show that injective monotones exist for a large class of preorders, including the uncertainty preorder.

Relating monotones to multi-utilities
Although it is not possible to capture all information about a non-total preorder using a single real-valued function, a family of functions may be used instead. A family V of real-valued functions v : X ! R is called a multi-utility (representation) of " if x " y () vðxÞ vðyÞ 8v 2 V : Whenever a multi-utility consists of strict monotones, it is called a strict monotone (or Richter-Peleg Alcantud et al., 2016) multi-utility (representation) of ".
Analogously, if the multi-utility consists of injective monotones, we call it an injective monotone multi-utility (representation) of ".
It is straightforward to see that every preordered space ðX; "Þ has the multiutility ðv iðxÞ Þ x2X , where v A denotes the characteristic function of a set A and iðxÞ:¼fy 2 Xjx " yg (Ok, 2002). Moreover, if there exists a strict monotone u, then a multi-utility U only consisting of strict monotones can easily be constructed from a given multi-utility V by U:¼fv þ aug v2V;a [ 0 (Alcantud et al., 2013). Even though this construction does not work directly in the case of injective monotones, a simple modification does, where special care is given to incomparable elements. Proposition 4 Let ðX; "Þ be a preordered space. There exists an injective monotone if and only if there exists an injective monotone multi-utility.
Proof Consider w.l.o.g. an injective monotone c : X ! ð0; 1Þ and i.e., the part of X that has incomparable elements y with strictly larger values of c.
For all x 2 A c , let c x :¼c þ v iðxÞ . Notice, by construction c x ðyÞ ¼ cðyÞ\1 c x ðxÞ for Fig. 2 Example for when the maximum entropy principle does not yield all maximal elements of " U in some B P X . Here, we show the usual visualization of the 2-simplex, that is, the set of all probability distributions in P X for jXj ¼ 3. Let the energy function E be given by Eðx 1 Þ:¼1, Eðx 2 Þ:¼ À 1, and Eðx 3 Þ:¼0, and let B be given by the constraint hEi ¼ 1 4 , represented by the vertical line. The distribution p ¼ ð1=2; 1=4; 1=4Þ is a maximal element in B, because any other element of B is either smaller than p (belongs to an outer blue region) or incomparable (belongs to the white region). However, q ¼ ð9=20; 4=20; 7=20Þ is in B and HðpÞ\HðqÞ. As a result, p is a maximal element of B which is not obtained via the maximum entropy principle all y 2 X with x ffl y. Using cðXÞ ð0; 1Þ and the fact that c is an injective monotone, it is straightforward to see that C:¼fcg [ fc x g x2A c is an injective monotone multi-utility. h Note that the injective monotone multi-utility in the proof of Proposition 4 can be chosen to have cardinality of at most c, the cardinality of the continuum, because it is enough to have one c x per equivalence class ½x 2 X= $ , and, whenever an injective monotone exists, jX= $ j c.
The cardinality of multi-utilities plays an important role. In particular, special interest lies in preordered spaces with countable multi-utilities. In practice, countable multi-utilities are often used to define preordered spaces. For example, the uncertainty preorder " U is defined in (1) by a countable (finite) multi-utility. Also, many applications in multicriteria optimization (Bevilacqua et al., 2018b;Ehrgott 2005) rely on preordered spaces defined by countable multi-utilities. It turns out that for the existence of strict monotones, such as entropy for " U , it is sufficient to have a countable multi-utility (Alcantud et al., 2016, Section 4). Here, we show that countable multi-utilities actually imply the existence of injective monotones, which, due to Proposition 1, improves upon (Alcantud et al., 2016).
Proposition 5 If, for a given preordered space ðX; "Þ, there exists a countable multi-utility, then there exists an injective monotone.
This means that the class of preordered spaces where countable multi-utilities exist is contained in the class of preordered spaces where an injective monotone exists (cf. Fig. 1). However, there exist preordered spaces with injective monotones, i.e., by Proposition 4, with injective monotone multi-utilities of cardinality c, but without countable multi-utilities (see Proposition 8).
For the uncertainty preorder " U , which is defined in (1) through a finite multiutility, Proposition 5, therefore, guarantees the existence of injective monotones. Moreover, we can see a possible construction in (3) below.
By a slight adaptation of the proof of Proposition 5, we obtain the stronger.
Proposition 6 For a given preordered space ðX; "Þ, there exists a countable multiutility if and only if there exists a countable multi-utility only consisting of injective monotones.
This improves upon (Alcantud et al., 2016, Proposition 4.1), where it is shown that a countable multi-utility exists if and only if a countable strict monotone multiutility exists. Notice, however, while for the proof in Alcantud et al. (2016), one can simply modify each member of a given multi-utility separately-similarly as we did for the construction in Proposition 4-our proof of Proposition 6 relies on a more indirect technique, where each member of the resulting injective monotone multiutility does not have a direct relationship to a non-injective member of the given multi-utility.
For the proofs of Propositions 5 and 6, we rely on the following basic facts, the proofs of which can be found in the appendix.
Lemma 1 Let X be a set. Given r 2 ð0; 1 2 Þ and a countable family ðA n Þ n ! 0 of subsets A n X, define the function c : X ! R by Then, cðxÞ\cðyÞ if and only if, for the first The following characterizations of injective monotones and countable multiutilities follow by straightforward manipulations of their definitions. Lemma 2 Let ðX; "Þ be a preordered space. A monotone u is an injective monotone if and only if x 0 y ) uðxÞ\uðyÞ and x ffl y ) uðxÞ 6 ¼ uðyÞ : A collection U of monotones is a multi-utility if and only if Mehta, 1986a). We say a family ðA n Þ n2N of subsets A n X separates x from y, if there exists n 2 N with x 6 2 A n and y 2 A n .
Lemma 3 Let ðA n Þ n ! 0 be a family of increasing sets.
(i) If, for all x; y 2 X with x 0 y, ðA n Þ n ! 0 separates x from y, then the function c : X ! R defined in (3) is a strict monotone for all r 2 ð0; 1Þ. (ii) If in addition, for all x; y 2 X with x ffl y, ðA n Þ n ! 0 separates x from y, or y from x, then c is an injective monotone for all r 2 ð0; 1 2 Þ.
Notice, the construction of strict and injective monotones in Lemma 3 is based on Lemma 1 and is analogous to constructions that appear in the literature, where one typically uses a value of r ¼ 1 2 (e.g., Alcantud et al., 2016;Mehta, 1977;Ok, 2002). The requirement of r\ 1 2 in Lemmas 1 and 3 ensures that the resulting monotone is injective. In fact, as can be seen from the proof of Lemma 1 in the appendix, for r 2 ð0; 1Þ we have r m ¼ r 1Àr P n [ m r n . A value of r 2 ð0; 1 2 Þ thus enables the strict estimate r m [ P n [ m r n , which is exactly where the injectivity up to equivalence of c in Lemma 3 rests.
Proof of Proposition 5 For a countable multi-utility ðu m Þ m2M and q 2 Q, consider the increasing sets A m;q :¼u À1 m ð½q; 1ÞÞ. It suffices to show that ðA n Þ n ! 0 , where A n :¼A m n ;q n for some enumeration n7 !ðm n ; q n Þ of M Â Q, satisfies (i) and (ii) in Lemma 3. If x 0 y or x ffl y, then, by (5), in both cases there exists m 2 M with u m ðxÞ\u m ðyÞ. Hence, we can choose q 2 Q with u m ðxÞ\q\u m ðyÞ, in particular, x 6 2 A m;q and y 2 A m;q . h Proof of Proposition 6 Let ðu m Þ m2M be a countable multi-utility and let c be an injective monotone of the form (3) constructed from the increasing sets A n in the proof of Proposition 5. We define, for any pair ðm; pÞ 2 N such that m\p, u m;p : N ! N which permutes m and p without changing any other natural number. For each u m;p , we define an injective monotone c m;p of the form (3) constructed from ðA u m;p ðnÞ Þ n ! 0 , the increasing sets used to define c reordered by u m;p . Since fcg [ fc m;p g ðm;pÞ2N 2 ;m\p is composed of injective monotones, it suffices to show (5) holds to conclude there exists a countable multi-utility composed of injective monotones. Consider, thus, x; y 2 X such that :ðy " xÞ. If x 0 y, then cðxÞ\cðyÞ by definition.
Assume now x ffl y. If cðxÞ\cðyÞ, then we have finished. Otherwise, we have x 2 A m and y 6 2 A m for the first m 2 N such that v A m ðxÞ 6 ¼ v A m ðyÞ by Lemma 1.
Since there exists some p 2 N p [ m such that y 2 A p and x 6 2 A p , the first n 2 N such that v A u m;p ðnÞ ðxÞ 6 ¼ v A u m;p ðnÞ ðyÞ is n ¼ m. We conclude c m;p ðxÞ\c m;p ðyÞ by Countable separating families such as the ones in Lemma 3 have been used to characterize preordered spaces with continuous utility functions (Herden, 1989), generalizing theorems of Peleg and Mehta (1981). In a similar spirit, Alcantud et al. Countable separating families are a useful tool to improve the classification of preordered spaces by monotones. In particular, we use them in Proposition 8 to show the converse of Proposition 5 is false, that is, there are preordered spaces where injective monotones exist and countable multi-utilities do not.
Proposition 8 There are preordered spaces with injective monotones and without countable multi-utilities.
Proof Consider X:¼½0; 1 [ ½2; 3 equipped with " where x " y () x ¼ y x 2 ½0; 1; y 2 ½2; 3 and y 6 ¼ 8x; y 2 X (see Fig. 3 for a representation of "). Notice ðX; "Þ is a preordered space and the identity map i d : X ! R is an injective monotone. We will show any family ðA i Þ i2I , where A i X is increasing 8i 2 I and 8x; y 2 X such that :ðy " xÞ there exists some i 2 I such that x 6 2 A i and y 2 A i , is uncountable. Since the existence of some ðA i Þ i2I with those properties and countable I is equivalent to the existence of a countable multi-utility (Alcantud et al., 2013, Proposition 2.13), we will get there is no countable multi-utility for X. Consider a family ðA i Þ i2I with the properties above and, for each x 2 ½0; 1, y x :¼x þ 2. Since x ffl y x by definition, there exists some A x 2 ðA i Þ i2I such that x 2 A x and y x 6 2 A x . We fix such an A x for each x 2 ½0; 1 and consider the map f : ½0; 1 ! ðA i Þ i2I , x7 !A x . Given x; z 2 ½0; 1 x 6 ¼ z, if we assume z 2 A x , then, since A x is increasing and z 0 y x as y x 6 ¼ z þ 2, we would have y x 2 A x , a contradiction. Notice, analogously, we get a contradiction if we assume x 2 A z and, therefore, A x 6 ¼ A z . Thus, A x ¼ A z implies x ¼ z and we have, by injectivity of f, j½0; 1j jðA i Þ i2I j. As a consequence, X has no countable multi-utility. h As we have seen in this section, the concept of separating families is closely related to the existence of monotones. In particular, this link is apparent when considering sets of the form u À1 ð½q; 1ÞÞ for some monotone u and q 2 Q, allowing to translate the two concepts into each other (see the proofs of Propositions 5 and 6). There is another rich class of separability properties of preordered spaces providing Fig. 3 Representation of a preordered space, defined in Proposition 8, where injective monotones exist and countable multi-utilities do not. In particular, we show A:¼½0; 1, B:¼½2; 3 and how x; y; z 2 A, x\y\z, are related to x þ 2; y þ 2; z þ 2 2 B. Notice an arrow from an element w to an element t represents w 0 t necessary conditions for the existence of monotones, which could collectively be described by the term order separability. Many important results from mathematical economics fall into this category, such as the Debreu Open Gap Lemma (Debreu, 1964), the Nachbin Separation Theorem (Nachbin, 1965), Szpilrajn's theorem (Szpilrajn, 1930), and Fishburn's theorem (Fishburn, 1970, Theorem 3.1). We discuss the role of injective monotones relative to order separability in the following section.
6 Relating monotones to order separability A subset Z X, such that x 0 y implies that there exists z 2 Z with x 0 z 0 y is called order dense (Bridges & Mehta, 2013;Ok, 2002), and Z is called order dense in the sense of Debreu (or Debreu dense for short) if x " z " y. Accordingly, we say that ðX; "Þ is order separable if there exists a countable order dense set (Mehta, 1986a), and Debreu separable if there exists a countable Debreu dense set in ðX; "Þ. Notice, our definition of order separability is also known as weak separability (Ok, 2002).
It is well known that a total preorder " has a utility function if and only if it is Debreu separable (e.g., Bridges & Mehta, 2013, Theorem 1.4.8). Moreover, if " is non-total, then Debreu separability still implies the existence of strict monotones (Bridges & Mehta, 2013;Debreu, 1954;Herden & Levin, 2012). The converse, however, is not true, i.e., there are preordered spaces with strict monotones that are not Debreu separable. For example, any Debreu dense subset of ðP X ; " U Þ is uncountable (if jXj [ 2)-see Appendix A.1 for a proof. While Debreu separability is concerned with elements satisfying x 0 y, an analogous condition that is sufficient for the existence of injective monotones must also consider incomparable elements.
We call a subset Z X upper dense if x ffl y implies that there exists a z 2 Z such that x ffl z 0 y, and it is called upper dense in the sense of Debreu (or Debreu upper dense for short) if x ffl z " y. 4 Accordingly, ðX; "Þ is called upper separable if there exists a countable subset of X which is both order dense and upper dense (Ok, 2002), and ðX; "Þ is called Debreu upper separable if there exists a countable subset which is both Debreu dense and Debreu upper dense. We list all mentioned order denseness and separability properties in Table 1. Proposition 9 If ðX; "Þ is a Debreu upper separable preordered space, then there exists a countable multi-utility; in particular, there exists an injective monotone.
Proof Consider a countable set D given by Debreu upper separability. We will show x " y () where iðdÞ:¼fy 2 Xjd " yg and rðdÞ:¼fy 2 Xjd 0 yg. By transitivity x " y implies v iðdÞ ðxÞ v iðdÞ ðyÞ and v rðdÞ ðxÞ v rðdÞ ðyÞ 8d 2 D. If :ðx " yÞ then either y 0 x or y ffl x. If y 0 x, then there exists some d 2 D such that either v iðdÞ ðxÞ [ v iðdÞ ðyÞ or v rðdÞ ðxÞ [ v rðdÞ ðyÞ. If y ffl x then there exists some d 2 D such that y ffl d " x which means v iðdÞ ðxÞ [ v iðdÞ ðyÞ. Since there exists a countable multi-utility, as we just showed, there is an injective monotone by Proposition 5. h Since Debreu upper separability still requires a countable Debreu dense set, the converse of Proposition 9 is again false due to the uncertainty preorder not being Debreu separable (Appendix A.1). However, as can be seen from the proof, if we remove Debreu denseness as a requirement, i.e., if we only require D to be Debreu upper dense, then the only part of the proof that does not work is to follow from y 0 x that there exists an element v of the multi-utility with vðxÞ [ vðyÞ. Since a strict monotone has exactly this property, we obtain the following proposition.
Proposition 10 Consider ðX; "Þ a preordered space. If there exists a countable Debreu upper dense set, then the following are equivalent: (i) There exists a strict monotone. (ii) There exists an injective monotone. (iii) There exists a countable multi-utility.
Proof Assume there exists a countable Debreu upper dense set D X. It is enough to show that (i) implies (iii), which follows along the same lines as the proof of Proposition 9, but with the multi-utility consisting of fug [ fv iðdÞ g d2D , where u is a strict monotone. h Table 1 Separability properties of preordered spaces ðX; "Þ

Name Object Definition
Order dense Z X 8x; y 2 X x 0 y ) 9z 2 Z: x 0 z 0 y Debreu dense Z X 8x; y 2 X x 0 y ) 9z 2 Z: x " z " y Upper dense Z X 8x; y 2 X x ffl y ) 9z 2 Z: x ffl z 0 y Debreu upper dense Z X 8x; y 2 X x ffl y ) 9z 2 Z: x ffl z " y The situation in Proposition 10 corresponds exactly to the situation of the uncertainty preorder, which has a countable Debreu upper dense set (Appendix A.1) and, e.g., Shannon entropy as a strict monotone.

Discussion
In this paper, we are mainly concerned with the introduction of injective monotones, their relation to other monotones, optimization, multi-utilities and order separability, and the application to the uncertainty preorder. The key contributions of our work are the following. First, we refine the classification of preordered spaces based on the existence of monotones. In particular, by extending known results for strict monotones to injective monotones, we find conditions for their existence from different perspectives: other classes of monotones, optimization principles, separating families of increasing sets, and (in particular, countable) multi-utilities. An overview of our conditions in relation to previous work can be found in Fig. 4. Second, we introduce the notion of upper Debreu separability, an order separability Fig. 4 Classification of preordered spaces ðX; "Þ in terms of representations by real-valued functions (boxes) and order properties (ellipses). We include known relations in black and our contributions in red. Notice, by Proposition 10, the blue area is empty whenever there exists a countable Debreu upper dense set in X property that allows to extend well-known results about the existence of monotones on Debreu separable spaces to countable multi-utilities and injective monotones. Finally, we apply our general results to the uncertainty preorder, defined on the space of probability distributions over finite sets, in particular, by establishing order separability properties.
Hierarchy of preordered spaces. A number of scientific disciplines rely on preordered spaces and their representation by monotones, as was already pointed out in Campión et al. (2018), Candeal et al. (2001) and Minguzzi (2010). In Fig. 1, we classify the space of preorders in terms of the existence of certain monotones relevant in various disciplines, which leads to a hierarchy of classes of preordered spaces. The conception of injective monotones then allows for a refinement of this hierarchy of preorders.
Historically, much of the early development of real-valued representations has focused on total preordered spaces that allow for the existence of utility functions. In particular, in the field of mathematical economics, utility theory has pioneered the axiomatic study of conditions that ensure the existence of utility functions for a preordered set ðX; "Þ, where X is a set of commodities and " is some total preference relation, a total preorder (Debreu, 1954;Rébillé, 2019). Similarly, we can consider statistical estimation, where the aim is to infer the distribution of a random variable X from some of its realizations. Assuming the distribution belongs to a family fp h g h2R N for some N [ 0, a loss function ' : R N ! R allows rating distributions according to how well they fit with the observed data: p h " ' p h 0 if and only if À'ðhÞ À 'ðh 0 Þ where h; h 0 2 R N (Hennig & Kutlukaya, 2007). Choosing a loss function ' corresponds, thus, to defining a total preorder with a utility representation " ' on fp h g h2R N .
Another example of a preorder with a utility function is equilibrium thermodynamics. Given a thermodynamic system, we consider ðX; " A Þ where X is the set of all equilibrium states for the system and x " A y if and only if y is adiabatically accessible from x 8x; y 2 X (Lieb & Yngvason, 1999), that is, one can turn x into y using a device and a weight, with the device returning to its initial configuration at the end and the weight being allowed to change position in some gravitational field. The main concern in the area is the so-called entropy representation problem (Candeal et al., 2001), that is, the existence of a utility function, called entropy function, for ðX; " A Þ (Lieb & Yngvason, 1999).
Assuming a total preorder as in the previous examples is necessary for the existence of a utility function, but renders injective monotones uninteresting, as they become equivalent to strict monotones. When the totality assumption is dropped, the classes of preorders with these monotones can be distinguished. A well-known instance of non-total preorders with injective monotones is our running example, the uncertainty preorder. One of its relevant applications lies in the study of quantum entanglement, as it characterizes the possible transformations using local operations and classical communications (Nielsen, 1999, Theorem 1). In physics, the uncertainty preorder given by majorization has recently also been extended. Given ' 1 1 ðR þ Þ :¼ fðp i Þ i2N j0 p i 1; P 1 i¼1 p i ¼ 1g, we define infinite majorization " IM (Li & Busch, 2013) for any p; q 2 ' 1 1 ðR þ Þ like p " IM q :, where p # represents p ordered in a decreasing way. Since " IM is defined through a countable multi-utility, there exist injective monotones by Proposition 5. Finally, the uncertainty preorder is also an instance of multicriteria optimization (Ehrgott, 2005), also known as vector optimization (Jahn, 2009), since it is concerned with the simultaneous optimization of a finite number of objective functions (1). Notice strict and injective monotones belong to the scalarization techniques (Bevilacqua et al., 2018b;Ehrgott, 2005;Jahn, 2009) in vector optimization and always exist, again by Proposition 5. Preordered spaces from the next general class, the ones with strict monotones, include general relativity. Spacetime can be studied as a pair ðM; " C Þ where M is a set of events and " C is a causal relation, a partial order specifying which events can influence others, which lie to the future of others (Bombelli et al., 1987). A usual question is to establish sufficient conditions on ðM; " C Þ for the existence of strict monotones, which are referred to as time functions (Minguzzi, 2010) and are usually required to be continuous according to some topology. The study of physically plausible conditions from which countable multi-utilities or injective monotones can be constructed has, to our knowledge, not been addressed yet in the field. Notice, spacetime was originally approached through a differentiable structure (M, g), where M is a manifold and g a metric, and was only later studied as a partial order (Bombelli et al., 1987).
A final example from the most general class of preorders, the one where only monotones exist, are social welfare relations (SWR) in economics. A SWR is a partial order " S defined on the countably infinite product of the unit interval X :¼ Q n2N Â 0; 1 Ã . A SWR is said to be ethical if (1) given x; y 2 X with some i; j 2 N such that x i ¼ y j , y i ¼ x j and x k ¼ y k 8k 6 2 fi; jg we have x $ S y and (2) given x; y 2 X where x i y i 8i 2 N and x j \y j for some j 2 N then x 0 S y. Any ethical SWR is an example of a preordered space without strict monotones (Banerjee & Dubey, 2010, Proposition 1) and, thus, without both injective monotones and countable multi-utilities.
Monotones and topology. While we have focused on preordered spaces and left some brief comments regarding topology for Appendix A.3, in the past they have been often studied together. The original interest in functions representing order structures was concerned with (continuous) utility representations of total topological preordered spaces (Debreu, 1954(Debreu, , 1964Eilenberg, 1941). Of particular importance were results concerning the existence of a continuous utility function for both connected and separable total topological preordered spaces (Eilenberg, 1941) and for second countable total topological preordered spaces (Debreu, 1954). Among the classical results we also find the existence of an order isomorphism between a subset of the real numbers and any total order with countably many jumps whose order topology is second countable (Fleischer, 1961). Based on the work of Nachbin (1965) relating topology and order theory, in particular a generalization of Urysohn's separation theorem, the classical results where reproved and sometimes generalized for example in Bosi et al. (2020b), Herden (1989) and Mehta (1977Mehta ( , 1986aMehta ( , b, 1988.
Multi-utility representations. The study of non-total order structures was introduced in Aumann (1962). Representation of non-total preorders by multiutilities came later and was remarkably developed in Evren and Ok (2011). Although strict monotones can be traced back to Peleg (1970) and Richter (1966), there continue to be advances in the field (Bosi et al., 2020a;Herden & Levin, 2012;Rébillé 2019). In fact, it was only recently in Minguzzi (2013) where strict monotone multi-utilities were introduced and later in Alcantud et al. (2013Alcantud et al. ( , 2016 where they were further studied. The relation of these ideas with optimization and the existence of maximal elements is also present in the literature (Bevilacqua et al., 2018a, b;Bosi & Zuanon, 2017;Bosi et al., 2018;White, 1980). Countable multiutilities where studied particularly in Alcantud et al. (2016), Bevilacqua et al. (2018c), while finite multi-utility representations were notably advanced in Kaminski (2007) and Ok (2002) and, in vector optimization, in Jahn (2009. Open questions. While we have shown the existence of injective monotones for the widely studied class of preorders with countable multi-utilities, our construction is impractical since it relies on an infinite sum. For specific applications, injective monotones with a simpler representation are of interest. In general, any of the disciplines where these ideas are applied would benefit from a better understanding of the classification of preordered spaces in terms of real-valued monotones. For example, regarding the maximum entropy principle, the classification could be useful to reconsider the reasoning behind the choice of Shannon entropy. Even though there have been many principled approaches to ''derive'' Shannon entropy as a measure of uncertainty in the past, such as Aczél et al. (1974) and Shore and Johnson (1980), and for many practical purposes its appealing properties overweigh the bias in choosing this particular strict monotone, the question remains whether one should maximize entropy or maximize uncertainty. Quantum physics could also benefit as, for instance, the preorder underlying entanglement catalysis, trumping, is not well understood (Müller & Pastena, 2016). Many relevant open questions related to our work can also be found in Bosi et al. (2020a), for example, while we have focused mostly on preordered spaces and made some remarks on semicontinuity, it would be important to study continuous injective monotones in terms of topological properties of the underlying spaces, as in the classical works on utility functions.

Entropy and the uncertainty preorder
In the following, we provide proofs for statements regarding the uncertainty preorder " U and entropy H that appear throughout the main part of this article, in particular, all results are stated with respect to the preordered space ðP X ; " U Þ, for a finite set X.

Lemma 4 (Basic facts)
(i) Shannon entropy is a strict monotone. If jXj ! 3 then it is not an injective monotone. (ii) If jXj ! 3 then for all c 2 ð0; log jXjÞ, there is an uncountable set S c such that HðsÞ ¼ c 8s 2 S c . In particular, there are p; q 2 P X with c ¼ HðpÞ ¼ HðqÞ but p ffl q for all c 2 ð0; log jXjÞ.

Proof
(i) Strict monotonicity of H comes from the fact HðpÞ ¼ P jX i¼1 f ðp i Þ where f ðxÞ ¼ Àx logðxÞ is a strictly convex function. Given any other strictly convex f, strict monotonicity will still hold. One can find the details in (Marshall et al., 1979, C.1.a). H is not an injective monotone for jXj ! 3 by (ii). (ii) Given p; q 2 P X , we denote by pq the segment with endpoints p, q. Consider u 2 P X the uniform distribution, e i ; e j 2 P X Dirac distributions for two different elements i; j 2 X and some c 2 À 0; log jXj Á . Consider some c 0 s.t. 0\c 0 \minfc; HðmÞg where m is the middle point of e i e j . By the intermediate value theorem, there exists some r 2 e i m such that HðrÞ ¼ c 0 . Consider now a parametrization of e i r: fr t g t2½0;1 and define ' t :¼r t u for each t 2 ½0; 1. Again by the intermediate value theorem, since Hðr t Þ\c 8t 2 ½0; 1, there exists some p t 2 ' t such that Hðp t Þ ¼ c 8t 2 ½0; 1. By construction, given t; t 0 2 ½0; 1 t 6 ¼ t 0 we have p t 6 ¼ p t 0 since ' t \ ' t 0 ¼ fug whenever t 6 ¼ t 0 which means fp t g t2½0;1 is uncountable. In particular, there are t c ; t 0 c 2 ½0; 1 t c 6 ¼ t 0 c such that p t c ffl p t 0 c and Hðp t c Þ ¼ Hðp t 0 c Þ ¼ c for for every c 2 ð0; log jXjÞ. h Lemma 5 (Debreu separability) (i) If jXj ¼ 2 then ðP X ; " U Þ is order separable. In particular, ðP X ; " U Þ is Debreu separable for jXj ¼ 2. (ii) If jXj ! 3 then any subset Z P X which is Debreu dense in ðP X ; " U Þ has the cardinality of the continuum jZj ¼ c. (iii) For any jXj\1, there exists a countable upper dense set Z P X . Proof For simplicity of notation, in the following, we omit the subscript U and thus write " for " U (analogously for ffl and 0).

(i)
Consider p; q 2 P such that p 0 q. By definition, we have q # 1 \p # 1 . Consider some s 2 Q such that q # 1 \s\p # 1 . Notice by normalization 1 2 q # 1 \s and by normalization again 1 À s\s. Thus, p 0 r 0 q where r # :¼ ðs; 1 À sÞ and Q 2 \ P X is countable and order dense in ðP X ; " U Þ for jXj ¼ 2. In particular, ðP X ; "Þ is Debreu separable for jXj ¼ 2 which we could have known applying Theorem 1.4.8 in Bridges and Mehta (2013) since for jXj ¼ 2 there is a utility function, u 1 .
Notice q 0 p. Notice for any x 2 À 1 2 ; 1 Á we can define a pair q x ; p x 2 P X such that q x 0 p x as we did before where for any t 2 P X such that q x " t " p x we have t # 1 ¼ x. Given Z P X a subset which is Debreu dense in ðP X ; "Þ there exists for any x 2 À 1 2 ; 1 Á some z x 2 Z such that q x " z x " p x . Fix for every x 2 À 1 2 ; 1 Á some z x . Notice, given x; y 2 À 1 2 ; 1Þ, then z x ¼ z y implies x ¼ ðz x Þ # 1 ¼ ðz y Þ # 1 ¼ y which means that u : ð 1 2 ; 1Þ ! Z; x7 !z x is injective, implying c jZj. Since Z P X and jP X j ¼ c we have jZj ¼ c. In case jXj [ 3, any Debreu dense subset would also be Debreu dense in the subset with jXj ¼ 3. We can thus follow the above lines and get the same conclusion for any jXj ! 3.
(iii) Consider x; y 2 P X such that x ffl y. Since x ffl y, there exist n; m jXj À 1 such that P n i¼1 x # i \ P n i¼1 y # i and P m i¼1 x # i [ P m i¼1 y # i . Notice y # i \1 8i jXj since in the opposite case y " x 8x 2 P X . Consider 1\k jXj the largest integer such that y # k [ 0 and define f i g kÀ1 i¼1 where Notice m\k since the opposite case leads to \ Q such that q i ! q iþ1 and q k ¼ 1 À P kÀ1 i¼1 q i . Then z :¼ ðq 1 ; q 2 ; ::; q k ; 0; ::; 0Þ has jXj À k zeros, the same number of zeros as y, and z ¼ z # , since q k \1 À P kÀ1 i¼1 y # i ¼ y k y kÀ1 \q kÀ1 . By construction, we have where in the first inequality, we applied z # j ¼ q j \y # j þ j 8j m and in the second, we applied the definition of m by which P m j¼1 j \ P m j¼1 x # j À P m j¼1 y # j . Thus, x ffl z 0 y. We have shown Q jXj \ P X is a countable upper dense set in ðP X ; " U Þ for any jXj\1. h

Proofs
Preorders without strict monotones (Alcantud et al., 2016, Corollary 2.2) For example, consider the power set of the reals equipped with set inclusion, ðPðRÞ; Þ. Since is reflexive, transitive, and antisymmetric (i.e., a partial order), by Szpilrajn extension theorem, there exists a totally ordered space ðPðRÞ; "Þ extending ðPðRÞ; Þ, respecting the relations that already exist and relating the incomparable elements (e.g., overlapping intervals). Hence, if there was a strict monotone v : PðRÞ ! R, then vðUÞ ¼ vðVÞ for some U; V R would imply that U ¼ V, because w.l.o.g. U " V, and U 0 V cannot hold since v is a strict monotone. This contradicts Cantor's theorem by which the cardinality of the power set PðRÞ is strictly greater than that of R.
(ii) Relying on Lemma 3 and (i), we can take ðA n Þ n2N defined as in (i) assuming v is an injective monotone. Given x; y 2 X such that x ffl y, we have vðxÞ 6 ¼ vðyÞ which implies there exists some q n 2 Q between v(x) and v(y), i.e., either x 2 A n and y 6 2 A n or y 2 A n and x 6 2 A n .

Semicontinuity
Much of the economic literature on utility representations in preordered spaces is concerned with topological questions, in particular, under which conditions on the preordered space one can expect that monotones and utilities satisfy certain continuity properties (e.g., Alcantud et al., 2016;Debreu, 1964;Mehta, 1986a). This is particularly important for optimization, since continuous functions attain their maximal elements on compact sets. Therefore, in this section, we collect the continuity properties of the injective monotones that appear in the main part of this article. Given a topology s, a triple ðX; "; sÞ is called a preordered topological space. A function f : ðX; sÞ ! ðR; s nat Þ, where s nat is the topology given by the Euclidean metric, is said to be upper semicontinuous if f À1 ððÀ1; rÞÞ 2 s 8r 2 R.
Upper semicontinuous functions retain the property of continuous functions that they assume their maxima on compact sets, that is, they are effective on any compact set B X.
Similarly, we say ðX; "; sÞ is upper semicontinuous if iðxÞ ¼ fz 2 Xjx " zg is closed 8x 2 X. We may abuse notation and say that " is upper semicontinuous whenever X and s are clear. Notice, the uncertainty preorder " U is upper semicontinuous with respect to the Euclidean topology, since iðpÞ ¼ fq 2 P X jp " u qg ¼ \ jXjÀ1 i¼1 fq 2 P X ju i ðpÞ u i ðqÞg ¼ \ jXjÀ1 i¼1 u À1 i ð½u i ðpÞ; 1ÞÞ where u À1 i ð½u i ðpÞ; 1ÞÞ is closed, because all u i are upper semicontinuous.

Proof
(i) If there exists an upper semicontinuous injective monotone, then we can construct w.l.o.g. an upper semicontinuous injective monotone c : X ! ð0; 1Þ. Since " is upper semicontinuous, we know v iðxÞ is upper semicontinuous, and given the fact the class of upper semicontinuous functions is closed under addition by Proposition 1.5.12 in Pedersen (2012), c x in the proof of Proposition 4 is upper semicontinuous 8x 2 A c . Thus, fcg [ fc x g x2A c is an upper semicontinuous injective monotone multi-utility of ðX; "; sÞ. (ii) Take ðu m Þ m2M , ðA n Þ n2N and c defined as in the proof of Proposition 5. If ðu m Þ m2M is upper semicontinuous then 8n 2 N A n ¼ u À1 m n ð½q n ; 1½Þ 2 s c and v A n ðxÞ is upper semicontinuous 8n 2 N. The class of upper semicontinuous function is closed under addition, product by positive scalars and uniform convergence by Proposition 1.5.12 in Pedersen (2012). By the first two c N :¼ P N n¼0 3 Àn v A n is upper semicontinuous 8N 2 N and by the third c ¼ lim N!1 c N is upper semicontinuous. (iii) Following (ii), we get fcg [ fc n 1 ;n 2 g n 1 \n 2 defined as in the proof of Proposition 6 consists of upper semicontinuous injective monotones. (iv) Notice whenever ðA n Þ n2N in the proof of Proposition 7 is defined through an upper semicontinuous function, either a strict monotone or an injective monotone, then A n is closed 8n 2 N. Conversely, we can follow the proof of (ii) to get upper semicontinuity for both a strict monotone and an injective monotone constructed as in Lemma 3. (v) We again only show (i) implies (iii) in Proposition 10. If " is upper semicontinuous then v iðdÞ is upper semicontinuous 8d 2 D and since we can choose u to be an upper semicontinuous strict monotone by hypothesis we get fug S fv iðdÞ g is an upper semicontinuous countable multi-utility. h Availability of data and material Not applicable.
Code availability Not applicable.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.
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