Debreu-like properties of utility representations

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Journal of Mathematical Economics 44 (2008) 1161–1179

Debreu-like properties of utility representations

A. Caserta a, A. Giarlotta b,∗, S. Watson c

a Department of Mathematics, Second University of Naples, Via Vivaldi, Caserta 81100, Italyb Department of Economics and Quantitative Methods, University of Catania, Corso Italia 55, Catania 95129, Italy

c Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto M3J 1P3, Canada

Received 12 April 2007; received in revised form 29 December 2007; accepted 22 January 2008Available online 3 February 2008

Abstract

Traditionally the codomain of a utility function is the set of real numbers. This choice has the advantage of ensuring theexistence of a continuous representation but does not allow to represent many preference structures that are relevant to utility theory.Recently, some authors have started a systematic study of utility representations that are not real-valued, introducing the notion of aDebreu chain. We continue their analysis defining two Debreu-like properties, which are connected to a local continuity of a utilityrepresentation. The classes of locally Debreu and pointwise Debreu chains here introduced enlarge the class of Debreu chains. Wegive several examples and analyze some properties of these two classes of chains, with particular attention to lexicographic products.© 2008 Elsevier B.V. All rights reserved.

JEL classification: C60; D11

Keywords: Debreu chain; Locally Debreu chain; Pointwise Debreu chain; Utility function; Continuous representation; Lexicographic ordering

1. Introduction

Let (X,≺) and (Y, <) be linear orders (also called chains). We say that (X,≺) is representable in (Y, <) if there existsan order-embedding f : X ↪→ Y , i.e., for each x, y∈X, x ≺ y implies f (x) < f (y). (Note that since X and Y are chains,also the reverse implication holds, namely, f (x) < f (y) implies x ≺ y.) In utility theory, the chain (Y, <) is called thebase of the representation, the order-embedding f a utility function and the pair ((Y, <), f ) a utility representation of(X,≺).

Traditionally, the literature on utility theory has been dealing with utility representations such that the base chain(Y, <) is the set R of real numbers, endowed with its natural order <. Specifically, much attention has been devotedto the following topics: (i) the existence of an R-valued utility function; (ii) the continuity of such a function. In thissection, we present a brief outline of these topics. With this respect, our goals are: (i) to justify alternative types ofutility representations; (ii) to suggest suitable continuity properties that a utility representation should satisfy.

∗ Corresponding author.E-mail addresses: [email protected] (A. Caserta), [email protected] (A. Giarlotta), [email protected] (S. Watson).

0304-4068/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2008.01.003

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dx.doi.org/10.1016/j.jmateco.2008.01.003

1162 A. Caserta et al. / Journal of Mathematical Economics 44 (2008) 1161–1179

1.1. Utility representations of a chain

The first characterization of the representability of a chain in R most likely appeared in Milgram (1939): A linearorder (X,≺) is order-embeddable in R if and only if it includes a countable subset that is ≺-dense in X. (Recall thatY ⊆ X is ≺-dense in X if for each x1, x2 ∈X such that x1 ≺ x2 there exists y∈Y with the property that x1 � y � x2.)Similar characterizations were given by Birkhoff (1948).

Nevertheless, due to an imperfect communication in the scientific community, until the early 50’s economistsconsidered all preference relations (hence all chains) as representable in R. In his famous paper on the Open GapLemma, Debreu (1954) finally exhibited an example of a natural preference relation that fails to be representable in R:the lexicographic power (R2,≺lex).

Later on, other authors obtained similar characterizations of the representability of a chain in R. For example,Fleischer (1961a) proved the following result: A chain (X,≺) is representable in R if and only if it has at mostcountably many jumps and the topological space (X, τ≺) is separable. (A jump in a chain (X,≺) is a pair (a, b)∈X2

such that a ≺ b and the open interval (a, b) is empty. The order topology τ≺ on X is the topology generated by all openintervals; the topological space (X, τ≺) is separable if it contains a countable set D that is topologically dense in X, i.e.,D intersects each non-empty open set of X.) For other characterizations of the representability in R and an overviewof the literature on the topic, the reader is referred to the book by Bridges and Mehta (1995).

In dealing with the existence of utility representations, a natural question arises: Why do we consider onlyR-valuedutility functions as meaningful representations of a chain? As discussed in Mehta (1998), the literature on utilityrepresentations mostly deals with utility functions having (R, <) as a codomain. Regrettably, the very same literaturelacks a systematic and convincing discussion that explains the choice of R as the (unique) base chain of a utilityrepresentation. Most likely the reasons for this choice are both historical and technical. Indeed, most economists havealways considered somehow natural that utility is a real number. Furthermore, the choice of real-valued utility functionsmight be linked to the fact that the linearly ordered topological space (R, <, τ<) is a simple and amenable mathematicalobject, with a lot of important properties (metrizability, completeness, separability, etc.).

Nevertheless, as Herden and Mehta (2004) sharply point out, these arguments are not so inexpugnable as theyappear to be. The two authors list some unpleasant consequences of the practice of considering only R-valued utilityrepresentations and excluding any other type of representation. In syntony with their classification, we can identifytwo kinds of problems: (a) mathematical, which in turn can be order-theoretical or cardinal; (b) theoretical.

(a) If we insist in using only real-valued utility functions, then many preference relations that arise quite naturally inmathematical economics fail to be representable. The mathematical obstruction to their representability in R canbe of two types: order-theoretical and cardinal. Historically, the most significant example of the first kind is thelexicographic power (R2,≺lex); this chain is not representable in R because it does not satisfy the countable chaincondition (i.e., there are uncountably many pairwise disjoint non-empty open intervals). But there are several otherchains relevant to utility theory that fail to be representable inR only for structural reasons. The long line, which isthe lexicographic product ω1×lex[0, 1) with his first point (0, 0) removed, is such an instance. The importance ofthis chain in economic theory is widely acknowledged (Monteiro, 1987; Estevez and Herves, 1995). The structuralreason for which the long line cannot be order-embedded intoR is that it contains a copy of ω1, the first uncountableordinal. For a throughout analysis of order-theoretical obstructions to the representability of a chain in R, we referthe reader to two papers by Beardon et al. (2002a, b).

Cardinal obstructions to representability inR are quite frequent as well. In fact, some chains, which are relevantto economic theory, are simply too large to be embedded in R. Herden and Mehta (2004) give some examples ofcommodity spaces recently studied in economic theory, which have a cardinality greater than the cardinality c ofthe continuum and therefore cannot be embedded in R. A first example of this kind is the infinite-dimensionalcommodity space L∞(μ) of essentially bounded measurable functions on a measure space; in most models usedin general equilibrium theory (Bewley, 1972), this chain is too large to be embedded in R. Another example ofa chain that cannot be embedded in R for cardinality reasons is the space (Rn)R of all functions from R to thecommodity space Rn, used in capital theory (Diamond, 1965).

(b) Apart from mathematical reasons, there are also theoretical reasons to consider utility representations that use abase chain different fromR. In fact, the use ofR to represent preferences might even clash with the very concept ofutility. In a paper on the foundations of utility (Chipman, 1960), the author argues that utility is not a real number,

A. Caserta et al. / Journal of Mathematical Economics 44 (2008) 1161–1179 1163

but a vector that is inherently lexicographic in nature. He proposes to use the lexicographic power (2α,≺lex) as abase of utility representations. (Here 2 = {0, 1} is the chain with two elements, and α is a suitably chosen ordinalnumber.) Chipman points out that the use of a transfinite sequence of length α in place of a real number to representpreferences is convenient from both a mathematical and an economic point of view. Mathematically, every chainbecomes representable if we use (2α,≺lex) as a base of the representation, provided that the exponent α is suitablychosen. Furthermore, from an economic point of view, the concept of utility might be easier to understand if wecodify it as a vector rather than as a real number. Last but not least, representability of a chain (X,≺) in R requiresthat the topological space (X, τ≺) has a countable base, which has no intuitive meaning from the economic point ofview (Chipman, 1971). For an extensive analysis of a notion of lexicographic utility and alternative types of utilityrepresentations (including those involving non-standard analysis), the reader is referred to the excellent survey byFishburn (1974).

The interest for alternative utility representations is also witnessed by a systematic analysis of the family of chainsthat are not representable in R. The most important result in this direction is a subordering classification of this familyof chains (Beardon et al., 2002a):

Theorem 1. A chain X is non-representable inR if and only if it contains a subchain that is order-isomorphic to either(i) the first uncountable ordinal ω1 or its reverse ordering ω∗1, or (ii) a non-representable subchain of the lexicographicplane (R2,≺lex), or (iii) an Aronszajn line.

(An Aronszajn line is an uncountable chain, which contains no uncountable subchain that is order-isomorphic toeither ω1 or ω∗1 or a subchain of R.)

Other results in this direction aim at refining this classification. In Giarlotta (2004a, Chapter 5), we introduce thenotion of small chain, i.e., a linear order that has no subchain that is order-isomorphic to either ω1 or ω∗1 or an Aronszajnline. We describe a suitable hierarchy of small chains, which are ranked according to their degree of “lexicographiccomplexity”.

In the light of the above discussion, it seems natural to consider other possible utility representations. The alternativerepresentations examined in the literature are essentially of two types: (a) representations that use non-standard tools;(b) representations that use a base chain M different from R. Concerning option (a), we have already mentioned thepossibility of using non-standard analysis to ensure representability of preference relations that would otherwise fail tobe representable in a standard way (see Fishburn, 1974 and references therein). More recently, Campion et al. (2006a)have also suggested the possibility of using fuzzy real numbers (endowed with some useful ordering) instead of theusual non-fuzzy real numbers to represent some classes of preferences.

Option (b) has been examined in much more detail. With this respect, most base chains M proposed in the literatureare lexicographic products. We have already discussed the choice of a suitable lexicographic power (2α,≺lex) as a baseof utility representations. Wakker (1988) proves some interesting results about the lexicographic product (R× 2,≺lex),which make this linear order a possible choice as base chain of a representation. More recently, Knoblauch (2000)considers the lexicographic power (Rn,≺lex) as a suitable choice to represent a certain class of preference relations.Even more recently, Campion et al. (2006b, c) prove that the long line (which is a special lexicographic product) is themost suitable continuous chain defined on a separably connected topological space (see Candeal et al., 1998 for thisnotion). More generally, one can think of developing a theory of utility representations in which the base chain is alexicographic product of linear orders. This choice is motivated by the following economic considerations.

In the process of modeling multidimensional preferences, it is necessary to endow the Cartesian product of somegiven chains with a linear order. In this context, lexicographic utility structures arise quite naturally (Knoblauch,2000). These types of structures are linked to the existence of some factors which are “overwhelmingly more impor-tant” than others. For example, assume that there are n factors X1, . . . , Xn of concern to the decision maker. Anelement xi ∈Xi is a “level of the factor Xi” (e.g., in an allocation problem, xi may be the resources allocated to thei-th activity). Then X:=X1 × · · · ×Xn is the set on which a preference-order ≺ has to be established by the deci-sion maker. A lexicographic modeling of utilities requires finding whether there exist n individual utility functionsui : X→ R, i = 1, . . . , n, such that for each x = (x1, . . . , xn), y = (y1, . . . , yn)∈X, we have x ≺ y if and only if(u1(x), . . . , un(x))≺lex(u1(y), . . . , un(y)), where ≺lex is the lexicographic ordering on Rn.

In an attempt to study lexicographic products of chains from the perspective of utility theory, we have to distinguishtwo types of situations: (i) lexicographic products that are representable in the sense of classical utility theory (i.e., they


can be order-embedded intoR); (ii) lexicographic products that are non-representable inR. The economic significanceof those chains that belong to the first class is that in these cases the set of real numbers suffices to represent lexicographicmultidimensional preferences. Representable lexicographic products have been analyzed and characterized (Giarlotta,2004b).

On the other hand, there are cases in which lexicographic multidimensional preferences can be represented onlyby using chains that are more complicated than R. In these cases, a feasible approach is to classify chains accordingto a measure of their lexicographic complexity, taking the point of view that a chain that can be order-embedded in alexicographically power of the real numbers is representable, even if in a weaker sense. Such an ordering is realized ina way that is more complex than for suborderings of R, but which still fits within the general utility concept.

We call a chain (X,≺) α-representable in R if it can be order-embedded into the lexicographic power (Rα,≺lex),where α is an ordinal number (Giarlotta, 2005). This corresponds to having a representation of the preference ordering≺ in X by a well-ordered family of utility functions fξ : X→ R indexed by the ordinal numbers ξ < α; for any x, y∈X

one has x ≺ y if and only if fβ(x) < fβ(y) holds, where β is the least ordinal number below α at which fβ(x) and fβ(y)differ. One can think of the ordinal indices as determining the relative importance of the utility functions fξ . The leastordinal number α for which a chain L is α-representable in R is called the representability number of X in R. Moregenerally, for any pair of chains (X, Y ), we define the representability number of X in Y as the least ordinal number α

such that X can be order-embedded into (Yα,≺lex); this ordinal number is denoted by reprY (X). In Giarlotta (2005),we determine the value of reprY (X) for several pairs of chains (X, Y ), with particular attention to the case Y = R.

1.2. Continuous utility representations and Debreu chains

Within the theory of continuous utility representation, a cornerstone is given by the famous Open Gap Lemma:

Theorem 2 (Debreu, 1954). For each subset S of the extended real line R, there exists an order-embedding f : S ↪→ Rwhose image f [S] has only open gaps.

(Recall that a gap of Y ⊆ R in R is a maximal interval, disjoint from Y, which has both a lower bound and an upperbound in Y.) Many different proofs of the Open Gap Lemma have been given. This is justified by the importance of thisresult not only in mathematical utility theory and the theory of orders, but also in other fields of research. For instance,the Open Gap Lemma has found applications in mathematical psychology, ordinal data analysis and thermodynamics.For an account of this topic, see Herden and Mehta (2004) and references therein.

The most important consequence of the Open Gap Lemma in the theory of continuous utility representations is theso called Debreu Representation Lemma:

Corollary 3. A chain X is representable in R if and only if it is continuously representable in R.

(Here “continuous” means continuous with respect to the order topology on both X and R.) In other words, theDebreu Representation Lemma states that the chain (R, <) has the Debreu property, i.e., for each linear order (X,≺),there exists a utility representation of (X,≺) in (R, <) if and only if there exists a continuous utility representation of(X,≺, τ≺) in (R, <, τ<). This additional property of the set of real numbers might be a further justification to employonly R-valued utility representations.

In this context, it is natural to ask whether there are chains, different fromR, which satisfy the Debreu property. Thisquestion was first put by Beardon (1988, 1993, 1994) and then originated the work by Herden and Mehta (2004). Theissue is quite relevant in utility theory, because a positive answer to the above question would motivate the developmentof a theory of continuous utility representations, which use a base chain different from R.

In their paper, Herden and Mehta (2004) extensively study the Debreu property. They show that only few chains dosatisfy this property: apart from R, the most interesting Debreu chain is the long line. On the other hand, many linearorders that might be possible candidates as base chains for utility representations fail to satisfy the Debreu property.For example, the following lexicographic products are not Debreu chains:

• (R× 2,≺lex);• (Rα,≺lex) for each ordinal number α ≥ 2;• (Xα,≺lex) for each non-trivial chain X and uncountable ordinal α;• (Xα,≺lex) for each totally disconnected chain X and infinite ordinal α.


Thus, in particular, the lexicographic power (2α,≺lex) does not satisfy the Debreu property for each infiniteordinal α.

Admittedly, the limited amount of (economically significant) linear orders that satisfy the Debreu property mightbe an additional argument to support the choice of R-valued utility representations. Nevertheless, this fact canalso be interpreted as a suggestion that the Debreu property is somehow too narrow, because identifies existenceof utility functions with existence of continuous ones. Indeed, in the process of choosing a suitable base chainY for utility representations, we detect a tension between two facts: (i) continuity properties satisfied by Y; (ii)type and amount of linear orders that are representable in Y. This paper is motivated by a possible way to find acompromise between (i) and (ii). To this aim, we consider some meaningful weakenings of the Debreu property,which are desirable from the point of view of utility representations but are more likely to hold then the Debreuproperty.

The Debreu-like properties introduced in this paper, namely, the pointwise Debreu and locally Debreu properties,still link existence and continuity of a utility representation, but in a local sense rather than in a global one. Recallthat a chain (X,≺) is Debreu if for each Y ⊆ X, there exists a continuous order-embedding f : (Y, τ≺) ↪→ (X, τ≺).On the other hand, we say that (X,≺) is locally Debreu if for each Y ⊆ X and y∈Y , there exists an order-embeddingfy : (Y, τ≺) ↪→ (X, τ≺), which is continuous at y. Further, X is pointwise Debreu if for each Y ⊆ X and y∈Y , thereexists an order-preserving function fy : (Y, τ≺)→ (X, τ≺), which is continuous at y and injective at y (i.e., the preimageof f (y) is the singleton {y}). Note that the class of pointwise Debreu chains properly contains the class of locally Debreuchains, which in turn enlarges the class of Debreu chains.

We carry out our analysis trying to keep notation as simple as possible. In fact, we avoid using more sophisticatedmathematical tools, which would give more general results, but at the price of making the exposition complicated.However, some results obtained in this paper can be extended using the tool of resolutions of topological spaces,because lexicographic products can be seen as order resolutions of chains. (See Watson (1992) for an overview ofresolutions of topological spaces, and Caserta et al. (2006) for order-resolutions of linearly ordered topological spaces.)

The paper is organized as follows. In Section 2, we introduce some basic terminology and results. Section 3 is rathertechnical and collects most of the mathematical tools needed in the remainder of the paper. Section 4 is the core of thepaper: here we introduce the notions of locally Debreu and pointwise Debreu, give several examples and prove somerelated results. Section 5 groups conclusions and future directions of research.

2. Basic definitions

In this section, we collect some elementary notions about linear orderings, ordinal numbers and topological spaces.Related definitions and properties can be found in most book on the topics: see Rosenstein (1982) for linear orderings,Kunen (1980) for ordinal numbers and Engelking (1989) for topological spaces. We also prove a simple set-theoreticresult about ordinal numbers, which will be useful in the next section.

A chain is a linearly ordered set (X,≺). In this paper, we assume, unless otherwise specified, that all chains underexamination have at least two elements. For each a, b∈ (X,≺), the notation a � b stands for a ≺ b or a = b. We useindifferently a ≺ b and b a to denote that a is a predecessor of b (b is a successor of a). In particular, we say thata is an immediate predecessor of b (b is an immediate successor of a) if a ≺ b and there exists no c∈X such thata ≺ c ≺ b; in this case, the pair (a, b)∈X2 is called a jump of X.

The reverse ordering of a chain (X,≺) is the chain (X,≺∗), whose linear order is defined by a≺∗b if and only ifb ≺ a; we often simplify notation and write X∗. Note that X has a minimum (maximum) element if and only if X∗ hasa maximum (minimum) element. Further, we have (X∗)∗ = X.

A subchain of (X,≺) is a set Y ⊆ X, endowed with the induced order. For any two subchains A and B of (X,≺),the notation A ≺ B means that a ≺ b for each a∈A and b∈B. In the case that A = {a}, we simplify notation andwrite a ≺ B in place of {a} ≺ B; a similar meaning has the notation A ≺ b. We say that (X,≺) is Dedekind completeif each subchain of X that has an upper (respectively, lower) bound also has a least upper bound (respectively, greatestlower bound). The Dedekind completion of X is a Dedekind complete chain (C,≺) such that X is a subchain of C andeach Dedekind complete subchain of C does not contain X. The Dedekind completion of X is essentially unique (seeRosenstein, 1982, p.37).

Intervals and rays in (X,≺) are denoted as usual. Namely, for each a, b∈X such that a � b, let (a, b):={x∈X : a ≺x ≺ b} and [a, b]:={x∈X : a � x � b}. (Observe that an open interval can possibly be empty and each singleton is


a closed interval.) The half-open intervals [a, b) and (a, b] are defined similarly. Further, let (←, b):={x∈X : x ≺ b}and (a,→):={x∈X : a ≺ x}. The closed rays (←, b] and [a,→) are defined similarly. If Y is a subchain of X anda, b∈Y are such that a � b, then we denote by (a, b)Y the corresponding open interval in Y, i.e., (a, b)Y :=(a, b) ∩ Y .The notations [a, b)Y , [a,→)Y , etc. have a similar meaning.

A homomorphism is an order-preserving function f : X→ Y between two chains, i.e., for all a, b∈X, if a � b

then f (a) � f (b). To say that Y is a homomorphic image of X means that there exists a surjective homomorphism ofX onto Y. An embedding is an injective homomorphism, i.e., for all a, b∈X, if a ≺ b then f (a) ≺ f (b). The notationX ↪→ Y stands for embeddability of X into Y. (Thus, a utility function for a chain X is an embedding f : X ↪→ R, andX is representable if X ↪→ R.) An isomorphism is a bijective homomorphism. We use the notation X ∼= Y to mean thatthere exists an isomorphism from X onto Y. (Sometimes we abuse notation and identify isomorphic chains, writingX = Y in place of X ∼= Y .) For each isomorphism class of chains, we choose a canonical representative, which iscalled the order-type of that class. An automorphism of X is an isomorphism of X onto itself. To avoid ambiguity (orsimply to emphasize the order-theoretic nature of a function), sometimes we write order-homomorphism (respectively,order-embedding, etc.) in place of homomorphism (respectively, embedding, etc.).

By R, Q, Z and N we mean the chains (R, <), (Q, <), (Z, <) and (N, <), where < is their natural order. Observethat R ∼= R∗, Q ∼= Q∗ and Z ∼= Z∗.

A set T is transitive if every element of T is a subset of T. An ordinal number (for short, an ordinal) is a transitiveset, which is well-ordered by the ∈ -relation. Ordinal numbers will be denoted by α, β, γ , etc. A non-zero ordinal α isa successor if it has an immediate predecessor, i.e., α = β + 1 for some ordinal β; it is a limit otherwise. As usual, anordinal α is identified with the set of all ordinals below it. Thus, for all ordinals α and β, we have β < α if and onlyif β∈α; in particular, α+ 1 = α ∪ {α}. Recall that for any well-ordered set (W,≺), there exists a unique ordinal thatis isomorphic to it; this is the order-type of the isomorphism class of (W,≺), for simplicity called the order-type of(W,≺).

Since X ∼= X∗ for any finite chain X, we identify the natural number n and its reverse ordering n∗. The reverseordering α∗ of an infinite ordinal α is called a reverse ordinal. An anti-well-ordered set is the reverse ordering of awell-ordered set; its order-type is a reverse ordinal.

A cardinal number (for short, a cardinal) is an initial ordinal, i.e., an ordinal that cannot put in bijective correspon-dence with any ordinal below it. Note that an infinite cardinal is a limit ordinal. The first cardinal greater than a cardinalκ is denoted by κ+. The first infinite cardinal is the order-type of the natural numbers N and is denoted by ω. Thus, forexample, the set Z of integer numbers has order-type ω∗ + ω. The first uncountable ordinal (and cardinal) is denotedby ω1 (hence ω1 = ω+).

For any two ordinals α and β, we say that β maps cofinally into α if there exists a homomorphism f : β→ α suchthat for each ξ < α, there exists γ < β such that ξ ≤ f (γ). The cofinality of an ordinal α > 0 is the least ordinal β

that maps cofinally into α; this number, which is a cardinal, is denoted by cf(α). Therefore, a non-zero ordinal α is asuccessor if and only if cf(α) = 1, and is a limit if and only if cf(α) ≥ ω.

Further, we say that a chain Z embeds cofinally (respectively, coinitially) into another chain Y if there exists anembedding f : Z ↪→ Y such that for each y∈Y , there is z∈Z with the property that y � f (z) (respectively, f (z) � y).Note that if Y has a maximum (respectively, a minimum), then the chain with exactly one element embeds cofinally(respectively, coinitially) in Y.

A cardinal κ is regular if cf(κ) = κ, and is singular otherwise. In this paper, we consider 0 and 1 as regularcardinals with cofinality 0 and 1, respectively. Observe that the cofinality of an ordinal is a regular cardinal. Wedenote by Reg the class of all infinite regular cardinals; further, we set Reg:=Reg ∪ {0, 1}. For each ordinal α, letReg<α:={κ∈Reg : κ < α} and Reg<α:=Reg<α ∪ {0, 1}. A similar meaning have Reg≤α and Reg≤α. Besides, we

denote by Reg∗ the class of the reverse orderings of infinite regular cardinals. The notations Reg∗<α, Reg∗≤α, Reg∗,

Reg∗<α and Reg

∗≤α have an obvious meaning.

Let X be a non-empty set and α a non-zero ordinal. An element (xξ)ξ<α ∈Xα is a transfinite α-sequence in X; forsake of simplicity, we will slightly abuse terminology and call it a sequence, regardless of the countability of the ordinalnumber α. A sequence (xξ)ξ<α ∈Xα is eventually constant if there exist an element x∈X and an ordinal β < α suchthat for each ξ ≥ β, we have xξ = x; in this case, we also say that the sequence (xξ)ξ<α is eventually equal to x.

Let (Xi,≺)i∈ I be a family of chains indexed by a well-ordered set (I, <). The lexicographic product of (Xi)i∈ I

is the chain (∏

i∈ IXi,≺lex), where the order relation is defined as follows: for any a = (ai)i∈ I and b = (bi)i∈ I in


∏i∈ IXi, set a≺lexb if there exists an index i0 ∈ I such that ai0 ≺ bi0 and ai = bi for all i∈ I satisfying i < i0. We

denote this chain by∏lex

i∈ IXi. In particular, if the well-ordered set I is an ordinal α, we denote the correspondinglexicographic product by

∏lexξ<αXξ . Similarly, X×lexY denotes the lexicographic product of the two chains X and Y,

whereas Xαlex is the lexicographic product

∏lexξ<αXξ , with Xξ:=X for each ξ < α. In particular, X1

lex:=X and X0lex:=1

(the chain with exactly one element).If (Xi)i∈ I is a family of chains indexed by a linear order (I, <), then the sum of (Xi)i∈ I is the chain whose

underlying set is⋃

i∈ I{i} ×Xi and whose order ≺ is defined by (j, aj) ≺ (k, ak) if either j < k, or j = k and aj ≺ ak

in Xj . This chain is denoted by∑

i∈ IXi. (For a finite sum we use the notation X1 +X2 + · · · +Xn.) Observe that thelexicographic product X×lexY is equal to the sum

∑x∈XYx, where Yx = Y for each x∈X. Note also that for any two

chains X and Y, we have (X+ Y )∗ ∼= Y∗ +X∗.Let (X,≺) be a chain. The order ≺ induces a topology τ≺ on X, the so-called order topology. This is the topology

whose base is the family of all open intervals in X. Note that if Y is a subchain X, then there are two natural topologieson Y: (i) the subspace topology τY , obtained by considering Y as a subspace of the topological space (X, τ≺); and (ii)the order topology τ≺, induced by the restriction to Y of the linear order ≺. The subspace topology τY is finer than theorder topology τ≺.

Let f : Y → X be a function between two chains. We say that f is continuous if is continuous with respect to theorder topology on both chains. We are interested in continuous function that are, in addition, order-preserving. Inparticular, in the case that Y is a subchain of X, whenever we say that Y continuously embeds into X, we mean that thereexists a continuous order-embedding f : (Y, τ≺) ↪→ (X, τ≺).

We end this section with a set-theoretic lemma, which concerns homomorphic images of infinite ordinals and regularinfinite cardinals. This result will be useful in the next section.

Lemma 4. Let (X,≺) be a chain. The following statements hold:

(i) if α is an infinite ordinal and f : α→ X is a homomorphism, then its image im f is isomorphic to an ordinalβ ≤ α, which is either a successor ordinal or an infinite ordinal with the same cofinality as α;

(ii) if κ is a regular infinite cardinal and f : κ→ X is a homomorphism, then its image im f is isomorphic either toa successor ordinal β < κ or to κ.

Proof. We prove (i) only, since (ii) is an immediate consequence of (i). Let α be an infinite ordinal and f : α→ (X,≺)a homomorphism. First we show that the image of α under f is isomorphic to an ordinal β ≤ α. Indeed, the subchainimf ⊆ X is a well-ordered set, hence there exists an isomorphism ı : imf → β, where β is an ordinal. For each ξ < β,select αξ < α such that ı(f (αξ)) = ξ. By construction, the set {αξ : ξ < β} ⊆ α is isomorphic to β. It follows thatβ ≤ α. Now, if f is eventually constant, then there exists γ < α and x∈X such that for each ξ ≥ γ , we have f (ξ) = x.Thus, in this case β is a successor ordinal.

On the other hand, assume that f is not eventually constant. Let cf(α) = λ. We use transfinite recursion to constructa strictly increasing sequence (αξ)ξ<λ of ordinals smaller than α such that f restricted to the set {αξ : ξ < λ} is injectiveand (f (αξ))ξ<λ is cofinal in the image of f. This will imply that cf(β) = cf(α). Let γ < λ. We assume that αξ hasbeen defined for each ξ < γ , and define αγ . Since cf(α) = λ, the sequence (αξ)ξ<γ is not cofinal in α. Thus, thereexists δ < α such that αξ < δ for all ξ < γ . Since f is not eventually constant, there exists η < α such that δ < η andf (δ) ≺ f (η). Set αγ :=η. This completes the definition of the strictly increasing sequence (αξ)ξ<λ. It is easy to checkthat f restricted to the set {αξ : ξ < λ} is injective and (f (αξ))ξ<λ is cofinal in im f . Finally, we show that cf(β) = cf(α).Toward a contradiction, assume that cf(β) < λ = cf(α). Observe that cf(β) is an infinite cardinal. Let (xξ)ξ<cf(β) be

a strictly increasing sequence, which is cofinal in im f . For each ξ < cf(β), choose a point νξ ∈ f−1{xξ}. Since f isorder-preserving, it follows that (νξ)ξ<cf(β) is a cofinal sequence in α. This implies that cf(α) ≤ cf(β), which contradictsthe hypothesis. �

3. Coterminal pairs and hereditary coterminal pairs

In this section, we develop the machinery needed in the next section. Definitions and results are rather technical,but they will prove to be very useful in studying Debreu-like properties of linear orders.


First we introduce the notions of cofinality and coinitiality for both a chain (Definition 5) and an element of a chain(Definition 6). To avoid confusion, we use an upper case notation for chains (Cf(X), Ci(X), etc.) and a lower casenotation for elements of a chain (cf(x), ci(x), etc.).

Definition 5. Let (X,≺) be a chain and Y ⊆ X a subchain. The cofinality of Y /= ∅, denoted by Cf(Y ), is the uniqueregular cardinal κ that embeds cofinally into Y. Thus, Cf(Y ) = 1 if and only if Y has a maximum element. The coinitialityof Y, denoted by Ci(Y ), is the reverse ordering λ∗ of the unique regular cardinal λ such that λ∗ embeds coinitially intoY. Thus, Ci(Y ) = 1 if and only if Y has a minimum element. In particular, Cf(Y ) = Ci(Y ) = 1 for each finite subchainY. We set by definition Cf(∅) = Ci(∅):=0. Note that the definitions of cofinality and coinitiality of a subchain Y ⊆ X

are absolute (i.e., independent of the chain X).

Definition 6. Let (X,≺) be a chain, Y ⊆ X and y∈Y . The cofinality of y in Y, denoted by cfY (y), is equal to Cf(←, y)Y .(Recall that (←, y)Y :=(←, y) ∩ Y .) Similarly, the coinitiality of y in Y is ciY (y):=Ci(y,→)Y . We simplify notation incase Y = X, and write cf(y) in place of cfX(y); similarly, ci(y) stands for ciX(y). Note that, for each Y ⊆ X and y∈Y ,the four regular cardinals cfY (y), (ciY (y))∗, cf(y) and (ci(y))∗ might be all different.

More generally, a hereditary cofinality of y in Y is a regular cardinal κ (either infinite or equal to 0 or equal to 1) suchthat there exists a subchain A ⊆ Y such that A ≺ y and Cf(A) = κ. Equivalently, a hereditary cofinality of y in Y is aregular cardinal κ that embeds into the ray (←, y)Y . Thus, the cofinality of y in Y is a particular hereditary cofinality ofy in Y. Dually, a hereditary coinitiality of y in Y is a reverse regular cardinal λ∗ such that there exists a subchain B ⊆ Y

such that y ≺ B and Ci(B) = λ∗. Equivalently, a hereditary coinitiality of y in Y is a reverse regular cardinal λ∗ thatembeds into the ray (y,→)Y . The coinitiality of y in Y is a particular hereditary coinitiality of y in Y.

Note that 0 is both a hereditary cofinality and a hereditary coinitiality of each y in Y; further, 1 is a hereditarycofinality (respectively, coinitiality) of y in Y if and only if y is not the minimum of Y (respectively, the maximum ofY).

Remark 7. We keep the standard notation in the case that X is an ordinal α, and denote the cofinality of the “chain”α by cf(α), in place of Cf(α) (as it should be, according to Definition 5). This is justified by the fact that α is equal tothe set of all ordinals less than α, and so cf(α) can be seen as the cofinality of the “element” α in the class Ord of allordinal numbers, i.e., cf(α) = cfOrd(α).

Before introducing some additional notation, we prove two set-theoretic results, which are related to preservationof cofinalities and coinitialities under particular homomorphisms.

Lemma 8. Let κ be an infinite regular cardinal, (X,≺) a chain and x∈X. If there exists a strictly increasing sequence(xξ)ξ<κ of elements ≺ x converging to x, t hen cf (x) = κ.

Proof. Assume that such a sequence exists. The sequence (xξ)ξ<κ is a homomorphism f : κ→ X, where f (ξ):=xξ

for each ξ < κ. By hypothesis, the image of κ under f cannot be a successor ordinal. Therefore, Lemma 4(ii) impliesthat the image of κ is isomorphic to κ. This proves the claim. �

A function f : Y → Z between sets is said to be injective at y∈Y if the preimage of f (y) is the singleton {y}.(Equivalently, if y′ ∈Y is such that y /= y′, then f (y) /= f (y′).)

Lemma 9. Let Y and Z be chains, y an element of Y and f : Y → Z a homomorphism. Assume that f is injective at yand continuous at y. We have:

(i) if cf(y) is infinite, then cf(y) = cf(f (y));(ii) if ci(y) is infinite, then ci(y) = ci(f (y)).

Proof. We prove (i) only, since (ii) is dual to (i). Assume that cf(y) = κ is an infinite regular cardinal. Let (yξ)ξ<κ be astrictly increasing sequence converging to y. Since f is order-preserving and injective at y, it follows that the sequence(f (yξ))ξ<κ is increasing (not necessarily strictly) and is such that f (yξ) ≺ f (y) for each ξ < κ. Further, continuity off at y implies that (f (yξ))ξ<κ converges to f (y). Thus, cf(f (y)) = κ, using Lemma 8. �


Next, we associate to each element y of a subchain Y ⊆ X two sets of regular cardinals cfY [y] and hcfY [y], and twosets of reverse regular cardinals ciY [y] and hciY [y].

Definition 10. Let y∈Y ⊆ X. We define the set of cofinalities of y in Y, denoted by cfY [y], as the set {0, 1, cfY (y)} ify is not the minimum of Y, and the set {cfY (y)} = {0} if y = min Y . Dually, the set of coinitialities of y in Y, denotedby ciY [y], is the set {0, 1, ciY (y)} if y is not the maximum of Y, and the set {ciY (y)} = {0} if y = max Y . As before,cf[y] and ci[y] stand for cfX[y] and ciX[y], respectively. Further, we define the set of hereditary cofinalities of y in Yas the set

hcfY [y]:={κ∈ Reg : κ is a hereditary cofinality of y in Y}and the set of hereditary coinitialities of y in Y as the set

hciY [y]:={λ∗ ∈ Reg∗

: λ∗ is a hereditary coinitiality of y in Y}.The notations hcf[y] and hci[y] stand for hcfX[y] and hciX[y], respectively.

Remark 11. Let y∈Y ⊆ X. The definition of the set of cofinalities of y in Y yields:

cfY [y] =

⎧⎪⎨⎪⎩{0} if y is the minimum of Y

{0, 1} if y has an immediate predecessor in Y

{0, 1, cfY (y)} otherwise.

Dually, we have:

ciY [y] =

⎧⎪⎨⎪⎩{0} if y is the maximum of Y

{0, 1} if y has an immediate successor in Y

{0, 1, ciY (y)} otherwise.

Note also that the inclusions cfY [y] ⊆ hcfY [y] and ciY [y] ⊆ hciY [y] hold. In particular, we have hcfY [y] = {0} if andonly if y = min Y , and hciY [y] = {0} if and only if y = max Y .

In the next definition, we associate to a chain X two sets of ordered pairs, CT(X) and HCT(X). The first com-ponent of each ordered pair in these sets is a regular cardinal and its second component is a reverse regularcardinal.

Definition 12. A coterminal pair of X is an ordered pair (κ, λ∗) such that there exists x∈X with the property thatκ∈ cf[x] and λ∗ ∈ ci[x]. The set of coterminal pairs of X at x is CT(X, x):=cf[x]× ci[x]. We denote by CT(X) thecollection of all coterminal pairs of X, i.e., CT(X):=⋃

x∈XCT(X, x).Similarly, a hereditary coterminal pair of X is an ordered pair (κ, λ∗) such that that there exists x∈X with the property

that κ∈ hcf[x] and λ∗ ∈ hci[x]. Equivalently, (κ, λ∗) is a hereditarily coterminal pair of X if there exist A, B ⊆ X andx∈X with the property that A ≺ x ≺ B, Cf(A) = κ and Ci(B) = λ∗, i.e., there exists x∈X such that κ ↪→ (←, x) andλ∗ ↪→ (x,→). The set of hereditary coterminal pairs of X at x is HCT(X, x):=hcf[x]× hci[x]. We denote by HCT(X)the collection of all hereditary coterminal pairs of X, i.e., HCT(X):=⋃

x∈XHCT(X, x).

Remark 13. Observe that for each chain X, we have CT(X) ⊆ HCT(X), and equality holds if X is finite (but not ifand only if, see Example 14(iv)). In particular, if X has at least three elements, then {0, 1}2 ⊆ CT(X) ⊆ HCT(X).

The remainder of this section is devoted to the study of some properties of the sets CT(X) and HCT(X). We startwith some examples.

Example 14. We compute CT(X) and HCT(X) for some linear orders X.


(i) Let X:=R. For each x∈R, we have cf(x) = ω and ci(x) = ω∗, hence CT(R, x) = HCT(R, x) = {0, 1, ω} ×{0, 1, ω∗}. Therefore, we have

CT(R) = HCT(R) = {0, 1, ω} × {0, 1, ω∗}.

(ii) Let X:=[0, 1]. For each x∈ (0, 1), we have CT([0, 1], x) = HCT([0, 1], x) = {0, 1, ω} × {0, 1, ω∗}. Furthermore,we have CT([0, 1], 0) = HCT([0, 1], 0) = {0} × {0, 1, ω∗} and CT([0, 1], 1) = HCT([0, 1], 1) = {0, 1, ω} × {0}.Thus, we obtain

CT([0, 1]) = HCT([0, 1]) = {0, 1, ω} × {0, 1, ω∗}.

(iii) Let X:=α, where α is an infinite ordinal. We have cf[0] = {0}, cf[β] = {0, 1} for each successor ordinal β < α,and cf[β] = {0, 1,cf(β)} for each limit ordinal β < α. Further, for each ordinal β < α, we have ci[β] = {0, 1} if β

is not the maximum of α, and ci[β] = {0} otherwise. A simple calculation shows that if α /= κ + 1, with κ regularcardinal, then we have

CT(α) = HCT(α) = Reg<α × {0, 1}.

In particular, CT(ω) = HCT(ω) = {0, 1}2 and CT(ω1) = HCT(ω1) = {0, 1, ω} × {0, 1}. On the other hand, ifα = κ + 1 is the immediate successor of a regular cardinal κ, then we have

CT(κ + 1) = HCT(κ + 1) = (Reg≤κ × {0, 1}) \ {(κ, 1)}.

(iv) Let X:=κ + λ∗, where κ and λ are infinite regular cardinals. In this case, the set of all coterminal pairs is properlycontained in the set of all hereditary coterminal pairs. Indeed, we have:

CT(κ + λ∗) = (Reg<κ × {0, 1}) ∪ ({0, 1} × Reg∗<λ)

HCT(κ + λ∗) = (Reg≤κ × Reg∗≤λ) \ {(κ, λ∗)}.

In particular,

CT(ω + ω∗) = {0, 1}2 � HCT(ω + ω∗) = ({0, 1, ω} × {0, 1, ω∗}) \ {(ω, ω∗)}.

(v) Let X:=κ + 1+ λ∗, where κ and λ are infinite regular cardinals. Note that X is the Dedekind completion of thechain in (iv). We have:

CT(κ + 1+ λ∗) = (Reg≤κ × {0, 1}) ∪ ({0, 1} × Reg∗≤λ) ∪ {(κ, λ∗)}

HCT(κ + 1+ λ∗) = Reg≤κ × Reg∗≤λ.

In particular, CT(ω + 1+ ω∗) = HCT(ω + 1+ ω∗) = {0, 1, ω} × {0, 1, ω∗}.

Remark 15. The sets of all coterminal pairs and of all hereditary coterminal pairs behave differently with respectto subchains of a given chain X. In fact, the operator HCT is always monotone on the lattice of all subchains of X


(ordered by containment), i.e., for all Y, Z ⊆ X, if Y ⊆ Z then HCT(Y ) ⊆ HCT(Z). On the other hand, Y ⊆ X doesnot imply CT(Y ) ⊆ CT(X). There are chains X such that the operator CT is monotone with respect to X, i.e., if Y ⊆ X

then CT(Y ) ⊆ CT(X). These chains, called pointwise Debreu, will play a main role in this paper (see Definition 22and Theorem 24).

The next result shows that the operator HCT is the hereditary closure (with respect to subchains) of the operator CT.

Lemma 16. For each chain X, we have HCT(X) = ⋃Y⊆XCT(Y ).

Proof. [⊆]. Let (κ, λ∗) be a hereditary coterminal pair of X. Thus, there exist A, B ⊆ X and x∈X such that A ≺ x ≺ B,Cf(A) = κ and Ci(B) = λ∗. Set Y :=A ∪ {x} ∪ B. It follows that cfY (x) = κ and ciY (x) = λ∗. Thus (κ, λ∗) is a coterminalpair of Y ⊆ X.

[⊇]. The result is obvious if |X| = 2. Thus, we can assume that |X| ≥ 3. By Remark 13, we have {0, 1}2 ⊆CT(X) ∩ HCT(X). Therefore, it suffices to show that for all regular cardinals κ and λ such that max{κ, λ} ≥ ω, if(κ, λ∗)∈CT(Y ) for some Y ⊆ X, then (κ, λ∗)∈HCT(X). Fix Y ⊆ X and assume that (κ, λ∗) is a coterminal pair of Ysuch that κ or λ is an infinite (regular) cardinal. Without loss of generality, we can assume that κ ≥ ω. By hypothesis,there exists y∈Y such that κ∈ cfY [y] and λ∗ ∈ ciY [y]. Since κ is infinite, it follows that cfY (y) = Cf(←, y)Y = κ.

In the sequel we define two subchains A, B ⊆ X and an element x∈X such that A ≺ x ≺ B, Cf(A) = κ andCi(B) = λ. This will show that (κ, λ∗) is a hereditary coterminal pair of X. Set A:=(←, y) and x:=y. If λ = 0, then letB:=∅. If λ = 1, then y is not the maximum of Y, and so there exists z∈Y such that y ≺ z; let B:={z}. If λ ≥ ω, thenciY (y) = λ; let B:=(y,→)Y . The subchains A, B ⊆ X and the element x∈X satisfy the claim. �

Finally, we prove a result that clarifies the order-theoretic and topological nature of the two sets CT(X) and HCT(X).

Theorem 17. Let (X,≺) be an infinite chain. For each κ, λ∈ Reg, we have:

(a) (κ, λ∗)∈HCT(X) if and only if κ + 1+ λ∗ embeds into X;(b) (κ, λ∗)∈CT(X) if and only if κ + 1+ λ∗ embeds into X with an embedding continuous at the middle point 1.

Proof. Part (a) is easy and is left to the reader. We prove part (b). Assume that (κ, λ∗) is a coterminal pair of X.Therefore, there exists a point x∈X such that κ∈ cf[x] and λ∗ ∈ ci[x]. It suffices to consider the case in which both κ

and λ are infinite. Then, we have cf(1) = κ and ci(1) = λ∗. Thus, the definition of cofinality and coinitiality of a pointyields the existence of an embedding f : κ + 1+ λ∗ ↪→ X, as claimed.

For the reverse implication, assume that there exists an embedding f : κ + 1+ λ∗ ↪→ X, which is continuous atthe middle point 1. We claim that the point x:=f (1) is such that κ∈ cf[x] and λ∗ ∈ ci[x]. This will complete the proofof (b). To prove the claim, observe that the hypothesis implies that there exists an embedding g : κ + 1 ↪→ X, whichis continuous at the maximum point 1, namely, g:=f � (κ + 1). If κ is either 0 or 1, then it is immediate to show thatκ belongs to the set of cofinalities of g(1). So assume that κ is infinite. Then Lemma 9 yields cf(g(1)) = cf(1) = κ,hence κ∈ cf[g(1)] = cf[f (1)]. Using an argument dual to the one give above, one can show that λ∗ ∈ ci[h(1)], whereh is the restriction of f to 1+ λ∗. �

Recall that a chain X is short if neither ω1 nor ω∗1 embeds into X. The next fact is an immediate consequence ofTheorem 17(a).

Corollary 18. If X is a short chain, then HCT(X) ⊆ {0, 1, ω} × {0, 1, ω∗}.We end this section with a result about the relationship between cofinalities of points and injectivity/continuity

of a homomorphism at a point. This result will be useful in the next section (Theorem 24). We first introduce someterminology.

Definition 19. Let f : Y → Z be a map between chains endowed with the order topology, and y an element of Y. Wesay that f is left-continuous at y if either (i) y is the minimum of Y, or (ii) for each z′ ∈Z such that z′ ≺ f (y), thereexists y′ ≺ y such that f (y′, y) ⊆ (z′, f (y)). (Thus, if y has an immediate predecessor, then f is left-continuous at yby (ii).) The notion of right-continuous at y is defined dually. Furthermore, we say that f : Y → Z is left-injective aty∈Y if for each y′ ∈Y , y′ ≺ y implies f (y′) ≺ f (y). The notion of right-injective at y is defined dually.


Observe that f is continuous at y if and only if it is both left-continuous and right-continuous at y. Similarly, f isinjective at y if and only if it is both left-injective and right-injective at y.

Lemma 20. Let Y and Z be chains endowed with the order topology.

(i) If there exist y∈Y and z∈Z such that cf(y)∈ cf[z], then there exists a homomorphism g : Y → Z such thatg(y) = z, and g is left-continuous at y and left-injective at y.

(ii) If there exist y∈Y and z∈Z such that ci(y)∈ ci[z], then there exists a homomorphism h : Y → Z such thath(y) = z, and h is right-continuous at y and right-injective at y.

(iii) If there exist y∈Y and z∈Z such that cf(y)∈ cf[z] and ci(y)∈ ci[z], then there exists a homomorphism f : Y → Z

such that f (y) = z, and f is continuous at y and injective at y.

Proof. We prove (i). Assume that there exist y∈Y and z∈Z satisfying the hypothesis. If cf(y) = 0, then y is theminimum of Y, hence any homomorphism g : Y → Z is left-continuous and left-injective at y. In particular, the constanthomomorphism g(y′):=z for each y′ ∈Y satisfies the claim. If cf(y) = 1∈ cf[z], then y has an immediate predecessory′ ∈Y and z is not the minimum of Z. Let z′ ∈Z be such that z′ ≺ z. The function g : Y → Z, defined by g(←, y′]:=z′and g[y,→):=z, satisfies the claim. Next, assume that cf(y) = cf(z) = κ is an infinite regular cardinal. By definition,we have that Cf(←, y) = κ and Cf(←, z) = κ. Let (yξ)ξ ∈ κ and (zξ)ξ ∈ κ be two strictly increasing sequences, whichare cofinal, respectively, in (←, y) and (←, z). Define a function g : Y → Z as follows. Let g(yξ):=zξ for each ξ < κ.Further, map the ray (←, y0) to the point z0, and for each ξ < κ, map the interval [yξ, yξ+1) to the point zξ . Finally, letg[y,→):=z. The function g satisfies the claim. This finishes the proof of (i).

Part (ii) is dual to (i). To prove (iii), assume that there exist y∈Y and z∈Z such that cf(y)∈ cf[z] and ci(y)∈ ci[z]. By(i) and (ii), there exist homomorphisms g : Y → Z and h : Y → Z such that g(y) = h(y) = z, g is left-continuous andleft-injective at y, and h is right-continuous and right-injective at y. The function f : Y → Z, defined by f (y′):=g(y′)for each y′ � y and f (y′′):=h(y′′) for each y′′ y, is a homomorphism that satisfies the claim. �

4. Locally Debreu and pointwise Debreu chains

In Herden and Mehta (2004), the authors introduce the notion of a Debreu chain. We recall the definition.

Definition 21. A linear order (X,≺) is a Debreu chain if each subchain Y ⊆ X continuously embeds into X, i.e., thereexists a continuous order-embedding f : (Y,≺, τ≺) ↪→ (X,≺, τ≺). Therefore, (X,≺) fails to be a Debreu chain if thereexists Z ⊆ X such that any order-embedding of Z into X is not continuous; we call the linear order Z a discontinuoussubchain of X.

As already pointed out, the Debreu property of a chain X is related to the continuity of a utility representationhaving X as base chain. In fact, X is Debreu if for each chain L, we have that L is representable in X if and only if L iscontinuously representable in X. Herden and Mehta (2004) show that only few chains are Debreu. In view of extendingthe notion of representability of linear orders by choosing a suitable base chain for a utility representation, we examinesome Debreu-like properties that a base chain might satisfy. These properties are still linked to the continuity of autility representation, but are not so unlikely to hold as the Debreu property. To this aim, in this section, we describetwo classes of linear orders, which properly contain the class of Debreu chains.

Definition 22. A linear order (X,≺) is locally Debreu if for each Y ⊆ X and y∈Y , there exists an order-embeddingfy : (Y, τ≺) ↪→ (X, τ≺), which is continuous at y. Further, X is pointwise Debreu if for each Y ⊆ X and y∈Y , thereexists an order-homomorphism fy : (Y, τ≺)→ (X, τ≺), which is continuous at y and injective at y.

Thus, a chain (X,≺) is not locally Debreu if there exist Y ⊆ X and y∈Y such that any order-embedding of Y intoX is not continuous at y; we call the linear order Y a locally discontinuous subchain of X. Further, X is not pointwiseDebreu if there exist Y ⊆ X and y∈Y with the property that any order-homomorphism f of Y into X such that thepreimage of f (y) is a singleton fails to be continuous at y; we call the linear order Y a pointwise discontinuous subchainof X.


Clearly, a Debreu chain is locally Debreu, and a locally Debreu chain is pointwise Debreu. The converse of theseimplications does not hold. In the following we exhibit an example of a locally Debreu chain that fails to be Debreu.This linear order is important in the representation of preferences, because it is (isomorphic to) the lexicographic power2ω

lex (Chipman, 1960). Later on, we will give another example (relevant from the point of view of utility theory) of achain that is locally Debreu but not Debreu (see Example 36). Furthermore, we will provide several instances of linearorders that are pointwise Debreu but not locally Debreu (Examples 23, 26, 27 and 28).

Example 23. The Cantor set C is locally Debreu but not Debreu. The topological space C can be identified with thelexicographic power 2ω

lex endowed with the order topology τ≺. Since 2ωlex contains a subchain that is order-isomorphic

to [0, 1] and any order-embedding of ([0, 1], τ≺) into (2ωlex, τ≺) is not continuous, it follows that C is not Debreu.

Recall that in the Cantor set there are three classes of points: (i) rational numbers that are isolated from the left,identified with those binary sequences that are eventually equally to 0; (ii) rational numbers that are isolated from theright, identified with those binary sequences that are eventually equally to 1; (iii) irrational numbers, which are isolatedneither from the left nor from the right and are identified with all binary sequences that are not eventually constant.

To show that C is locally Debreu, let Y ⊆ C and y∈Y . We consider the case in which y is neither the maximumnor the minimum of Y, and y has neither immediate predecessor nor immediate successor. (All other cases can be dealtwith in a similar way.) Therefore, we have cfY (y) = ω and ciY (y) = ω∗. Let (yn)n∈ω be a strictly increasing sequencein Y that is cofinal in (←, y)Y , and (y′n)n∈ω a strictly decreasing sequence in Y that is coinitial in (y,→)Y .

We define a function fy : Y → C as follows. Let fy(y):=x, where x∈C is an arbitrary irrational number. Sincecf(x) = ω and ci(x) = ω∗, there exist a strictly increasing sequence (xn)n∈ω in C and a strictly decreasing sequence(x′n)n∈ω in C, which are, respectively, cofinal in (←, x) and coinitial in (x,→). Without loss of generality, we canassume that x0 /= 0, x′0 /= 1 and no points in the two sequences are consecutive. For each n∈ω, let fy(yn):=xn andfy(y′n):=x′n. Further, for each n∈ω, map any point in the interval (yn, yn+1)Y into the interval (xn, xn+1) with anorder-embedding. Such a function exists because C order-embeds into the interval (xn, xn+1). Further, map the initialray (←, y0)Y into [0, x0) ⊆ C with an order-embedding. Dually, we can use an order-embedding to map any point inthe interval (y′n+1, y

′n)

Yinto the interval (x′n+1, x

′n) and any point in the final ray (y′0,→)

Yinto (x′0, 1]. This completes

the definition of the function fy : Y → C. By construction, fy is an order-embedding, which is continuous at y. Thisproves that C is locally Debreu.

In Chipman (1960), the author argues that the lexicographic power 2αlex, with α being a suitably chosen infinite

ordinal number, is a natural candidate as the codomain of a utility function. As Herden and Mehta (2004) point out, thelinear orderings 2α

lex are not Debreu. On the other hand, Example 23 shows that 2ωlex does have a Debreu-like property,

which makes it a suitable base chain for utility representations.The next result gives a characterization of pointwise Debreu linear orders.

Theorem 24. The following statements are equivalent for an infinite chain (X,≺):

(i) X is pointwise Debreu;(ii) for each A, B ⊆ X such that A ≺ c ≺ B for some c∈X, there exists x∈X such that Cf(A)∈ cf[x] and Ci(B)∈ ci[x];

(iii) CT(Y ) ⊆ CT(X) for each Y ⊆ X;(iv) CT(X) = HCT(X);(v) for all κ, λ∈ Reg, if κ + 1+ λ∗ embeds into X, then κ + 1+ λ∗ embeds into X with an embedding continuous at

the middle point 1.

Proof. (i)⇒ (ii). Assume that X is pointwise Debreu. Let A and B be subsets of X such that A ≺ c ≺ B for some c∈X.Set Y :=A ∪ {c} ∪ B. By hypothesis, there exists a homomorphism fc : Y → X that is injective at c and continuous atc. We show that x:=fc(c)∈X satisfies (ii). By duality, it suffices to prove that Cf(A)∈ cf[x]. If A = ∅, then Cf(A) =0∈ cf[x] by definition of extended cofinality of a point. If A /= ∅ has no maximum, then Lemma 9 yields that Cf(A) =Cf(←, c)Y = cfY (c) = cf(x)∈ cf[x]. If A /= ∅ has a maximum element a, then fc(a) ≺ x, because fc is order-preservingand injective at c. Thus x is not the minimum of X, and so Cf(A) = 1∈ cf[x].

(ii)⇒ (iii). Assume that (ii) holds. Let Y ⊆ X and (κ, λ∗)∈CT(Y ). Thus, there exists y∈Y such that κ∈ cfY [y] andλ∗ ∈ ciY [y]. Since X is infinite, we have that {0, 1} ⊆ cf[x] ∩ ci[x] for some x∈X, hence {0, 1}2 ⊆ CT(X). Therefore,


we can assume that at least one of the regular cardinals κ and λ is infinite. Without loss of generality, let κ ≥ ω. Bydefinition of extended cofinality, we have that cfY (y) = Cf(←, y)Y = κ. Next, we define two subchains A, B ⊆ X,which satisfy the hypothesis of (ii). Set A:=(←, y)Y . If λ = 0, set B:=∅. If λ = 1, then y is not the maximum of Y, hencethere exists z∈Y ⊆ X such that y ≺ z; set B:={z}. If λ is infinite, then ciY (y) = Ci(y,→)Y = λ∗; set B:=(y,→)Y . Thesubchains A and B are such that A ≺ y ≺ B. By hypothesis, there exists x∈X with the property that κ = Cf(A)∈ cf[x]and λ∗ = Ci(B)∈ ci[x]. This proves that (κ, λ∗) is a coterminal pair of X.

(iii)⇒ (i). Assume that (iii) holds. Let Y ⊆ X and y∈Y . The hypothesis yields that there exists x∈X such thatcfY (y)∈ cf[x] and ciY (y)∈ ci[x]. By Lemma 20(iii), there is a homomorphism f : Y → X, which is continuous at yand injective at y. Thus, X is pointwise Debreu.

(iii)⇔ (iv)⇔ (v). These equivalences follow from Lemma 16 and Theorem 17, respectively. �

Corollary 25. Let (X,≺) be a short chain. If there exists an embedding of ω + 1+ ω∗ into X that is continuous atthe middle point 1, then X is pointwise Debreu.

Proof. Assume that ϕ : ω + 1+ ω∗ ↪→ X is an embedding, which is continuous at the middle point 1. Lemma 9implies that cf(ϕ(1)) = ω and ci(ϕ(1)) = ω∗, hence CT(X, ϕ(1)) = {0, 1, ω} × {0, 1, ω∗}. Since X is short, Corollary 18yields HCT(X) ⊆ {0, 1, ω} × {0, 1, ω∗} = CT(X) ⊆ HCT(X). Thus, CT(X) = HCT(X), and so X is pointwise Debreuby Theorem 24. �

Corollary 25 provide us with a simple tool to show that a short chain is pointwise Debreu. We use this result inthe next examples to exhibit four short chains, which are pointwise Debreu but not locally Debreu. The first example,despite its simplicity, clearly illustrates why a pointwise Debreu linear order might fail to be locally Debreu.

Example 26. The chain X:=ω + 1+ ω∗ + ω + ω∗ is pointwise Debreu but not locally Debreu. Since X is short,Corollary 25 yields that X is pointwise Debreu. To prove that X is not locally Debreu, consider the subchain Y :=ω + ω +1. Denote by x the middle point of ω + 1+ ω∗ ⊆ X and by y the maximum of Y. Let fy : Y → X be a homomorphism,which is continuous at y. Assume by contradiction that fy is injective. Since fy is continuous at y and cfY (y) = ω,Lemma 9 yields that cf(fy(y)) = ω. It follows that fy(y) = x. Since fy is an embedding, fy should map injectivelythe ray (←, y)Y = ω + ω into the ray (←, x) = ω, which is impossible. This proves that Y is a locally discontinuoussubchain of X.

Next, we deal with a linear order that is historically important in utility theory: the lexicographic plane R2lex.

Example 27. The chain R2lex is pointwise Debreu but not locally Debreu. The chain R2

lex is short (Fleischer, 1961b,Theorem 1) hence Corollary 25 implies that it is pointwise Debreu. To show that R2

lex is not locally Debreu, considerthe subchain Y :=A ∪ {(0, 0)} ∪ B, where A:={(x, y)∈R2

lex : x < 0} and B:={(x, y)∈R2lex : x > 0}. Assume that f :

Y ↪→ R2lex is an embedding and let f (0, 0) = (x0, y0). We claim that the f-preimage of the open interval ((x0, y0 −

1), (x0, y0 + 1)) ⊆ R2lex is the singleton {(0, 0)}. This will prove that Y is a locally discontinuous subchain of R2

lex,because f cannot be continuous at (0, 0).

We prove the claim by contradiction. Assume that f−1((x0, y0 − 1), (x0, y0 + 1)) contains a point (a, b) /= (0, 0). Byduality, we can assume that (a, b)≺lex(0, 0), i.e., (a, b)∈A. Since f is order-preserving, it follows that [(a, b), (0, 0)] ⊆f−1((x0, y0 − 1), (x0, y0 + 1)). Observe that a < 0 implies that the open interval ((a, b), (0, 0)) is isomorphic to thewhole lexicographic power R2

lex. Injectivity of f implies that ((a, b), (0, 0)) embeds into ((x0, y0 − 1), (x0, y0 + 1)),which contradicts the fact that R2

lex does not embed into R. This proves the claim.

The next example deals with another lexicographic power of the real numbers, the chainRω+ωlex . Its interest lies in the

fact that this chain is an instance of a wide class of linear orders that are pointwise Debreu but not locally Debreu. Thisclass of chains comprises all lexicographic powers of the real numbers having as exponent a decomposable countablelimit ordinal, i.e., a countable limit ordinal α that can be written as α = β + γ , with β, γ < α.

Example 28. The lexicographic power Rω+ωlex is pointwise Debreu but not locally Debreu. The chain Rω+ω is short,

because it is the lexicographic product of countably many short chains (Giarlotta, 2004a, Corollary 5.1.8). Therefore,


Rω+ω is pointwise Debreu by Corollary 25. Next, we show directly thatRω+ωlex is not locally Debreu. For each n∈ω \ {0},

let Yn:={x∈Rω+ωlex : x(0) = −(1/n)}. Note that each Yn is isomorphic to the full powerRω+ω

lex . Set Y :=⋃1≤n<ωYn ∪ {0},

where 0 is the function of Rω+ωlex that is identically equal to zero. Observe that for each y∈Y \ {0}, the ray (y,→) =

(y, 0] ⊆ Y is an open neighborhood of 0 in Y, which embeds Rω+ωlex .

We claim that Y is a locally discontinuous subchain ofRω+ωlex . It suffices to show that any embedding from Y intoRω+ω

lexis not continuous at 0∈Y . Assume that f : Y ↪→ Rω+ω

lex is an embedding. Let s and t be the elements of Rω+ωlex defined

as follows. Select s, t ∈R such that s < f (0)(ω) < t. Set s(ξ) = t(ξ):=f (0)(ξ) for ξ /= ω, s(ω):=s and t(ω):=t. Theinterval (s, t) is an open neighborhood of f (0) inRω+ω

lex , which embeds intoRωlex. Toward a contradiction, assume that f is

continuous at 0. Thus, f−1(s, t) is an interval (y, 0], where y∈Y \ {0}. It follows thatRω+ωlex ↪→ (y, 0] ↪→ (s, t) ↪→ Rω

lex,which contradicts the fact that Rω+ω

lex does not embed into Rωlex (Kuhlmann, 1995, Corollary 2.4).

Example 28 can be generalized as follows. We call a chain X hierarchical if for all ordinals α, β, we have β < α ifand only if X

βlex ↪→ Xα

lex. Then, one can show that all lexicographic powers having a hierarchical short chain as baseand a countable decomposable ordinal as exponent are pointwise Debreu but not locally Debreu. Note that Example28 is a particular case of this result, since R is hierarchical (Kuhlmann, 1995; Giarlotta, 2004a).

Next, we introduce an order-theoretic property, called interval-embeddability property, which is linked to Debreu-like properties of linear orders. Many chains that are important in utility theory do satisfy this property (see Examples30 and 32).

Definition 29. We say that a chain (X,≺) has the interval-embeddability property (for short, i-e property) if it satisfiesthe following conditions:

• for each x1, x2, y1, y2 ∈X such that x1 ≺ x2 and y1 ≺ y2, if the interval (x1, x2) is non-empty, then (y1, y2) embedsinto (x1, x2);• for each x, y∈X, if (x,→) is non-empty, then (y,→) embeds into (x,→);• for each x, y∈X, if (←, x) is non-empty, then (←, y) embeds into (←, x).

Example 30. The chains R, Q and Rωlex have the i-e property. The Cantor set C has the i-e property as well, because

all non-empty open intervals and open rays in C embed the interval (0, 1). On the other hand, all infinite chains withat least three consecutive points do not have the i-e property. In particular, all infinite ordinals and reverse ordinalsdo not have the i-e property. The lexicographic power R2

lex does not have the i-e property, because the open interval((0, 0), (1, 0)) cannot be embedded into the open interval ((0, 0), (0, 1)). More generally, for each ordinal α ≥ 1, thelexicographic power Rα+1

lex fails to have the i-e property.

Now we recall the definition of the long line and some related linear orders.

Definition 31. The long line is the lexicographic product ω1×lex[0, 1) with its initial point (0, 0) deleted. Further,the symmetric long line is the chain (ω∗1×lex(−1, 0])+ (ω1×lex[0, 1)) with the two middle points identified. (Note thatCampion et al. (2006d) call this chain the double long line.) More generally, for any uncountable cardinal κ, the κ-long line is the lexicographic product κ×lex[0, 1) with its initial point (0, 0) deleted, and the symmetric κ-long line isthe chain (κ∗×lex(−1, 0])+ (κ×lex[0, 1)) with the two middle points identified. Note that the (symmetric) long line isthe (symmetric) ω1-long line.

The long line has been widely studied in utility theory. For example, Monteiro (1987) establishes a connectionbetween the long line and path connected topological spaces. The long line is also used by Estevez and Herves (1995)to construct an example of a continuous total order on any non-separable metric space that has no real-valued utilityrepresentation. The symmetric κ-long line and the κ-long line are, in our opinion, suitable alternatives to the set ofreal numbers as a codomain of utility functions. In fact, they have, among the other properties, the Debreu property(Herden and Mehta, 2004, Theorem 3). The reader is referred to Campion et al. (2006b, c) for further comments aboutthe suitability of the long line as a codomain of utility functions.

Example 32. The long line has the i-e property, because all of its open intervals are isomorphic to the open unitinterval (0, 1). Similarly, the symmetric long line has the i-e property. On the other hand, if κ is a cardinal greater than


ω1, then the (symmetric) κ-long line does not have the i-e property, because it contains intervals isomorphic to (0, 1)and intervals isomorphic to the long line.

Remark 33. If X has the i-e property, then it does not follow that X embeds into each of its non-empty open intervals.Consider, e.g., the long line.

The notions of locally Debreu and pointwise Debreu coincide within the class of linear orders that have the i-eproperty.

Theorem 34. Let (X,≺) be a chain that has the interval-embeddability property. Then X is locally Debreu if andonly if it is pointwise Debreu.

Proof. It suffices to show that if X is pointwise Debreu then it is locally Debreu. Assume that X is pointwise Debreu.Let Y ⊆ X and y∈Y . By hypothesis, there exists a homomorphism f : Y → X, which is injective at y and continuousat y. In the sequel, we define an embedding ϕ : Y ↪→ X, which is continuous at y. This will show that X is locallyDebreu.

Let cfY (y) = κ and ciY (y) = λ∗. By duality, we can assume that κ ≥ λ. First let κ be infinite; thus, cf(f (y)) = κ

using Lemma 9. Set ϕ(y):=f (y). Next, we define ϕ on the ray (←, y)Y . Let (yξ)ξ<κ be a strictly increasing sequencein Y converging to y. The subsequence (yξ+1)ξ<κ still converges to y and has order-type κ. Similarly, let (xξ)ξ<κ

be a strictly increasing sequence in X converging to f (y), and (xξ+1)ξ<κ its subsequence. By possibly passing toanother subsequence, we can assume that the elements of the sequence (xξ+1)ξ<κ are not consecutive. The i-e propertyof X yields the existence of an embedding ϕξ+1 : (yξ+1, yξ+2)Y ↪→ (xξ+1, xξ+2) for each ξ < κ, and an embeddingϕ0 : (←, y1)Y ↪→ (←, x1). We define ϕ on (←, y)Y as the pasting of all these embeddings. Thus, for each ξ < κ, letϕ(yξ+1):=xξ+1 and ϕ � (yξ+1, yξ+2)Y :=ϕξ+1. Finally, set ϕ � (←, y1)Y :=ϕ0.

To complete the definition of ϕ in the case that κ is an infinite ordinal, we now define ϕ on the ray (y,→)Y . Wedistinguish the following three cases: λ is infinite, λ = 0 and λ = 1. If λ ≥ ω, then the definition of ϕ on the ray(y,→)Y is dual to the definition of ϕ on the ray (←, y)Y . If λ = 0, then y = max Y , hence ϕ is already fully defined.If λ = 1, then y is the immediate predecessor of some y′′ ∈Y . Since the homomorphism f is injective at y, it followsthat f (y) ≺ f (y′′). Thus, we can use the i-e property of X to embed the ray (y,→)Y into the ray (f (y),→).

The function ϕ is an embedding by construction. To show that ϕ is continuous at y, it suffices to prove that ϕ isleft-continuous in the case that λ = ci(y) is infinite. Let z′ ∈X be such that z′ ≺ f (y). Then there exists ξ < κ suchthat z′ ≺ xξ+1. By the definition of ϕ, we obtain that ϕ(yξ+1) = xξ+1 and ϕ(yξ+1, y) ⊆ (z′, f (y)), as claimed. Thus, ϕ

is an embedding that is continuous at y.Finally, let κ and λ be finite. By duality, it suffices to prove the claim in the following two cases: (i) κ = λ = 1; (ii)

κ = 1 and λ = 0. In both cases, set ϕ(y):=f (y). In case (i), there exists y′, y′′ ∈Y that are, respectively, the immediatepredecessor and successor of y in Y. Since f is injective at y, there exist x′, x′′ ∈X such that x′ = f (y′) ≺ f (y) ≺f (y′′) = x′′. Using the i-e property of X, we can embed the ray (y,→)Y into the ray (f (y),→) and the ray (←, y)Yinto the ray (←, f (y)). The function ϕ is an embedding that is continuous at y.

In case (ii), we have that y = max Y and y is the immediate successor of some y′ ∈Y . Since f is injective at y,it follows that f (y′) ≺ f (y). Using the fact that X has the i-e property, we can embed the ray (←, y)Y into the ray(←, f (y)). The function ϕ satisfies the claim. �

Remark 35. In Example 23, we showed that the Cantor set C is locally Debreu. Here, is an alternative proof of thisfact. The Cantor set is a short chain such that there exists an embedding ϕ : ω + 1+ ω∗ ↪→ C, which is continuous atthe middle point 1. Thus, C is pointwise Debreu by Corollary 25. Since C has the i-e property, Theorem 34 yields thatC is locally Debreu.

Recall that a chain (X,≺) is doubly transitive if for each y1, y2, x1, x2 ∈X such that y1 ≺ y2 and x1 ≺ x2, thereexists an automorphism f : X→ X such that f (y1) = x1 and f (y2) = x2. Clearly, a doubly transitive chain has thei-e property (but the converse does not hold, consider, e.g., the Cantor set).

In the next example, we examine the Debreu-like properties of another lexicographic product, which is significantwhen we deal with utility representations: the chain Zω

lex. Note that if we endow Zωlex with the order topology, then the

linearly ordered topological space that we obtain is order-isomorphic and homeomorphic to the space P of irrational


numbers (Rosenstein, 1982, p. 129). The chain P ∼= Zωlex is a short linear order, which is doubly transitive, in particular

it has the i-e property. Observe also that P is isomorphic to a subchain of the Cantor set 2ωlex (see Example 23).

Example 36. The space P of irrational numbers is locally Debreu but not Debreu. Since there exists an embeddingf : ω + 1+ ω∗ ↪→ P that is continuous at the middle point 1, Corollary 25 implies that P is pointwise Debreu. Thus,P is locally Debreu by Theorem 34. On the other hand, P ∼= Zω

lex contains a subchain Y that is order-isomorphic to[0, 1]. Since any order-embedding of (Y, τ≺) into (Zω

lex, τ≺) is not continuous, it follows that P is not Debreu.

Campion et al. (2006d) consider P as a possible codomain for utility functions. They motivate their choice by thefact that P is an example of what they call a semicontinuous Debreu chain. Example 36 gives additional evidence inthis direction.

5. Conclusion and future directions of research

In this paper, we have analyzed some properties of linear orders, which are related to the topic of continuous utilityrepresentations. Our goal is to continue developing a theory of continuity of utility functions that are not necessarilyreal-valued. In their seminal paper, Herden and Mehta (2004) started this topic of research by introducing the notion ofDebreu chain and studying related properties. Here, we have extended their analysis to weaker versions of the Debreuproperty by introducing the notions of locally Debreu and pointwise Debreu chains. In fact, the classes of chains thatare locally Debreu and pointwise Debreu are an enlargement of the class of Debreu chains. We have given severalexamples of linear orders that are not Debreu but satisfy some Debreu-like properties, e.g., the Cantor set C, the setof irrational numbers P, the lexicographic power R2

lex, etc. The notion of pointwise Debreu chain is weaker than thatof locally Debreu chain. We have proved that these two notions coincide within a special class of chains, those thatsatisfy the interval embeddability property.

There are interesting problems still to be analyzed. A first topic is the hereditariety of the Debreu property, mentionedby Herden and Mehta (2004): a chain is hereditarily Debreu if each of its subchains is Debreu. Herden and Mehta(2004) give the following subordering characterization of hereditarily Debreu chains (Theorem 9):

Theorem 37. A chain is hereditarily Debreu if and only if it does not embed ω + ω∗.

This result suggests that the class of hereditarily Debreu chains is very small. Indeed, one can prove that the onlylinear orders that are hereditarily Debreu are suitable sums of ordinals and reverse ordinals:

Theorem 38. A chain is hereditarily Debreu if and only if it is isomorphic to one of the following linear orders:

(i)∑

n∈ωα∗n + β, where the αn’s and β are ordinal numbers;(ii) α∗ +∑

n∈ω∗βn, where α and the βn’s are ordinal numbers.

Furthermore, one can also show that the class of hereditarily Debreu chains coincides with the (seemingly larger)class of hereditarily pointwise Debreu chains (i.e., those chains whose subchains are pointwise Debreu).

A second topic that is worth an attentive analysis concerns Debreu and Debreu-like properties of lexicographicpowers. Herden and Mehta (2004) prove the following result:

Theorem 39. For any non-trivial chain X and infinite uncountable ordinal α, the lexicographic power Xαlex is not

Debreu.

Thus, most lexicographic powers fail to be Debreu chains. On the other hand, as we have argued in Section 1, itmight be interesting to use lexicographic orders as a codomain of a utility function. From this perspective, it becomesmeaningful the fact that many important lexicographic powers, such as the Cantor set C ∼= 2α

lex, the set of irrationalnumbers P ∼= Zα

lex, the lexicographic planeR2lex, the chainRω+ω

lex , etc., do satisfy a Debreu-like property despite failingto be Debreu.

We are currently working on a systematic study of Debreu-like properties of lexicographic powers. Our first goal isto refine Theorem 39 and provide a better understanding of the key reasons for which most lexicographic powers failto be Debreu. We have already obtained some partial results in this direction. For example, the following fact holds:

Theorem 40. For any non-trivial chain X and limit ordinal α, we have:


• if X is short and α is countable, then Xαlex is pointwise Debreu;

• if α is uncountable, then Xαlex is not pointwise Debreu.

Theorem 40 is proved using the implication (iv)⇒ (i) in Theorem 24. The difficulties in the analysis of Debreu-likeproperties of a lexicographic power Xα

lex are hidden in the complex form of the sets HCT(Xαlex) and CT(Xα

lex). Themost difficult cases are those such that the exponent α is an infinite successor ordinal. With this respect, we conjecturethat Xα+1

lex is not pointwise Debreu for all sufficiently large exponents α.Finally, another interesting topic of research could be the relationship between some properties of the base chain X

(such as the interval-embeddability property and the property of being hierarchical) and Debreu-like properties of thelexicographic power Xα

lex.

Acknowledgment

The authors would like to thank two anonymous referees for several helpful comments and suggestions.

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Debreu-like properties of utility representations