Abstract

We present different conditions for the existence of a pair of upper semicontinuous functions representing an interval order on a topological space without imposing any restrictive assumptions neither on the topological space nor on the representing functions. The particular case of second countable topological spaces, which is particularly interesting and frequent in economics, is carefully considered. Some final considerations concerning semiorders finish the paper.

Keywords

Interval Order, Upper Semicontinuous Numerical Representation, Semiorder

1. Introduction

An interval order on a set X can be thought of as the simplest model of a binary relation on X whose associated preference-indifference relation is not transitive. Indeed, under certain conditions, it can be fully represented by means of a pair of real-valued functions on X, in the sense that is equivalent to for all. Therefore, interval orders are particularly interesting in economics and social sciences. Whenever the set X is endowed with a topology, it is interesting to look for representations of an interval order on that satisfy suitable continuity conditions.

The existence of numerical representations of interval orders was first studied by Fishburn [1] [2] and then by other authors (see, e.g., Bosi et al. [3] and Doignon et al. [4] ).

When the set X is endowed with a topology, it may be of interest to look for continuous or at least semicontinuous representations of an interval order on. Results in this direction were presented by Bosi et al. [5] [6] , Chateauneuf [7] and, in the particular case of semiorders, by Candeal et al. [8] [9] .

For many purposes, the existence of a representation with u and v both upper semicontinuous is satisfactory. In particular, if such a representation exists and the topology is compact, then there exist maximal elements for the interval order which are obtained by maximizing u or v. Also the existence of undominated maximal elements can be guaranteed by means of an approach of this kind (see, e.g., Alcantud et al. [10] ). This kind of semicontinuous representability of interval orders was first studied by Bridges [11] and then by Bosi and Zuanon [12] [13] .

In this paper, we present different results concerning the representability of an interval order on a topological space by means of a pair of upper semicontinuous real-valued functions.

2. Notations and Preliminaries

An interval order on a set X is an irreflexive binary relation on X which in addition satisfies the following condition for all:

An interval order is in particular a partial order (i.e., is an irreflexive and transitive binary relation). The preference-indifference relation associated to an interval order on set X is defined as follows for all:

It is well know that if is an interval order, then is total (i.e., for all, either or). On the other hand, is not transitive in general.

Fishburn [2] proved that if is an interval order on a set X, then the following two binary relations and (the traces of the original interval order) are weak orders (i.e., asymmetric and negatively transitive binary relations on X):

The following proposition was proved, for example, by Alcantud et al. ([10] , Lemma 3).

Proposition 2.1. Let be an irreflexive binary relation on a set X. Then is an interval order if and only if is a asymmetric.

An interval order on a set X is a weak order if and only if. The preference-indifference relations and associated to the binary relations and are defined as follows for all:

Therefore, we have that

An interval order on a set X is said to be i.o. separable (see Bosi et al. [3] and Doignon et al. [4] ) if there exists a countable subset D of X such that for all with there exists such that. In this case D is said to be an i.o. order dense subset of.

From Chateauneuf [7] , an interval order on a set X is said to be strongly separable if there exists a countable set such that, for every with, there exist with. D is said to be a strongly order dense subset of X. It is clear that strong separability implies i.o. separability. Further, strong separability occurs, for example, whenever an interval order is representable by means of a pair of nonnegative positively homogeneous functions on a cone in a topological vector space. This kind of representability, in the more general setting of acyclic binary relations, was studied, for example, by Alcantud et al. [14] and in the case of not necessarily total preorders by Bosi et al. [15] .

If R is a binary relation on a set X, then denote by and the lower section and respectively the upper section of any element, i.e., for every,

A subset A of a related set is said to be R-decreasing if for every.

A real-valued function u on X is said to be a weak utility function for a partial order on a set X if, for all,

The following characterization of the existence of an upper semicontinuous weak utility for a partial order on a topological space is well known (see e.g. Alcantud and Rodríguez-Palmero ([16] , Theorem 2)).

Proposition 2.2. Let be a partial order on a topological space. Then the following conditions are equivalent:

1) There exists an upper semicontinuous weak utility function u for;

2) There exists a countable family of open -decreasing subsets of X such that if then there exists such that,.

A real-valued function u on X is said to be a utility function for a partial order on a set X if, for all,

If a partial order admits a representation by means of a utility function, then is a weak order or equivalently the associated preference-indifference relation is a total preorder (i.e. is total and transitive).

The following proposition is well known and easy to be proved.

Proposition 2.3. Let be a weak order on a topological space. Then the following conditions are equivalent:

1) There exists an upper semicontinuous utility function u for;

2) There exists a countable family of open-decreasing subsets of X such that if then there exists such that,.

A pair of real-valued functions on X represents an interval order on X if, for all,

If is a representation of an interval order, then it is easily seen that u and v are weak utility functions for and, respectively, while it is not in general guaranteed that u and v are utility functions for and, respectively.

We say that a pair of real-valued functions on X almost represents an interval order on X if, for all,

An interval order on a topological space is said to be upper (lower) semicontinuous if

is an open subset of X for every. If is both upper and lower semicontinuous, then it is said to be continuous.

If there exists a representation of an interval order on a topological space and u and v are both upper semicontinuous, then is necessarily upper semicontinous, due to the fact that

is open for every. In this case, also the associated weak order is upper semicontinuous, since

is expressed as union of open sets.

On the other hand, the existence of an upper semicontinuous representation does not imply that the weak order is upper semicontinuous. The following example, that was already presented in Bosi and Zuanon [13] , illustrates this fact.

Example 2.4. Let X be the set endowed with the natural induced topology on the real line and consider the interval order on X defined as follows for all:

If we define and for every, then it is clear that is an (upper semi) continuous representation of. We can easily verify that the associated weak order is not upper semicontinuous. Indeed, consider for example that is not an open set. Notice that for all, since but for no we have that because this would imply the existence of such that.

A weak order on a topological space is said to be weakly upper semicontinuous if for every that is not a minimal element there exists a uniquely determined -decreasing open subset of X such that and (see Bosi and Zuanon [13] ). This definition was presented by Bosi and Herden [17] in the context of preorders (i.e., reflexive and transitive binary relations). If a weak order on a topological space admits an upper semicontinuous weak utility u then it is weakly upper semicontinuous (just define, for every,). Further, it is clear that an upper semicontinuous weak order is also weakly upper semicontinuous.

If is a topological space and S is a dense subset of such that, then we say that a family of open subsets of X is a quasi scale in if the following conditions hold:

1);

2) for every such that.

The following proposition is a particular case of Theorem 4.1 in Burgess and Fitzpatrick [18] .

Proposition 2.5. If is a quasi scale in a topological space, then the formula

defines an upper semicontinuous function on with values in.

3. Conditions for the Semicontinous Representability of Interval Orders

In the following theorem we present some conditions that are equivalent to the existence of an upper semicontinuous representation of an interval order on a topological space.

Theorem 3.1. Let be an interval order on a topological space. Then the following conditions are equivalent:

1) There exists a pair of upper semicontinuous real-valued functions on representing the interval order;

2) The following conditions are verified:

a) The interval order on X is representable by means of a pair of real-valued functions;

b) is upper semicontinuous;

c) There exists an upper semicontinuous weak utility for;

3) The following conditions are verified:

a) The interval order on X is i.o.-separable;

b) is upper semicontinuous;

c) There exists a countable family of open -decreasing subsets of X such that if then there exists such that,;

4) There exists a countable family of pairs of upper semicontinuous real-valued functions on

almost representing such that for every with there exists with .

5) There exists a countable family of pairs of open subsets of X satisfying the following conditions:

a) and imply for all and for all;

b) and imply for all and for all;

c) for all such that there exists such that,;

6) There exist two quasi scales and in such that the family satisfies the following conditions:

a) and imply for every and;

b) for every such that there exist such that, ,.

Proof. The equivalence 1) Û 5) was proved in Bosi and Zuanon ([12] , Theorem 3.1), while the equivalences 1) Û 2) and 1) Û 3) were proved in Bosi and Zuanon ([13] , Theorem 3.1).

Let us prove the equivalence of conditions 1) and 4). It is clear that 1) implies 4). In order to show that 4) implies 1), assume that there exists a countable family of pairs of upper semicontinuous real-valued functions on almost representing such that for every with there exists with. Without loss of generality, assume that and take values in for every index n. Define functions u and v on X as follows:

in order to immediately verify that is an upper semicontinuous representation of the interval order on the topological space.

Finally, let us show that also the equivalence of conditions 1) and 6) is valid. In order to show that 1) implies 6), assume without loss of generality that there exists a pair of upper semicontinuous real-valued functions with values in representing the interval order on the topological space. Then just define , , for every, and in order to immediately verify that and are two quasi scales in such that the family satisfies the above subconditions a) and b) of condition 6).

In order to show that 6) implies 1), assume that there exist two quasi scales and such that the family satisfies the above subconditions a) and b) in condition 6). Then define two functions

as follows:

We claim that is a pair of continuous functions on with values in representing the interval order.

From the definition of the functions u and v, it is clear that they both take values in. Let us first show that the pair represents the interval order. First consider any two elements such that. Then, by condition b), there exist such that, ,. Hence, we have , which obviously implies that. Conversely, consider any two elements such that, and observe that, for every, if then it must be by the above condition a). Hence, it must be from the definition of u and v.

Finally, observe that u and v are upper semicontinuous real-valued functions on with values in as an immediate consequence of Proposition 2.5. This consideration completes the proof. QED It has been noticed that if is a representation of an interval order on a set X, then not necessarily u is a utility function for the trace. The following immediate corollary to Theorem 3.1 concerns this particular case.

Corollary 3.2. Let be an interval order on a topological space. Then the following conditions are equivalent:

1) There exists a pair of upper semicontinuous real-valued functions on representing the interval order such that u is a utility function for the associated weak order;

2) The following conditions are verified:

a) The interval order on X is representable by means of a pair of real-valued functions;

b) is upper semicontinuous;

c) is upper semicontinuous;

3) The following conditions are verified:

a) The interval order on X is i.o.-separable;

b) is upper semicontinuous;

c) There exists a countable family of open -decreasing subsets of X such that if then there exists such that,.

Since Bridges ([11] , Proposition 2.3) proved that an interval order on a second countable topological space is representable by a pair of (nonnegative) real-valued function, we have that the following corollary is an immediate consequence of the previous theorem.

Corollary 3.3. Let be an interval order on a second countable topological space. Then the following conditions are equivalent:

1) There exists a pair of upper semicontinuous real-valued functions on representing the interval order;

2) The following conditions are verified:

a) is upper semicontinuous;

b) There exists an upper semicontinuous weak utility for.

The following corollary is a consequence of both Corollary 3.2 and Corollary 3.3.

Corollary 3.4. Let be an interval order on a second countable topological space. Then the following conditions are equivalent:

1) There exists a pair of upper semicontinuous real-valued functions on representing the interval order such that u is a utility function for the associated weak order;

2) The following conditions are verified:

a) is upper semicontinuous;

b) is upper semicontinuous.

The following corollary is found in Bosi and Zuanon ([13] , Proposition 3.1).

Corollary 3.5. Let be a strongly separable interval order on a topological space. Then the following conditions are equivalent:

1) There exists a pair of upper semicontinuous real-valued functions on representing the interval order;

2) The following conditions are verified:

a) is upper semicontinuous;

b) is weakly upper semicontinuous.

We finish this paper by presenting some applications of the previous results to the semiorder case. We recall that a semiorder on an arbitrary nonempty set X is a binary relation on X which is an interval order and in addition verifies the following condition for all:

If is a semiorder, then the binary relation is a weak order (see e.g. Fishburn [2] ). The following proposition was proved by Bosi and Isler ([19] , Proposition 3).

Proposition 3.6. Let be an interval order on a set X. Then is a semiorder if and only if is asymmetric.

Clearly, this happens in the particular case when. More generally, we have that the following proposition holds. The easy proof of it is left to the reader.

Proposition 3.7. Let be an interval order on a set X. If there is a real-valued function u on X that is a weak utility for both and, then is semiorder.

Since it was already observed that upper semicontinuity of an interval order always implies upper semicontinuity of the associated weak order, we obtain the following corollaries as other immediate consequences of Theorem 3.1.

Corollary 3.8. Let be a semiorder on a topological space. If is upper semicontinuous and, then there exists a pair of upper semicontinuous real-valued functions on representing provided that there exists a pair of real-valued functions on X representing.

Corollary 3.9. Let be a semiorder on a second countable topological space. If is upper semicontinuous and, then there exists a pair of upper semicontinuous real-valued functions on representing.

References