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Continuous representability of semiorders
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Journal of Mathematical Psychology 52 (2008) 48–54 www.elsevier.com/locate/jmp
Continuous representability of semiorders Marı´ a J. Campio´na, Juan C. Candealb, Esteban Indura´ina,, Margarita Zudairea a
Universidad Pu´blica de Navarra, Departamento de Matema´ticas, Campus Arrosadı´a. Edificio ‘‘Las Encinas’’, E-31006 Pamplona, Spain b Universidad de Zaragoza, Facultad de Ciencias Econo´micas y Empresariales, Departamento deAna´lisis Econo´mico, c/ Doctor Cerrada 1-3. E-50005, Zaragoza, Spain Received 29 March 2006; received in revised form 7 September 2007 Available online 30 October 2007
Abstract In the framework of the analysis of orderings whose associated indifference relation is not transitive, we study the structure of a semiorder, paying attention to the problem of its continuous representability through a real-valued function and a positive threshold. For the case of connected topological spaces, we obtain a full characterization of the continuous representability of semiorders without extremal elements. r 2007 Elsevier Inc. All rights reserved. MSC: 54F05; secondary: 06A06, 91B16 Keywords: Totally preordered structures; Semiorders; Continuous numerical representations
1. Introduction The concept of a semiorder was introduced by Luce (1956) in order to build a model to interpret situations of intransitive indifference. Classical studies on semiorders appeared in Fishburn (1970a, 1970b, 1973, 1982, 1985), Manders (1981), Gensemer (1987a, 1987b, 1988), Pirlot (1990, 1991), Beja and Gilboa (1992), Bosi and Isler (1995), Gilboa and Lapson (1995), or Pirlot and Vincke (1997). In some of those studies were obtained either necessary or else sufficient conditions for the existence of a numerical representation of a semiorder defined on a nonempty set X by means of a real-valued function f : X ! R and a nonnegative constant or discrimination threshold KX0 such that a b3f ðaÞ þ Kof ðbÞ ða; b 2 X Þ. A characterization of the existence of this kind of representation appears in Candeal, Indura´in, and Zudaire (2002).
Corresponding author. Fax: +34 948 166057.
E-mail addresses: [email protected] (M.J. Campio´n), [email protected] (J.C. Candeal), [email protected] (E. Indura´in), [email protected] (M. Zudaire). 0022-2496/$ - see front matter r 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2007.09.006
This kind of numerical representations of semiordered structures is also encountered in a wide range of applications, as extensive measurement in Mathematical Psychology (see Krantz, 1967), choice theory under risk (see Fishburn, 1968), decision-making under risk (see Rubinstein, 1988), modellization of choice with errors (see Agaev & Aleskerov, 1993), social welfare theory (see Ng, 1975), general equilibrium theory in Economics (see Jamison & Lau, 1977), and expected utility theory in mixture spaces (see Vincke, 1980). Perhaps the main difficulty to get a characterization of the numerical representability of semiorders is due to the following remarkable fact: the known characterizations of the representability of other classical ordered structures such as total preorders (see e.g. the first chapter in Bridges & Mehta, 1995) and interval orders (see Bosi, Candeal, Indura´in, Olo´riz, & Zudaire, 2001; Doignon, Ducamp, & Falmagne, 1984; Fishburn, 1973; Olo´riz, Candeal, & Indura´in, 1998; or Candeal, Indura´in, & Zudaire, 2004) have always been given in terms of the existence of suitable countable subsets. A consequence is that countable total preorders and countable interval orders are (trivially) representable. But the analogous property is no longer true for semiorders: There exist countable semiorders that are
ARTICLE IN PRESS M.J. Campio´n et al. / Journal of Mathematical Psychology 52 (2008) 48–54
not representable (see e.g. Manders, 1981; Scott & Suppes, 1958; or Candeal et al., 2002). Suppose now that a semiorder has been defined on a nonempty set X endowed with a topology t. In this situation, it is natural to look for the existence of continuous representations of the semiordered structure ðX ; Þ, that is, we will be mainly interested in characterizing the existence of a real-valued function f : X ! R and a non-negative constant or threshold KX0 such that a b3f ðaÞ þ Kof ðbÞ ða; b 2 X Þ, where, in addition, the map f : ðX ; tÞ ! ðR; usual topologyÞ is continuous. The analogous problem for total preorders was already solved by Debreu (1954, 1964). If " is a total preorder on a topological space ðX ; tÞ, the preorder " is said to be continuous with respect to the topology t (or, equivalently, the topology t is said to be natural for the preorder ") if for every x 2 X it happens that the sets fy 2 X : x"yg and fz 2 X : z"xg are t-closed. Debreu proved that if ðX ; "Þ is a totally preordered structure endowed with a natural topology t, then the existence of a numerical representation (not necessarily continuous!) for " is equivalent to the existence of another numerical representation for " that is, in addition, continuous. Moreover, it is well known (see e.g. the first chapters in Bridges & Mehta, 1995) that a totally preordered structure ðX ; "Þ is representable through a (continuous, if necessary!) real-valued function if and only if there exists a countable subset D X such that for every x; y 2 X with :ðy"xÞ there exists d 2 D such that x"d"y. In this case, the structure ðX ; "Þ is said to be separable in the sense of Debreu, and the subset D is said to be order-dense in X. However, the general situation for semiorders is still an open problem. It is not known if the existence of a numerical representation through a real-valued function (not necessarily continuous!) and a threshold is equivalent to the existence of another numerical representation through a continuous real-valued function and a threshold. Also, despite some partial results were obtained by Gensemer (1987a, 1987b, 1988), no general characterization of the continuous representability of a semiorder is known. In the present paper we shall furnish a characterization, that is valid for connected topological spaces, of the continuous representability, through a real-valued map and a positive threshold, of a semiorder without extremal elements. 2. Preliminaries Let X be a nonempty set. Let be an asymmetric binary relation defined on X. Given x 2 X , the sets LðxÞ ¼ fy 2 X : y xg and GðxÞ ¼ fy 2 X : x yg are called, respectively, the lower contour set and the upper contour set of x relative to . Associated to we define the reflexive and total binary relation " given by x"y3:ðx yÞ ðx; y 2 X Þ, and the symmetric binary relation , called indifference, given by xy3½ð:ðx yÞÞ ^ ð:ðy xÞÞ ðx; y 2 X Þ.
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An element x 2 X is said to be maximal (respectively, minimal) with respect to the asymmetric binary relation defined on X if there is no y 2 X such that x y (respectively, there is no z 2 X such that z x). A maximal or minimal element with respect to the relation is said to be an extremal element with respect to . An interval order is an asymmetric binary relation such that ½ðx yÞ and ðz tÞ ) ½ðx tÞ or ðz yÞ ðx; y; z; t 2 X Þ. An interval order is said to be a semiorder if ½ðx yÞ and ðy zÞ ) ½ðx wÞ or ðw zÞ for every x; y; z; w 2 X . An interval order defined on X is said to be representable (as an interval order) if there exist two real valued maps u; v : X ! R such that x y3vðxÞouðyÞ ðx; y 2 X Þ. Also, a semiorder defined on X is said to be representable (now, as a semiorder!) in the sense of Scott and Suppes (see Scott & Suppes, 1958) if there exist a realvalued map u : X ! R and a nonnegative constant or ‘‘discrimination threshold’’ KX0 such that x y3 KouðyÞ uðxÞ ðx; y 2 X Þ. Obviously, if such a representation exists with K ¼ 0, the binary relation " associated to the semiorder is actually a total preorder. This is equivalent to say that the associated indifference is transitive. There exist interval orders that fail to be representable (as interval orders). Also, there exist semiorders that are not representable in the sense of Scott and Suppes. (See Candeal et al., 2002; Olo´riz et al., 1998 for further details.) Following Fishburn (1970a, 1970c), associated to an interval order defined on a nonempty set X, we shall consider two new binary relations and given by x y3x z"y for some z 2 X ðx; y 2 X Þ, and, similarly, x y3x"z y for some z 2 X ðx; y 2 X Þ. We denote x" y3:ðy xÞ, x y3x" y" x, x" y3:ðy xÞ and x y3x" y" x ðx; y 2 X Þ. Now it is straightforward to see that x" y3GðyÞ GðxÞ and also x" y3LðxÞ LðyÞ ðx; y 2 X Þ. As a matter of fact, both the binary relations " and " are total preorders on X. Moreover, the indifference relation associated to the interval order is transitive if and only if " , " and the binary relation " associated to the interval order coincide. In particular, in this case " is also a total preorder on X. (See Monjardet, 1978 for more details.) Moreover, a new binary relation "0 is defined on X with the help of and , by declaring that x"0 y3 ½ðx" yÞ ^ ðx" yÞ ðx; y 2 X Þ. This new binary relation "0 allows us to characterize semiorders among interval orders, as proved in Fishburn (1970a, 1970b), by means of the following useful lemma. Lemma 2.1. Let X be a nonempty set and an asymmetric binary relation on X. Then the binary relation is a semiorder if and only if "0 is a total preorder on X. The representability (not necessarily continuous!) of interval orders is characterized (see Proposition 8 in Doignon et al., 1984) by the existence of a countable
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subset D X , such that for every x; y 2 X with x y there exists an element d in D such that x d" y. Such a set is called a widely dense subset of X. Lemma 2.2. Let be an interval order on a nonempty set X. The relation is representable if and only if there exists a widely dense countable set D X . Remark 2.3. The path that we will follow here (in the next Section 3) to get a characterization of the continuous representability of semiorders, valid for connected topological spaces without extremal elements goes as follows, in three steps: 1. Since a semiorder defined on a nonempty set X is a very particular case of an interval order, first we will try to find representations of the semiorder but merely as an interval order through a pair of functions u; v : X ! R such that x y3vðxÞouðyÞ. 2. Assuming now that X is endowed with a topology t, the next step will be looking for a pair of continuous functions representing , again as an interval order. 3. In the final step we will look for a modification of the pair u; v to another pair of continuous functions U; V : ðX ; tÞ ! ðR; usual topologyÞ such that this new pair still represents as an interval order, but now there is a positive constant K40, or ‘‘threshold’’, such that V ðxÞ UðxÞ ¼ K ðx 2 X Þ. As commented before, the details of all this process will be analyzed in Section 3. 3. Continuous representability of semiorders on connected topological spaces To deal with continuous representations of semiorders, we introduce now some preparatory definitions and results. Definition. A semiorder defined on a nonempty topological space ðX ; tÞ is said to be typical if its associated indifference relation is not transitive, or equivalently if its associated relation " is not a total preorder. In what follows all the semiorders that we shall consider will be assumed to be typical.
y3KoUðyÞ UðxÞ ðx; y 2 X Þ). Then the semiorder is tcontinuous. Definition. Let be an interval order defined on a nonempty set X. An element x 2 X is said to be a singular point with respect to if for every y; z 2 X it holds that ½ðxyÞ ^ ðxzÞ ) ðyzÞ. Remark 3.2. Observe that in the particular case in which the associated relation " is a total preorder, then every element x 2 X is a singular point. When is a typical semiorder, the existence of many singular points tends to preclude the representability of the semiorder through a real-valued map u : X ! R and a positive threshold K40. Consider the following example: Let X ¼ ½0; 1 [ ½4; 6 and the semiorder defined by x y3½ðx; y 2 ½0; 1 and xoyÞ or ðxp1; yX4Þ or ðx; y 2 ½4; 6 and x þ 1oyÞ. Here all the elements in ½0; 1 are singular. Moreover is not representable through a real-valued map u : X ! R and a positive threshold K40 since, otherwise, the fact 0 2ðnþ1Þ 2n 1 ðnX1; n 2 NÞ would imply that uð0Þouð1Þ nK ðnX1; n 2 NÞ, which is impossible. Lemma 3.3. Let be an interval order defined on a connected topological space ðX ; tÞ. The following conditions are equivalent:
(i) The interval order admits a representation through a pair of continuous functions u; v : ðX ; tÞ ! ðR; usual topologyÞ such that x y3vðxÞouðyÞ ðx; y 2 X Þ. (ii) There exists a widely dense countable subset D X for , and for every x 2 X the lower contour set LðxÞ, the upper contour set GðxÞ, and the subsets fy 2 X : y xg, fy 2 X : y xg, fz 2 X : x zg and fz 2 X : x zg are t-open. (iii) The interval order admits a representation through a pair of continuous functions U; V : ðX ; tÞ ! ðR; usual topologyÞ such that x y3V ðxÞoUðyÞ ðx; y 2 X Þ and, in addition x y3UðxÞoUðyÞ; x y3V ðxÞoV ðyÞ ðx; y 2 X Þ. Proof. See the equivalences (i) 3 (viii) 3 (x) of Theorem 3.5 in Candeal et al. (2004). &
Definition. A semiorder defined on a nonempty topological space ðX ; tÞ is said to be t-continuous if for every x 2 X it holds that the lower contour set LðxÞ ¼ fy 2 X : y xg and the upper contour set GðxÞ ¼ fz 2 X : x zg of x relative to are t-open.
Lemma 3.4. Let I be an open real interval. Let h : I ! I a continuous and strictly increasing map. Then there exists a continuous and strictly increasing map F : I ! R that satisfies the Abel equation
The following fact is an immediate consequence of the last definition.
Proof. See Theorem 2.1 in Kuczma (1968), or alternatively pp. 133 and ff. in Acze´l (1987). &
Proposition 3.1. Let be a semiorder defined on a nonempty topological space ðX ; tÞ. Suppose that there exist a continuous map U : ðX ; tÞ ! (R, usual topology) and a positive threshold K40, that constitute a representation in the sense of Scott–Suppes of the semiorder (i.e.: x
Now we are ready to introduce the main result, that characterizes the existence of continuous Scott–Suppes representations with positive threshold for semiorders without extremal elements, defined on connected topological spaces.
F ðhðxÞÞ ¼ F ðxÞ þ 1
ðx 2 IÞ.
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Theorem 3.5. Let be a continuous semiorder without extremal elements, defined on a connected topological space ðX ; tÞ. Then there exist a positive threshold K40 and a continuous map u : ðX ; tÞ ! R such that x y3uðyÞ uðxÞ4K ðx; y 2 X Þ if and only if the following conditions hold: (i) There exists a widely dense countable subset D X for the binary relation considered as an interval order on X. (ii) The associated relations and coincide (i.e.: x y3x y ðx; y 2 X Þ). In particular the total preorders " ; " and "0 are identical. (iii) The binary relation has no singular points. Proof. Suppose that the semiorder is representable through the continuous real valued function u : X ! R and the threshold K40. Let v : X ! R be defined as vðxÞ ¼ uðxÞ þ K ðx 2 X Þ. Observe that the pair ðu; vÞ furnishes a representation of as an interval order, so that by Lemma 2.2, there exists a widely dense countable subset D X . Let now x; y 2 X such that x y. By definition of , there exists a 2 X such that x a"y. Thus we have uðxÞ þ KouðaÞpuðyÞ þ K, so that uðxÞouðyÞ. Since, by hypothesis, x is not a minimal element with respect to , there exists z 2 X such that z x. Thus uðzÞ þ KouðxÞouðyÞouðyÞ þ K. By continuity of u and connectedness of ðX ; tÞ there exists w 2 X such that uðxÞpuðwÞ þ KouðyÞ. Consequently, we get x"w y. Hence x y. The fact x y ) x y ðx; y 2 X Þ can be proved in an entirely analogous way. Therefore and coincide. Let now x 2 X . Since x is not an extremal element with respect to , there exist y; z 2 X such that z x y. Thus uðzÞ þ KouðxÞouðxÞ þ KouðyÞ. Again by continuity of u and connectedness of ðX ; tÞ, there exists s 2 X such that uðxÞpuðsÞ þ KouðxÞ þ KouðyÞ.In particular, we have that uðxÞpuðsÞ þ K. Thus x"s. Moreover, uðsÞ þ KouðxÞ þ K obviously implies uðsÞouðxÞ. Since s is not a minimal element, this implies as before that s x. But, as just proved, this is equivalent to say that s x. Therefore, there exists t 2 X such that x"s t"x. Hence uðxÞpuðsÞ þ KouðtÞpuðxÞ þ K. Now observe that xs and xt, but s t, so that x is not a singular point. For the converse, suppose now that the conditions (i)–(iii) hold. Given x; y 2 X , observe that if y x then there exists a 2 X such that y a"x. Thus [ fy 2 X : y ag. fy 2 X : y xg ¼ fa2X ;a"xg
Similarly fz 2 X : x zg ¼
[
fz 2 X : b zg.
fb2X ;x"bg
Since by condition (ii) the binary relations and coincide, and, by hypothesis, the semiorder is tcontinuous, it follows that the subsets fy 2 X : y xg and fz 2 X : x zg are t-open for every x 2 X . Consequently,
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by Lemma 3.3 it follows that admits a representation (as an interval order) through a pair of continuous functions u; v : ðX ; tÞ ! ðR; usual topologyÞ such that x y3 vðxÞouðyÞ ðx; y 2 X Þ and, in addition x y3uðxÞo uðyÞ; x y3vðxÞovðyÞ ðx; y 2 X Þ. Since is in particular asymmetric, it follows now that uðxÞpvðxÞ for every x 2 X . Suppose that there exists t 2 X such that uðtÞ ¼ vðtÞ. In this case, given y; z 2 X with ty and tz it follows that uðtÞpvðyÞ; uðyÞpvðtÞ; uðtÞpvðzÞ; uðzÞpvðtÞ: Consequently, we obtain that uðyÞpvðtÞ ¼ uðtÞpvðzÞ. Hence y"z. We also have that uðzÞpvðtÞ ¼ uðtÞpvðyÞ, which implies z"y. Thus we conclude that yz. This implies that t is a singular point and contradicts condition (iii). Therefore uðxÞovðxÞ, for every x 2 X . Since has no extremal elements and ðX ; tÞ is connected, and the maps u; v are continuous, it follows that the real subsets uðX Þ and vðX Þ are open intervals of R. Again because has no extremal elements, given x 2 X there exist elements y; z 2 X such that z x y. Thus uðzÞovðzÞouðxÞovðxÞouðyÞovðyÞ. Hence uðxÞ lies in the real interval vðX Þ, and similarly vðxÞ belongs to the real interval uðX Þ. Therefore uðX Þ and vðX Þ coincide. Denote I ¼ uðX Þ ¼ vðX Þ. Let h : I ! I be the map defined by hðuðxÞÞ ¼ vðxÞ ðx 2 X Þ. Given now x1 ; x2 2 X such that uðx1 Þouðx2 Þ, it follows that x1 x2 . Since and coincide by condition (ii), we obtain that x1 x2 . Thus vðx1 Þovðx2 Þ or, equivalently, hðuðx1 ÞÞohðuðx2 ÞÞ. Therefore the map h is strictly increasing. In the same way, we may prove that vðx1 Þovðx2 Þ ) uðx1 Þouðx2 Þ. Given a; b 2 I such that aob, since I ¼ vðX Þ we may choose elements xa ; xb 2 X such that vðxa Þ ¼ a; vðxb Þ ¼ b. Let now c 2 ðuðxa Þ; uðxb ÞÞ. Since uðX Þ has been proved to be an open real interval, we may choose an element xc 2 X such that uðxc Þ ¼ c. Moreover, by a previous argument we have that uðxa Þouðxc Þouðxb Þ ) a ¼ vðxa Þovðxc Þo vðxb Þ ¼ b. Hence vðxc Þ 2 ða; bÞ and c ¼ uðxc Þ 2 h1 ða; bÞ. Conversely, suppose now that an element d 2 uðX Þ satisfies that hðdÞ lies in ða; bÞ. If we choose an element xd 2 X such that uðxd Þ ¼ d, we have hðdÞ ¼ vðxd Þ and a ¼ vðxa Þo vðxd Þovðxb Þ ¼ b. As before, this implies that uðxa Þo uðxd Þ ¼ douðxb Þ. Thus we have that h1 ða; bÞ ¼ uðxa ; xb Þ. which is an open interval of the real line R. Therefore the map h is continuous. Now we can apply Lemma 3.4 to conclude that there exist a strictly increasing and continuous real function F : I ! R such that F ðhðtÞÞ ¼ F ðtÞ þ 1 for every t 2 I ¼ uðX Þ, or equivalently F ðvðxÞÞ ¼ F ðuðxÞÞ þ 1 for every x 2 X . Finally, it is straightforward to see that the continuous map F u : ðX ; tÞ ! R jointly with the threshold 1 furnish a Scott–Suppes representation of the semiorder . Indeed, since F is strictly increasing, we get x y3vðxÞouðyÞ3F ðvðxÞÞoF ðuðyÞÞ3F ðuðxÞÞþ 1oF ðuðyÞÞ, for every x; y 2 X . This concludes the proof. &
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Remark 3.6. In the statement of Theorem 3.5 there is no loss of generality in assuming that the given topology t on X is the order topology y generated by the semiorder . A subbasis for y is given by the family of contour sets fLðxÞ : x 2 X g [ fGðxÞ : x 2 X g. Since is t-continuous, the topology y must be coarser than t, hence connected as well. In addition, is plainly y-continuous. Suppose that is representable through a continuous map u : ðX ; yÞ ! ðR; usual topologyÞ and a positive threshold K40 such that x y3uðxÞ þ KouðyÞ ðx; y 2 X Þ. Since is t-continuous it is clear that u is also continuous if we consider on X the topology t and on R the usual topology. Theorem 3.5 completes the panorama of the Theorem on p. 288 of Gensemer (1987a) and Theorem 3.1 in Gensemer (1988). Through those results, some characterizations of the continuous representability of typical semiorders given on connected topological spaces were also achieved, so that it seems interesting, for the sake of completeness, to compare now our main result (Theorem 3.5 above) with Gensemer’s ones, pointing out the most noticeable differences. To start with, we should observe that Gensemer studies semiorders that may have extremal elements, whereas our Theorem 3.5 deals with semiorders without extremal elements. Apparently, this is a more restrictive condition. However, there is no loss of generality in assuming such condition of lacking of extremal elements, as the next Proposition 3.7 proves. Proposition 3.7. A continuous typical semiorder defined on a connected topological space ðX ; tÞ admits a continuous Scott–Suppes representation if and only if there exist a connected topological space ðY ; tY Þ that extends ðX ; tÞ, and an extension Y of the semiorder to Y, such that Y has no extremal elements and admits a continuous Scott–Suppes representation. Proof. Let us assume first that is representable through a continuous map u : ðX ; tÞ ! ðR; usual topologyÞ and a positive threshold K. For every a 2 RnuðX Þ we add an extra element ta to X. Let Y ¼ X [ fta : a 2 RnuðX Þg. Consider the map uY : Y ! R given by uY ðxÞ ¼ uðxÞ if x 2 X , uðta Þ ¼ a if a 2 RnuðX Þ. Let Y the semiorder defined on Y by declaring that y1 Y y2 3uY ðy1 Þ þ KouY ðy2 Þ ðy1 ; y2 2 Y Þ. It is straightforward to see that there exists on Y a topology tY that satisfies the following properties: (i) (ii) (iii) (iv) (v)
Y is connected with respect to tY . The restriction of tY to X is t. The restriction of Y to X is . The semiorder Y has no extremal elements. The semiorder Y is continuously representable through the map uY : ðY ; tY Þ ! ðR; usual topologyÞ and the positive threshold K.
Conversely, suppose now that there exist a connected topological space ðY ; tY Þ that extends ðX ; tÞ and an extension Y of the semiorder to Y, such that Y has no extremal elements and admits a continuous Scott–Suppes representation given by a map u : ðY ; tY Þ ! ðR; usual topologyÞ. Let uX : ðX ; tÞ ! ðR; usual topologyÞ be the restriction of u to X. Since t is the restriction of tY to X, it is clear that the map uX jointly with the positive threshold K constitute a continuous Scott–Suppes representation of the semiorder defined on X. & Remark 3.8. The semiorders that are characterized in Theorem 3.5 are isomorphic to the semiorder defined on the real line R by declaring that x y3x þ 1oy ðx; y 2 RÞ. However, by Proposition 3.7 we see that if a continuous typical semiorder defined on a connected topological space ðX ; tÞ admits a continuous Scott–Suppes representation, then it can be considered as a part of ðR; Þ. A clear example of this situation is the semiorder defined on the open interval ð0; 10Þ of the real line by declaring that x y3x þ 1oy ðx; y 2 ð0; 10ÞÞ. On ð0; 10Þ we shall consider the Euclidean topology, that does not coincide with the topology generated by the semiorder. Obviously, ðð0; 10Þ; Þ is not isomorphic to ðR; Þ since it does not contain any infinite sequence ðxn Þn2N of elements such that x0 x1 x2 xk xkþ1 . Observe in addition that all the points in ð0; 1 are minimal (respectively, all the elements in ½9; 10Þ are maximal) with respect to the semiorder defined on ð0; 10Þ, so that Theorem 3.5 does not apply (directly) to this example. But, as in Proposition 3.7, if we consider first ðð0; 10Þ; Þ embedded in ðR; Þ, where the real line R is also endowed with the Euclidean topology, the restriction to ð0; 10Þ of the (obvious) continuous Scott–Suppes representation of ðR; Þ furnishes a continuous Scott–Suppes representation of ðð0; 10Þ; Þ. Continuing our discussion, we may also observe that in the Theorem on p. 288 of Gensemer (1987a), some of the conditions required in the characterization of the representability of a continuous semiorder defined on a connected topological space ðX ; tÞ given there are more restrictive than the ones we use in our Theorem 3.5 in the present paper. Thus, in Gensemer (1987a) the associated total preorders " and " are asked to be continuous with respect to the given topology t (that is, the sets fy 2 X : :ðy" xÞg; fy 2 X : :ðx" yÞg; fy 2 X : :ðx" yÞg and fy 2 X : :ðy" xÞg must be t-open sets). We do not need to assume (a priori) any of those conditions involving continuity. Actually, we could obtain here those conditions from the continuity of and the coincidence of the total preorders " ; " and "0 , as in the proof of Theorem 3.5. Another condition used in the statement of the Theorem on p. 288 of Gensemer (1987a) is the strong separability of the binary relation . This condition means that there must exist a countable subset D X such that for every x; y 2 X with x y there exist a; b 2 D such that x a"b y. It is
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well known that the condition of strong separability is more restrictive than the condition of the existence of a widely dense countable subset, that we use in our Theorem 3.5. (See e.g., Candeal et al. 2004 for details.) More conditions used in the statement of the Theorem on p. 288 of Gensemer (1987a) are the so-called regularity and normality of the semiordered structure ðX ; Þ. They were introduced to tackle the problem of the existence of maximal or minimal elements. It can be proved that when no extremal element exists, then Gensemer’s condition of regularity plus normality amounts to our condition of coincidence of and . On the other hand, Theorem 3.1. in Gensemer (1988) also gives a characterization of the continuous representability of a semiorder defined on a connected topological space ðX ; tÞ. As in our approach, Gensemer handles first the binary relation as an interval order to deduct conditions about its Scott–Suppes representability as a semiorder. However, the main feature of Theorem 3.1 in Gensemer (1988) is that it starts from a particular tcontinuous representation of ðX ; Þ as an interval order, without saying which extra conditions could lead to the existence of such a kind of representation. As a matter of fact, it is not true, in general, that the condition of the existence of a widely dense countable subset, nor even the more restrictive one of strong separability, implies that the binary relation can be represented through a pair of functions u; v : X ! R such that x y3vðxÞouðyÞ ðx; y 2 X Þ satisfying also that, in addition, uðxÞovðxÞ for every x 2 X and also uðxÞouðyÞ3vðxÞovðyÞ ðx; y 2 X Þ. In other words, some additional conditions should be given to capture the last facts uðxÞovðxÞ for every x 2 X and also uðxÞouðyÞ3vðxÞovðyÞ for every x; y 2 X . To conclude our discussion, we point out that, except for the obvious fact of the semiorder being t-continuous, all the conditions that we use in the statement of Theorem 3.5 to study the continuous representability of a semiorder defined on a connected topological space ðX ; tÞ, are independent of the topology t. Unlike the conditions we use in our approach, the conditions required in the statement of the Theorem on p. 288 of Gensemer (1987a) as well as those on the statement of Theorem 3.1 in Gensemer (1988) depend on the given topology t. The reason is that they involve continuity of the associated binary relations ; or they ask for the existence (a priori) of a particular continuous representation of ðX ; Þ as an interval order. Acknowledgements Thanks are due to Prof. Dr. Susan H. Gensemer (Syracuse, NY), Prof. Dr. Sergei Ovchinnikov (San Francisco, CA) and an anonymous referee for their valuable suggestions and help. This work has been supported by the research project MTM 2006 - 15025 ‘‘Espacios Topolo´gicos Ordenados’’, of the Spanish Ministry of Culture.
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References Acze´l, J. (1987). A short course on functional equations based upon recent applications to the social and behavioral sciences. Dordrecht, The Netherlands: D. Reidel. Agaev, R., & Aleskerov, F. (1993). Interval choice: Classic and general cases. Mathematical Social Sciences, 26, 249–272. Beja, A., & Gilboa, I. (1992). Numerical representations of imperfectly ordered preferences (A unified geometric exposition). Journal of Mathematical Psychology, 36, 426–449. Bosi, G., Candeal, J. C., Indura´in, E., Olo´riz, E., & Zudaire, M. (2001). Numerical representations of interval orders. Order, 18, 171–190. Bosi, G., & Isler, R. (1995). Representing preferences with nontransitive indifference by a single real-valued function. Journal of Mathematical Economics, 24, 621–631. Bridges, D. S., & Mehta, G. B. (1995). Representations of preference orderings. Berlin: Springer. Candeal, J. C., Indura´in, E., & Zudaire, M. (2002). Numerical representability of semiorders. Mathematical Social Sciences, 43(1), 61–77. Candeal, J. C., Indura´in, E., & Zudaire, M. (2004). Continuous representability of interval orders. Applied General Topology, 5(2), 213–230. Debreu, G. (1954). Representation of a preference ordering by a numerical function. In R. Thrall, C. Coombs, & R. Davies (Eds.), Decision processes. Wiley: New York. Debreu, G. (1964). Continuity properties of Paretian utility. International Economic Review, 5, 285–293. Doignon, J. P., Ducamp, A., & Falmagne, J. C. (1984). On realizable biorders and the biorder dimension of a relation. Journal of Mathematical Psychology, 28, 73–109. Fishburn, P. C. (1968). Semiorders and risky choices. Journal of Mathematical Psychology, 5, 358–361. Fishburn, P. C. (1970a). Utility theory for decision-making. New York: Wiley. Fishburn, P. C. (1970b). Intransitive indifference in preference theory: A survey. Operations Research, 18(2), 207–228. Fishburn, P. C. (1970c). Intransitive indifference with unequal indifference intervals. Journal of Mathematical Psychology, 7, 144–149. Fishburn, P. C. (1973). Interval representations for interval orders and semiorders. Journal of Mathematical Psychology, 10, 91–105. Fishburn, P. C. (1982). Aspects of semiorders within interval orders. Discrete Mathematics, 40, 181–191. Fishburn, P. C. (1985). Interval orders and interval graphs. A study of partially ordered sets. New York: Wiley. Gensemer, S. H. (1987a). Continuous semiorder representations. Journal of Mathematical Economics, 16, 275–289. Gensemer, S. H. (1987b). On relationships between numerical representations of interval orders and semiorders. Journal of Economic Theory, 43, 157–169. Gensemer, S. H. (1988). On numerical representations of semiorders. Mathematical Social Sciences, 15(3), 277–286. Gilboa, I., & Lapson, R. (1995). Aggregation of semiorders: Intransitive indifference makes a difference. Economic Theory, 5, 109–126. Jamison, D. T., & Lau, L. J. (1977). The nature of equilibrium with semiordered preferences. Econometrica, 45, 1595–1605. Krantz, D. H. (1967). Extensive measurement in semiorders. Philosophy of Science, 34, 348–362. Kuczma, M. (1968). Functional equations in a single variable. Warsaw, Poland: Polska Akademia Nauk. Luce, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24, 178–191. Manders, K. L. (1981). On JND representations of semiorders. Journal of Mathematical Psychology, 24(3), 224–248. Monjardet, B. (1978). Axiomatiques et proprie´te´s des quasi-ordres. Mathe´matiques et Sciences Humaines, 63, 51–82. Ng, Y.-K. (1975). Bentham or Bergson? Finite sensibility, utility functions and social welfare functions. Review of Economic Studies, 42, 545–569.
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Olo´riz, E., Candeal, J. C., & Indura´in, E. (1998). Representability of interval orders. Journal of Economic Theory, 78(1), 219–227. Pirlot, M. (1990). Minimal representation of a semiorder. Theory and Decision, 28, 109–141. Pirlot, M. (1991). Synthetic representation of a semiorder. Discrete Applied Mathematics, 31, 299–308. Pirlot, M., & Vincke, Ph. (1997). Semiorders: Properties, representations, applications. Dordrecht: Kluwer.
Rubinstein, A. (1988). Similarity and decision-making under Risk (Is there a utility theory resolution to the Allais paradox)? Journal of Economic Theory, 46, 145–153. Scott, D., & Suppes, P. (1958). Foundational aspects of theories of measurement. Journal of Symbolic Logic, 23, 113–128. Vincke, Ph. (1980). Linear utility functions on semiordered mixture spaces. Econometrica, 48(3), 771–775.
Journal of Mathematical Psychology 52 (2008) 48–54 www.elsevier.com/locate/jmp
Continuous representability of semiorders Marı´ a J. Campio´na, Juan C. Candealb, Esteban Indura´ina,, Margarita Zudairea a
Universidad Pu´blica de Navarra, Departamento de Matema´ticas, Campus Arrosadı´a. Edificio ‘‘Las Encinas’’, E-31006 Pamplona, Spain b Universidad de Zaragoza, Facultad de Ciencias Econo´micas y Empresariales, Departamento deAna´lisis Econo´mico, c/ Doctor Cerrada 1-3. E-50005, Zaragoza, Spain Received 29 March 2006; received in revised form 7 September 2007 Available online 30 October 2007
Abstract In the framework of the analysis of orderings whose associated indifference relation is not transitive, we study the structure of a semiorder, paying attention to the problem of its continuous representability through a real-valued function and a positive threshold. For the case of connected topological spaces, we obtain a full characterization of the continuous representability of semiorders without extremal elements. r 2007 Elsevier Inc. All rights reserved. MSC: 54F05; secondary: 06A06, 91B16 Keywords: Totally preordered structures; Semiorders; Continuous numerical representations
1. Introduction The concept of a semiorder was introduced by Luce (1956) in order to build a model to interpret situations of intransitive indifference. Classical studies on semiorders appeared in Fishburn (1970a, 1970b, 1973, 1982, 1985), Manders (1981), Gensemer (1987a, 1987b, 1988), Pirlot (1990, 1991), Beja and Gilboa (1992), Bosi and Isler (1995), Gilboa and Lapson (1995), or Pirlot and Vincke (1997). In some of those studies were obtained either necessary or else sufficient conditions for the existence of a numerical representation of a semiorder defined on a nonempty set X by means of a real-valued function f : X ! R and a nonnegative constant or discrimination threshold KX0 such that a b3f ðaÞ þ Kof ðbÞ ða; b 2 X Þ. A characterization of the existence of this kind of representation appears in Candeal, Indura´in, and Zudaire (2002).
Corresponding author. Fax: +34 948 166057.
E-mail addresses: [email protected] (M.J. Campio´n), [email protected] (J.C. Candeal), [email protected] (E. Indura´in), [email protected] (M. Zudaire). 0022-2496/$ - see front matter r 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2007.09.006
This kind of numerical representations of semiordered structures is also encountered in a wide range of applications, as extensive measurement in Mathematical Psychology (see Krantz, 1967), choice theory under risk (see Fishburn, 1968), decision-making under risk (see Rubinstein, 1988), modellization of choice with errors (see Agaev & Aleskerov, 1993), social welfare theory (see Ng, 1975), general equilibrium theory in Economics (see Jamison & Lau, 1977), and expected utility theory in mixture spaces (see Vincke, 1980). Perhaps the main difficulty to get a characterization of the numerical representability of semiorders is due to the following remarkable fact: the known characterizations of the representability of other classical ordered structures such as total preorders (see e.g. the first chapter in Bridges & Mehta, 1995) and interval orders (see Bosi, Candeal, Indura´in, Olo´riz, & Zudaire, 2001; Doignon, Ducamp, & Falmagne, 1984; Fishburn, 1973; Olo´riz, Candeal, & Indura´in, 1998; or Candeal, Indura´in, & Zudaire, 2004) have always been given in terms of the existence of suitable countable subsets. A consequence is that countable total preorders and countable interval orders are (trivially) representable. But the analogous property is no longer true for semiorders: There exist countable semiorders that are
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not representable (see e.g. Manders, 1981; Scott & Suppes, 1958; or Candeal et al., 2002). Suppose now that a semiorder has been defined on a nonempty set X endowed with a topology t. In this situation, it is natural to look for the existence of continuous representations of the semiordered structure ðX ; Þ, that is, we will be mainly interested in characterizing the existence of a real-valued function f : X ! R and a non-negative constant or threshold KX0 such that a b3f ðaÞ þ Kof ðbÞ ða; b 2 X Þ, where, in addition, the map f : ðX ; tÞ ! ðR; usual topologyÞ is continuous. The analogous problem for total preorders was already solved by Debreu (1954, 1964). If " is a total preorder on a topological space ðX ; tÞ, the preorder " is said to be continuous with respect to the topology t (or, equivalently, the topology t is said to be natural for the preorder ") if for every x 2 X it happens that the sets fy 2 X : x"yg and fz 2 X : z"xg are t-closed. Debreu proved that if ðX ; "Þ is a totally preordered structure endowed with a natural topology t, then the existence of a numerical representation (not necessarily continuous!) for " is equivalent to the existence of another numerical representation for " that is, in addition, continuous. Moreover, it is well known (see e.g. the first chapters in Bridges & Mehta, 1995) that a totally preordered structure ðX ; "Þ is representable through a (continuous, if necessary!) real-valued function if and only if there exists a countable subset D X such that for every x; y 2 X with :ðy"xÞ there exists d 2 D such that x"d"y. In this case, the structure ðX ; "Þ is said to be separable in the sense of Debreu, and the subset D is said to be order-dense in X. However, the general situation for semiorders is still an open problem. It is not known if the existence of a numerical representation through a real-valued function (not necessarily continuous!) and a threshold is equivalent to the existence of another numerical representation through a continuous real-valued function and a threshold. Also, despite some partial results were obtained by Gensemer (1987a, 1987b, 1988), no general characterization of the continuous representability of a semiorder is known. In the present paper we shall furnish a characterization, that is valid for connected topological spaces, of the continuous representability, through a real-valued map and a positive threshold, of a semiorder without extremal elements. 2. Preliminaries Let X be a nonempty set. Let be an asymmetric binary relation defined on X. Given x 2 X , the sets LðxÞ ¼ fy 2 X : y xg and GðxÞ ¼ fy 2 X : x yg are called, respectively, the lower contour set and the upper contour set of x relative to . Associated to we define the reflexive and total binary relation " given by x"y3:ðx yÞ ðx; y 2 X Þ, and the symmetric binary relation , called indifference, given by xy3½ð:ðx yÞÞ ^ ð:ðy xÞÞ ðx; y 2 X Þ.
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An element x 2 X is said to be maximal (respectively, minimal) with respect to the asymmetric binary relation defined on X if there is no y 2 X such that x y (respectively, there is no z 2 X such that z x). A maximal or minimal element with respect to the relation is said to be an extremal element with respect to . An interval order is an asymmetric binary relation such that ½ðx yÞ and ðz tÞ ) ½ðx tÞ or ðz yÞ ðx; y; z; t 2 X Þ. An interval order is said to be a semiorder if ½ðx yÞ and ðy zÞ ) ½ðx wÞ or ðw zÞ for every x; y; z; w 2 X . An interval order defined on X is said to be representable (as an interval order) if there exist two real valued maps u; v : X ! R such that x y3vðxÞouðyÞ ðx; y 2 X Þ. Also, a semiorder defined on X is said to be representable (now, as a semiorder!) in the sense of Scott and Suppes (see Scott & Suppes, 1958) if there exist a realvalued map u : X ! R and a nonnegative constant or ‘‘discrimination threshold’’ KX0 such that x y3 KouðyÞ uðxÞ ðx; y 2 X Þ. Obviously, if such a representation exists with K ¼ 0, the binary relation " associated to the semiorder is actually a total preorder. This is equivalent to say that the associated indifference is transitive. There exist interval orders that fail to be representable (as interval orders). Also, there exist semiorders that are not representable in the sense of Scott and Suppes. (See Candeal et al., 2002; Olo´riz et al., 1998 for further details.) Following Fishburn (1970a, 1970c), associated to an interval order defined on a nonempty set X, we shall consider two new binary relations and given by x y3x z"y for some z 2 X ðx; y 2 X Þ, and, similarly, x y3x"z y for some z 2 X ðx; y 2 X Þ. We denote x" y3:ðy xÞ, x y3x" y" x, x" y3:ðy xÞ and x y3x" y" x ðx; y 2 X Þ. Now it is straightforward to see that x" y3GðyÞ GðxÞ and also x" y3LðxÞ LðyÞ ðx; y 2 X Þ. As a matter of fact, both the binary relations " and " are total preorders on X. Moreover, the indifference relation associated to the interval order is transitive if and only if " , " and the binary relation " associated to the interval order coincide. In particular, in this case " is also a total preorder on X. (See Monjardet, 1978 for more details.) Moreover, a new binary relation "0 is defined on X with the help of and , by declaring that x"0 y3 ½ðx" yÞ ^ ðx" yÞ ðx; y 2 X Þ. This new binary relation "0 allows us to characterize semiorders among interval orders, as proved in Fishburn (1970a, 1970b), by means of the following useful lemma. Lemma 2.1. Let X be a nonempty set and an asymmetric binary relation on X. Then the binary relation is a semiorder if and only if "0 is a total preorder on X. The representability (not necessarily continuous!) of interval orders is characterized (see Proposition 8 in Doignon et al., 1984) by the existence of a countable
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subset D X , such that for every x; y 2 X with x y there exists an element d in D such that x d" y. Such a set is called a widely dense subset of X. Lemma 2.2. Let be an interval order on a nonempty set X. The relation is representable if and only if there exists a widely dense countable set D X . Remark 2.3. The path that we will follow here (in the next Section 3) to get a characterization of the continuous representability of semiorders, valid for connected topological spaces without extremal elements goes as follows, in three steps: 1. Since a semiorder defined on a nonempty set X is a very particular case of an interval order, first we will try to find representations of the semiorder but merely as an interval order through a pair of functions u; v : X ! R such that x y3vðxÞouðyÞ. 2. Assuming now that X is endowed with a topology t, the next step will be looking for a pair of continuous functions representing , again as an interval order. 3. In the final step we will look for a modification of the pair u; v to another pair of continuous functions U; V : ðX ; tÞ ! ðR; usual topologyÞ such that this new pair still represents as an interval order, but now there is a positive constant K40, or ‘‘threshold’’, such that V ðxÞ UðxÞ ¼ K ðx 2 X Þ. As commented before, the details of all this process will be analyzed in Section 3. 3. Continuous representability of semiorders on connected topological spaces To deal with continuous representations of semiorders, we introduce now some preparatory definitions and results. Definition. A semiorder defined on a nonempty topological space ðX ; tÞ is said to be typical if its associated indifference relation is not transitive, or equivalently if its associated relation " is not a total preorder. In what follows all the semiorders that we shall consider will be assumed to be typical.
y3KoUðyÞ UðxÞ ðx; y 2 X Þ). Then the semiorder is tcontinuous. Definition. Let be an interval order defined on a nonempty set X. An element x 2 X is said to be a singular point with respect to if for every y; z 2 X it holds that ½ðxyÞ ^ ðxzÞ ) ðyzÞ. Remark 3.2. Observe that in the particular case in which the associated relation " is a total preorder, then every element x 2 X is a singular point. When is a typical semiorder, the existence of many singular points tends to preclude the representability of the semiorder through a real-valued map u : X ! R and a positive threshold K40. Consider the following example: Let X ¼ ½0; 1 [ ½4; 6 and the semiorder defined by x y3½ðx; y 2 ½0; 1 and xoyÞ or ðxp1; yX4Þ or ðx; y 2 ½4; 6 and x þ 1oyÞ. Here all the elements in ½0; 1 are singular. Moreover is not representable through a real-valued map u : X ! R and a positive threshold K40 since, otherwise, the fact 0 2ðnþ1Þ 2n 1 ðnX1; n 2 NÞ would imply that uð0Þouð1Þ nK ðnX1; n 2 NÞ, which is impossible. Lemma 3.3. Let be an interval order defined on a connected topological space ðX ; tÞ. The following conditions are equivalent:
(i) The interval order admits a representation through a pair of continuous functions u; v : ðX ; tÞ ! ðR; usual topologyÞ such that x y3vðxÞouðyÞ ðx; y 2 X Þ. (ii) There exists a widely dense countable subset D X for , and for every x 2 X the lower contour set LðxÞ, the upper contour set GðxÞ, and the subsets fy 2 X : y xg, fy 2 X : y xg, fz 2 X : x zg and fz 2 X : x zg are t-open. (iii) The interval order admits a representation through a pair of continuous functions U; V : ðX ; tÞ ! ðR; usual topologyÞ such that x y3V ðxÞoUðyÞ ðx; y 2 X Þ and, in addition x y3UðxÞoUðyÞ; x y3V ðxÞoV ðyÞ ðx; y 2 X Þ. Proof. See the equivalences (i) 3 (viii) 3 (x) of Theorem 3.5 in Candeal et al. (2004). &
Definition. A semiorder defined on a nonempty topological space ðX ; tÞ is said to be t-continuous if for every x 2 X it holds that the lower contour set LðxÞ ¼ fy 2 X : y xg and the upper contour set GðxÞ ¼ fz 2 X : x zg of x relative to are t-open.
Lemma 3.4. Let I be an open real interval. Let h : I ! I a continuous and strictly increasing map. Then there exists a continuous and strictly increasing map F : I ! R that satisfies the Abel equation
The following fact is an immediate consequence of the last definition.
Proof. See Theorem 2.1 in Kuczma (1968), or alternatively pp. 133 and ff. in Acze´l (1987). &
Proposition 3.1. Let be a semiorder defined on a nonempty topological space ðX ; tÞ. Suppose that there exist a continuous map U : ðX ; tÞ ! (R, usual topology) and a positive threshold K40, that constitute a representation in the sense of Scott–Suppes of the semiorder (i.e.: x
Now we are ready to introduce the main result, that characterizes the existence of continuous Scott–Suppes representations with positive threshold for semiorders without extremal elements, defined on connected topological spaces.
F ðhðxÞÞ ¼ F ðxÞ þ 1
ðx 2 IÞ.
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Theorem 3.5. Let be a continuous semiorder without extremal elements, defined on a connected topological space ðX ; tÞ. Then there exist a positive threshold K40 and a continuous map u : ðX ; tÞ ! R such that x y3uðyÞ uðxÞ4K ðx; y 2 X Þ if and only if the following conditions hold: (i) There exists a widely dense countable subset D X for the binary relation considered as an interval order on X. (ii) The associated relations and coincide (i.e.: x y3x y ðx; y 2 X Þ). In particular the total preorders " ; " and "0 are identical. (iii) The binary relation has no singular points. Proof. Suppose that the semiorder is representable through the continuous real valued function u : X ! R and the threshold K40. Let v : X ! R be defined as vðxÞ ¼ uðxÞ þ K ðx 2 X Þ. Observe that the pair ðu; vÞ furnishes a representation of as an interval order, so that by Lemma 2.2, there exists a widely dense countable subset D X . Let now x; y 2 X such that x y. By definition of , there exists a 2 X such that x a"y. Thus we have uðxÞ þ KouðaÞpuðyÞ þ K, so that uðxÞouðyÞ. Since, by hypothesis, x is not a minimal element with respect to , there exists z 2 X such that z x. Thus uðzÞ þ KouðxÞouðyÞouðyÞ þ K. By continuity of u and connectedness of ðX ; tÞ there exists w 2 X such that uðxÞpuðwÞ þ KouðyÞ. Consequently, we get x"w y. Hence x y. The fact x y ) x y ðx; y 2 X Þ can be proved in an entirely analogous way. Therefore and coincide. Let now x 2 X . Since x is not an extremal element with respect to , there exist y; z 2 X such that z x y. Thus uðzÞ þ KouðxÞouðxÞ þ KouðyÞ. Again by continuity of u and connectedness of ðX ; tÞ, there exists s 2 X such that uðxÞpuðsÞ þ KouðxÞ þ KouðyÞ.In particular, we have that uðxÞpuðsÞ þ K. Thus x"s. Moreover, uðsÞ þ KouðxÞ þ K obviously implies uðsÞouðxÞ. Since s is not a minimal element, this implies as before that s x. But, as just proved, this is equivalent to say that s x. Therefore, there exists t 2 X such that x"s t"x. Hence uðxÞpuðsÞ þ KouðtÞpuðxÞ þ K. Now observe that xs and xt, but s t, so that x is not a singular point. For the converse, suppose now that the conditions (i)–(iii) hold. Given x; y 2 X , observe that if y x then there exists a 2 X such that y a"x. Thus [ fy 2 X : y ag. fy 2 X : y xg ¼ fa2X ;a"xg
Similarly fz 2 X : x zg ¼
[
fz 2 X : b zg.
fb2X ;x"bg
Since by condition (ii) the binary relations and coincide, and, by hypothesis, the semiorder is tcontinuous, it follows that the subsets fy 2 X : y xg and fz 2 X : x zg are t-open for every x 2 X . Consequently,
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by Lemma 3.3 it follows that admits a representation (as an interval order) through a pair of continuous functions u; v : ðX ; tÞ ! ðR; usual topologyÞ such that x y3 vðxÞouðyÞ ðx; y 2 X Þ and, in addition x y3uðxÞo uðyÞ; x y3vðxÞovðyÞ ðx; y 2 X Þ. Since is in particular asymmetric, it follows now that uðxÞpvðxÞ for every x 2 X . Suppose that there exists t 2 X such that uðtÞ ¼ vðtÞ. In this case, given y; z 2 X with ty and tz it follows that uðtÞpvðyÞ; uðyÞpvðtÞ; uðtÞpvðzÞ; uðzÞpvðtÞ: Consequently, we obtain that uðyÞpvðtÞ ¼ uðtÞpvðzÞ. Hence y"z. We also have that uðzÞpvðtÞ ¼ uðtÞpvðyÞ, which implies z"y. Thus we conclude that yz. This implies that t is a singular point and contradicts condition (iii). Therefore uðxÞovðxÞ, for every x 2 X . Since has no extremal elements and ðX ; tÞ is connected, and the maps u; v are continuous, it follows that the real subsets uðX Þ and vðX Þ are open intervals of R. Again because has no extremal elements, given x 2 X there exist elements y; z 2 X such that z x y. Thus uðzÞovðzÞouðxÞovðxÞouðyÞovðyÞ. Hence uðxÞ lies in the real interval vðX Þ, and similarly vðxÞ belongs to the real interval uðX Þ. Therefore uðX Þ and vðX Þ coincide. Denote I ¼ uðX Þ ¼ vðX Þ. Let h : I ! I be the map defined by hðuðxÞÞ ¼ vðxÞ ðx 2 X Þ. Given now x1 ; x2 2 X such that uðx1 Þouðx2 Þ, it follows that x1 x2 . Since and coincide by condition (ii), we obtain that x1 x2 . Thus vðx1 Þovðx2 Þ or, equivalently, hðuðx1 ÞÞohðuðx2 ÞÞ. Therefore the map h is strictly increasing. In the same way, we may prove that vðx1 Þovðx2 Þ ) uðx1 Þouðx2 Þ. Given a; b 2 I such that aob, since I ¼ vðX Þ we may choose elements xa ; xb 2 X such that vðxa Þ ¼ a; vðxb Þ ¼ b. Let now c 2 ðuðxa Þ; uðxb ÞÞ. Since uðX Þ has been proved to be an open real interval, we may choose an element xc 2 X such that uðxc Þ ¼ c. Moreover, by a previous argument we have that uðxa Þouðxc Þouðxb Þ ) a ¼ vðxa Þovðxc Þo vðxb Þ ¼ b. Hence vðxc Þ 2 ða; bÞ and c ¼ uðxc Þ 2 h1 ða; bÞ. Conversely, suppose now that an element d 2 uðX Þ satisfies that hðdÞ lies in ða; bÞ. If we choose an element xd 2 X such that uðxd Þ ¼ d, we have hðdÞ ¼ vðxd Þ and a ¼ vðxa Þo vðxd Þovðxb Þ ¼ b. As before, this implies that uðxa Þo uðxd Þ ¼ douðxb Þ. Thus we have that h1 ða; bÞ ¼ uðxa ; xb Þ. which is an open interval of the real line R. Therefore the map h is continuous. Now we can apply Lemma 3.4 to conclude that there exist a strictly increasing and continuous real function F : I ! R such that F ðhðtÞÞ ¼ F ðtÞ þ 1 for every t 2 I ¼ uðX Þ, or equivalently F ðvðxÞÞ ¼ F ðuðxÞÞ þ 1 for every x 2 X . Finally, it is straightforward to see that the continuous map F u : ðX ; tÞ ! R jointly with the threshold 1 furnish a Scott–Suppes representation of the semiorder . Indeed, since F is strictly increasing, we get x y3vðxÞouðyÞ3F ðvðxÞÞoF ðuðyÞÞ3F ðuðxÞÞþ 1oF ðuðyÞÞ, for every x; y 2 X . This concludes the proof. &
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Remark 3.6. In the statement of Theorem 3.5 there is no loss of generality in assuming that the given topology t on X is the order topology y generated by the semiorder . A subbasis for y is given by the family of contour sets fLðxÞ : x 2 X g [ fGðxÞ : x 2 X g. Since is t-continuous, the topology y must be coarser than t, hence connected as well. In addition, is plainly y-continuous. Suppose that is representable through a continuous map u : ðX ; yÞ ! ðR; usual topologyÞ and a positive threshold K40 such that x y3uðxÞ þ KouðyÞ ðx; y 2 X Þ. Since is t-continuous it is clear that u is also continuous if we consider on X the topology t and on R the usual topology. Theorem 3.5 completes the panorama of the Theorem on p. 288 of Gensemer (1987a) and Theorem 3.1 in Gensemer (1988). Through those results, some characterizations of the continuous representability of typical semiorders given on connected topological spaces were also achieved, so that it seems interesting, for the sake of completeness, to compare now our main result (Theorem 3.5 above) with Gensemer’s ones, pointing out the most noticeable differences. To start with, we should observe that Gensemer studies semiorders that may have extremal elements, whereas our Theorem 3.5 deals with semiorders without extremal elements. Apparently, this is a more restrictive condition. However, there is no loss of generality in assuming such condition of lacking of extremal elements, as the next Proposition 3.7 proves. Proposition 3.7. A continuous typical semiorder defined on a connected topological space ðX ; tÞ admits a continuous Scott–Suppes representation if and only if there exist a connected topological space ðY ; tY Þ that extends ðX ; tÞ, and an extension Y of the semiorder to Y, such that Y has no extremal elements and admits a continuous Scott–Suppes representation. Proof. Let us assume first that is representable through a continuous map u : ðX ; tÞ ! ðR; usual topologyÞ and a positive threshold K. For every a 2 RnuðX Þ we add an extra element ta to X. Let Y ¼ X [ fta : a 2 RnuðX Þg. Consider the map uY : Y ! R given by uY ðxÞ ¼ uðxÞ if x 2 X , uðta Þ ¼ a if a 2 RnuðX Þ. Let Y the semiorder defined on Y by declaring that y1 Y y2 3uY ðy1 Þ þ KouY ðy2 Þ ðy1 ; y2 2 Y Þ. It is straightforward to see that there exists on Y a topology tY that satisfies the following properties: (i) (ii) (iii) (iv) (v)
Y is connected with respect to tY . The restriction of tY to X is t. The restriction of Y to X is . The semiorder Y has no extremal elements. The semiorder Y is continuously representable through the map uY : ðY ; tY Þ ! ðR; usual topologyÞ and the positive threshold K.
Conversely, suppose now that there exist a connected topological space ðY ; tY Þ that extends ðX ; tÞ and an extension Y of the semiorder to Y, such that Y has no extremal elements and admits a continuous Scott–Suppes representation given by a map u : ðY ; tY Þ ! ðR; usual topologyÞ. Let uX : ðX ; tÞ ! ðR; usual topologyÞ be the restriction of u to X. Since t is the restriction of tY to X, it is clear that the map uX jointly with the positive threshold K constitute a continuous Scott–Suppes representation of the semiorder defined on X. & Remark 3.8. The semiorders that are characterized in Theorem 3.5 are isomorphic to the semiorder defined on the real line R by declaring that x y3x þ 1oy ðx; y 2 RÞ. However, by Proposition 3.7 we see that if a continuous typical semiorder defined on a connected topological space ðX ; tÞ admits a continuous Scott–Suppes representation, then it can be considered as a part of ðR; Þ. A clear example of this situation is the semiorder defined on the open interval ð0; 10Þ of the real line by declaring that x y3x þ 1oy ðx; y 2 ð0; 10ÞÞ. On ð0; 10Þ we shall consider the Euclidean topology, that does not coincide with the topology generated by the semiorder. Obviously, ðð0; 10Þ; Þ is not isomorphic to ðR; Þ since it does not contain any infinite sequence ðxn Þn2N of elements such that x0 x1 x2 xk xkþ1 . Observe in addition that all the points in ð0; 1 are minimal (respectively, all the elements in ½9; 10Þ are maximal) with respect to the semiorder defined on ð0; 10Þ, so that Theorem 3.5 does not apply (directly) to this example. But, as in Proposition 3.7, if we consider first ðð0; 10Þ; Þ embedded in ðR; Þ, where the real line R is also endowed with the Euclidean topology, the restriction to ð0; 10Þ of the (obvious) continuous Scott–Suppes representation of ðR; Þ furnishes a continuous Scott–Suppes representation of ðð0; 10Þ; Þ. Continuing our discussion, we may also observe that in the Theorem on p. 288 of Gensemer (1987a), some of the conditions required in the characterization of the representability of a continuous semiorder defined on a connected topological space ðX ; tÞ given there are more restrictive than the ones we use in our Theorem 3.5 in the present paper. Thus, in Gensemer (1987a) the associated total preorders " and " are asked to be continuous with respect to the given topology t (that is, the sets fy 2 X : :ðy" xÞg; fy 2 X : :ðx" yÞg; fy 2 X : :ðx" yÞg and fy 2 X : :ðy" xÞg must be t-open sets). We do not need to assume (a priori) any of those conditions involving continuity. Actually, we could obtain here those conditions from the continuity of and the coincidence of the total preorders " ; " and "0 , as in the proof of Theorem 3.5. Another condition used in the statement of the Theorem on p. 288 of Gensemer (1987a) is the strong separability of the binary relation . This condition means that there must exist a countable subset D X such that for every x; y 2 X with x y there exist a; b 2 D such that x a"b y. It is
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well known that the condition of strong separability is more restrictive than the condition of the existence of a widely dense countable subset, that we use in our Theorem 3.5. (See e.g., Candeal et al. 2004 for details.) More conditions used in the statement of the Theorem on p. 288 of Gensemer (1987a) are the so-called regularity and normality of the semiordered structure ðX ; Þ. They were introduced to tackle the problem of the existence of maximal or minimal elements. It can be proved that when no extremal element exists, then Gensemer’s condition of regularity plus normality amounts to our condition of coincidence of and . On the other hand, Theorem 3.1. in Gensemer (1988) also gives a characterization of the continuous representability of a semiorder defined on a connected topological space ðX ; tÞ. As in our approach, Gensemer handles first the binary relation as an interval order to deduct conditions about its Scott–Suppes representability as a semiorder. However, the main feature of Theorem 3.1 in Gensemer (1988) is that it starts from a particular tcontinuous representation of ðX ; Þ as an interval order, without saying which extra conditions could lead to the existence of such a kind of representation. As a matter of fact, it is not true, in general, that the condition of the existence of a widely dense countable subset, nor even the more restrictive one of strong separability, implies that the binary relation can be represented through a pair of functions u; v : X ! R such that x y3vðxÞouðyÞ ðx; y 2 X Þ satisfying also that, in addition, uðxÞovðxÞ for every x 2 X and also uðxÞouðyÞ3vðxÞovðyÞ ðx; y 2 X Þ. In other words, some additional conditions should be given to capture the last facts uðxÞovðxÞ for every x 2 X and also uðxÞouðyÞ3vðxÞovðyÞ for every x; y 2 X . To conclude our discussion, we point out that, except for the obvious fact of the semiorder being t-continuous, all the conditions that we use in the statement of Theorem 3.5 to study the continuous representability of a semiorder defined on a connected topological space ðX ; tÞ, are independent of the topology t. Unlike the conditions we use in our approach, the conditions required in the statement of the Theorem on p. 288 of Gensemer (1987a) as well as those on the statement of Theorem 3.1 in Gensemer (1988) depend on the given topology t. The reason is that they involve continuity of the associated binary relations ; or they ask for the existence (a priori) of a particular continuous representation of ðX ; Þ as an interval order. Acknowledgements Thanks are due to Prof. Dr. Susan H. Gensemer (Syracuse, NY), Prof. Dr. Sergei Ovchinnikov (San Francisco, CA) and an anonymous referee for their valuable suggestions and help. This work has been supported by the research project MTM 2006 - 15025 ‘‘Espacios Topolo´gicos Ordenados’’, of the Spanish Ministry of Culture.
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