We characterize weak continuity of an interval order on a topological space by using the concept of a scale in a topological space.Weakly continuous interval order; continuous numerical representation

Bosi, Gianni

Zuanon, Magalì

English

Research Papers in Economics

Weak continuity of
preferences with
nontransitive indiﬀerence
Gianni BOSI
Dipartimento di Matematica Applicata
“Bruno de Finetti", Universit` a di Trieste,
Piazzale Europa 1,
34127 Trieste, Italy
Magal` ı E. ZUANON
Dipartimento di Metodi Quantitativi,
Universit` a degli Studi di Brescia,
Contrada Santa Chiara 50, 25122 Brescia, Italy
Abstract
We characterize weak continuity of an interval order - on a topolog-
ical space (X;) by using the concept of a scale in a topological space.
JEL Classiﬁcation: C60; D00.
Keywords: Weakly continuous interval order; continuous numerical
representation.
1 Introduction
An interval order - on a set X is in some sense the simplest kind of binary
relation with nontransitive indiﬀerence since, under certain conditions, it can
be represented by a pair (u;v) of real-valued functions on X (this means that,
for all x;y ∈ X, x - y if and only if u(x) ≤ v(y)). If in addition X is endowed
with a topology , then one may look for a pair (u;v) of continuous real-valued
functions representing an interval order - on (X;) (see e.g. Bosi, Candeal
and Indur´ ain [2] and Bosi, Candeal, Campi´ on and Indur´ ain [3]).
With a view to possible general conditions guaranteeing the existence of
such a continuous representation, Bosi [1] introduced the concept of a weakly
continuous interval order. In this paper, we characterize weak continuity of an
interval order by using the concept of a scale in a topological space.2 Gianni BOSI and Magal  E. ZUANON
2 Notation and preliminaries
We ﬁrst recall that an interval order - on an arbitrary nonempty set X is
a binary relation on X which is reexive and in addition veriﬁes the following
condition for all x;y;z;w ∈ X:
(x - z) and (y - w) ⇒ (x - w) or (y - z):
The irreexive part of an interval order - will be denoted by ≺ (i.e., for
all x;y ∈ X, x ≺ y if and only if (x - y) and not(y - x)).
Fishburn [6] showed that if - is an interval order on a set X, then each of
the following two binary relations -∗ and -∗∗ on X is a total preorder (i.e., a
total and transitive binary relation):
x -
∗ y ⇔ (z - x ⇒ z - y) for all z ∈ X;
x -
∗∗ y ⇔ (y - z ⇒ x - z) for all z ∈ X:
The irreﬂexive parts of -∗ and -∗∗ will be denoted by ≺∗ and ≺∗∗.
If - is an interval order on a set X, then denote by L≺(x) (U≺(x)) the
strict lower (upper) section of any element x ∈ X (i.e., for every x ∈ X,
L≺(x) = {y ∈ X : y ≺ x} and U≺(x) = {y ∈ X : x ≺ y}).
A pair (u;v) of real-valued functions on X is said to represent an interval
order - on X if, for all x;y ∈ X,
x - y ⇔ u(x) ≤ v(y):
We say that a pair (u;v) of real-valued functions on X almost represents
an interval order - on X if, for all x;y ∈ X,
(x - y ⇒ u(x) ≤ v(y)) and (x ≺ y ⇒ v(x) ≤ u(y)):
The following proposition holds which illustrates the importance of the
concept of a pair of continuous real-valued functions almost representing an
interval order in connection with the problem concerning the existence of a
representation by means of a pair of continuous real-valued functions.
Proposition 2.1 An interval order - on a topological space (X;) is rep-
resentable by means of a pair (u;v) of continuous real-valued functions with
values in [0;1] if and only if there exists a countable family {(un;vv)}n∈N\{0} of
pairs of continuous real-valued functions on (X;) with values in [0;1] almost
representing - such that for every x;y ∈ X with x ≺ y there exists n ∈ N\{0}
with vn(x) < un(y).Weakly continuous preferences 3
Proof. The “only if” part is clear. Hence, assume that there exists a countable
family {(un;vn)}n∈N\{0} of pairs of continuous real-valued functions on (X;)
with values in [0;1] almost representing - such that for every x;y ∈ X with
x ≺ y there exists n ∈ N \ {0} with vn(x) < un(y). Deﬁne functions u and v
on X as follows:
u(x) =
∞ ∑
n=1
2
−nun(x); v(x) =
∞ ∑
n=1
2
−nvn(x) (x ∈ X)
in order to immediately verify that (u;v) is a continuous representation of the
interval order - on the topological space (X;). 2
An interval order - on a topological space (X;) is said to be continuous
if L≺(x) and U≺(x)) are both open subsets of X for every x ∈ X. Further,
we say that it is strongly continuous if it is continuous and in addition the
associated total preorders -∗ and -∗∗ are both continuous.
We now recall the deﬁnition of a weakly continuous interval order presented
by Bosi [1].
Deﬁnition 2.2 (weakly continuous interval order) We say that an
interval order - on a topological space (X;) is weakly continuous if for every
x;y ∈ X such that x ≺ y there exists a pair (uxy;vxy) of continuous real-valued
functions on (X;) satisfying the following conditions:
(i) (uxy;vxy) almost represents -;
(ii) vxy(x) < uxy(y).
The concept of weak continuity described in Deﬁnition 2.2 is reminiscent
of the concept of weak continuity of a preorder on a topological space (see e.g.
Bosi and Herden [5]). Every interval order that is representable by means of a
pair of continuous functions (u;v) and at same time is such that the associated
total preorders -∗ and -∗∗ are not continuous provides an example of a weakly
continuous interval order which is continuous but not strongly continuous. For
example, this is the case of the interval order - on X = [3;5] ∪ [9;25] deﬁned
by x - y ⇔ x ≤ y2 (see Bosi, Candeal and Indur´ ain [2, Example 3.2]) when
X is endowed with the induced Euclidean topology on the real line.
3 Weak continuity of interval orders
In the sequel, we shall refer to the well known notion of a scale in a topo-
logical space (see e.g. Gillman and Jerison [7]).
Deﬁnition 3.1 If (X;) is a topological space and S is a dense subset of
[0;1] such that 1 ∈ S, then a family {Gr}r∈S of open subsets of X is said to be
a scale in (X;) if the following conditions hold:4 Gianni BOSI and Magal  E. ZUANON
(i) G1 = X;
(ii) Gr1 ⊆ Gr2 for every r1;r2 ∈ S such that r1 < r2.
We are now ready to characterize the weak continuity of an interval order
on a topological space.
Proposition 3.2 Let - be an interval order on a topological space (X;).
Then the following conditions are equivalent:
(i) - is weakly continuous;
(ii) For every pair (x;y) ∈ X × X such that x ≺ y there exist two scales
{G
∗(xy)
r }r∈S and {G
∗∗(xy)
r }r∈S in (X;) such that the family
{(G
∗(xy)
r ;G
∗∗(xy)
r )}r∈S satises the following conditions:
(a) z - w and w ∈ G
∗(xy)
r imply z ∈ G
∗∗(xy)
r for every z;w ∈ X and
r ∈ S;
(b) z ≺ w and w ∈ G
∗∗(xy)
r imply z ∈ G
∗(xy)
r for every z;w ∈ X and
r ∈ S;
(c) x ∈ G
∗(xy)
r and y ̸∈ G
∗∗(xy)
r for every r ∈ S \ {1}.
Proof. Consider a pair (x;y) ∈ X × X such that x ≺ y.
(i) ⇒ (ii). Since - is weakly continuous, there exists a pair (uxy;vxy) of con-
tinuous real-valued functions on (X;) such that (uxy;vxy) almost represents
- and in addition vxy(x) < uxy(y). Without loss of generality, we can assume
that both uxy and vxy take values in [0;1] and that vxy(x) = 0, uxy(y) = 1.
Deﬁne S = Q∩]0;1], G
∗(xy)
r = v−1
xy ([0;r[), G
∗∗(xy)
r = u−1
xy([0;r[) for every r ∈ S,
and G
∗(xy)
1 = G
∗∗(xy)
1 = X in order to immediately verify that {G
∗(xy)
r }r∈S and
{G
∗∗(xy)
r }r∈S are two scales in (X;) such that the family {(G
∗(xy)
r ;G
∗∗(xy)
r )}r∈S
satisﬁes the above conditions (a), (b) and (c).
(ii) ⇒ (i). From the assumptions, there exist two scales {G
∗(xy)
r }r∈S and
{G
∗∗(xy)
r }r∈S such that the family {(G
∗(xy)
r ;G
∗∗(xy)
r )}r∈S satisﬁes the above con-
ditions (a), (b) and (c). Deﬁne two functions uxy;vxy : X → [0;1] as follows:
uxy(z) = inf{r ∈ Q∩]0;1] : z ∈ G
∗∗(xy)
r } (x ∈ X);
vxy(z) = inf{r ∈ Q∩]0;1] : z ∈ G
∗(xy)
r } (x ∈ X):
We have that uxy and vxy are both continuous functions on (X;) with values
in [0;1] (see e.g. the proof of the lemma on pages 43-44 in Gillman and JerisonWeakly continuous preferences 5
[7]). We claim that the pair (uxy;vxy) almost represents the interval order -
and satisﬁes the condition vxy(x) < uxy(y).
From condition (c), we have that vxy(x) = 0 and uxy(y) = 1. It remains to
show that the pair (uxy;vxy) almost represents the interval order -. First con-
sider any two elements z;w ∈ X such that z ≺ w. Then, by condition (b),
we have that vxy(z) ≤ uxy(w). Finally, observe that if z;w ∈ X are any two
elements such that z - w, then we have that uxy(z) ≤ vxy(w) by condition
(a). This consideration completes the proof. 
References
[1] G. Bosi, A note on continuity and continuous representability of interval
orders, International Mathematical Forum 3 (2008), no. 32, 1563-1568.
[2] G. Bosi, J.C. Candeal and E. Indur´ ain, Continuous representability of
interval orders and biorders, Journal of Mathematical Psychology, 51
(2007), 122-125.
[3] G. Bosi, J.C. Candeal, M. J. Campi´ on and E. Indur´ ain, Interval-valued
representability of qualitative data: the continuous case, International
Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems, 15
(2007), 299-319.
[4] G. Bosi, J.C. Candeal, E. Indur´ ain, E. Oloriz and M. Zudaire, Numerical
representations of interval orders, Order 18 (2001), 171–190.
[5] G. Bosi and G. Herden, On a possible continuous analogue of the Szpilrajn
theorem and its strengthening by Dushnik and Miller, Order 23 (2006),
271-296.
[6] P.C. Fishburn, Interval Orders and Interval Graphs, Wiley, New York,
1985.
[7] L. Gillman and M. Jerison, Rings of continuous functions, Princeton, D.
Van Nostrand Company, 1960.

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Continuous representability of interval orders and biorders,

Interval Orders and Interval Graphs,

Numerical representations of interval orders,

On a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller,

Rings of continuous functions,

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