Further Results on the Continuous Representability of Semiorders

Abstract

We study necessary and sufficient conditions for the continuous Scott-Suppes representability of a semiorder through a continuous real-valued map and a strictly positive threshold. In the general case of a semiorder defined on topological space, we find several necessary conditions for the continuous representability. These necessary conditions are not sufficient, in general. As a matter of fact, the analogous of the classical Debreu's lemma for the continuous representability of total preorders is no longer valid for semiorders. However, and in a positive direction, we show that if the set is finite those conditions are indeed sufficient. In particular, we characterize the continuous Scott-Suppes representability of semiorders defined on a finite set endowed with a topology.

Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders

“…In this case, the order structure is represented by means of a single function u (also named utility function) and a strictly positive constant k, such that x ≺ y if and only if u(x) + k < u(y) [29]. Here is not possible to apply Debreu's Lemma directly on the function u, and some additional conditions for the continuous representability of the semiorder are necessary [21]. Anyway, with a good combination of the (Generalized) Open Gap Lemma and Abel equation (see [1]) new results could be achieved.…”

“…9 Given an interval order ≺, then x * * y ⇐⇒ z ≺ x implies z ≺ y, for any z ∈ X, so x ∼ * * y ⇐⇒ {z ≺ x ⇐⇒ z ≺ y}. See also Section 2 of [21]. biorder or with any other preorder Q defined on n k=1 X k (i.e.…”

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

“…Unfortunately, in these cases (namely interval orders and semiorders) an analogous result to the famous Debreu's open gap lemma (see Lemma 1 in Section 3 above) is, in general, no longer available (see, e.g., [127]). In fact, in the case of semiorders, the analogous of the Debreu's open gap lemma is false in general, as proved in Example 1 and Remark 7 in [128].…”

“…These are based on topological conditions different from the ones involved in the main results launched in [46]. Concerning the continuous representability of semiorders, in [128] some partial characterizations have recently been obtained, too.…”

Continuous Representability of Interval Orders: The Topological Compatibility Setting

“…Furthermore, unlike interval orders, in this case of semiorders the concept of a natural topology does not furnish good results, in general. 1,32 Some (partial) results about continuous representability of semiorders have also been introduced [32][33][34] But even with this negative output, there is an important and hopefully, quite positive fact: the analysis of the properties related to semiorders defined on a topological space have lead to a key concept, namely that of the topological compatibility with respect to the indifference of the main trace. Moreover, this new concept can be introduced not only for semiorders, but, actually, for the more general case of interval orders, and this constitutes the nucleus of our studies throughout the present manuscript.…”