Representability of Interval Orders

Untitled

“…A key concept, used in Olóriz et al [27] to get a characterization of the representability of interval orders, is that of interval order separability (henceforward i.o.-separability). An interval order ≺ on a set X is said to be i.o.-separable if there exists a countable subset D ⊆ X such that for every x, y ∈ X with x ≺ y there exists an element d in D such that x ≺ d * * y.…”

“…Following Lemma 1 in Olóriz et al [27], fix an element x 0 ∈ X and call u(x) = −F (x, x 0 ) ; v(y) = F (y, y) + u(y) (x, y ∈ X). It is straightforward to see now that (u, v) is a continuous representation for ≺.…”

“…Nothing is said about i.o.-separability, a condition less restrictive than strong separability, because such concept of interval order separability appeared later in the literature (in Olóriz et al [27]). ii) Several intermediate steps in the proof have appeared in previous works, usually as necessary lemmata to introduce the main Chateauneuf's theorem.…”

“…A different characterization was obtained by Doignon et al [14]. Furthermore, a new alternative solution using only one ordinal condition has been recently obtained by Olóriz et al [27]. Some further characterizations have appeared in Bosi et al [3].…”

“…ii) The second technique, used in Bosi and Isler [4], is based on measure theory. The functions u, v that represent the interval order are related to the measures of lower and upper contour sets, in some orderings related to ≺, of the elements of X. iii) The third technique, introduced in Olóriz et al [27], is based on the theory of functional equations. The idea is to construct a bivariate map F : X × X −→ R such that, for instance, x ≺ y ⇐⇒ F (x, y) > 0.…”