Searching for a Debreu’s open gap lemma for semiorders

In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was never achieved. In the present paper we propose a characterization for the existence of a continuous representation (in the sense of Scott-Suppes) for bounded semiorders. As a matter of fact, the weaker but more manageable concept of ε-continuity is properly introduced for semiorders. As a consequence of this study, a version of the Debreu’s Open Gap Lemma is presented (but now for the case of semiorders) just as a conjecture, which would allow to remove the open-closed and closed-open gaps of a subset S ⊆ R, but now keeping the constant threshold, so that x + 1 < y if and only if g(x) + 1 < g(y) (x, y ∈ S).