The Debreu Gap Lemma and some generalizations

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

“…In Herden and Mehta [12] the Debreu Open Gap Lemma has been discussed extensively. In particular, its value in any theory of continuous order-preserving functions the codomain of which is not necessarily the real line has been underlined.…”

“…Then a fundamental problem in mathematical utility theory is the problem of determining necessary and sufficient conditions that guarantee the existence of some real-valued orderpreserving (strictly increasing) function f on (X, ) that also preserves the mathematical structure on (X, ). In Herden and Mehta [12] it has been underlined by many examples that in a more applicable approach to mathematical utility theory the real line often is not the appropriate codomain of f . This means that the real line in many concrete cases should be replaced by a codomain that is more carefully adapted to the particular considered representation problem.…”

Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets

“…Also, from a purely mathematical point of view there is no reason or justification for studying only the very special case where R is the codomain. For further discussion of these ideas we refer the reader to [4] and [14]. Therefore, it is desirable to develop a general theory of representations of ordered structures in which the codomain is not necessarily the set of real numbers.…”

“…One way to proceed is to consider codomains such as the long line or the lexicographic plane as in [14]. In this paper we follow a somewhat different approach by studying codomains of strictly isotone functions that are subsets of R but not necessarily order-isomorphic to R. (For initial results in this direction see [8] where order-monomorphisms are studied with codomains consisting of the rational numbers and the real algebraic numbers.…”

“…As pointed out in [14], the most desirable codomains Y in studying continuity properties of strictly isotone functions are those for which the analogue of Debreu gap lemma holds, because in that case the existence of a strictly isotone function (which is not necessarily continuous) with such a codomain Y implies the existence of another strictly isotone function, that is continuous, and which also takes values in Y.…”