Pointwise Debreu Lexicographic Powers

A linear ordering (X,≺) is a Debreu chain (or has the Debreu property) if each subordering (Y,≺) of X can be embedded into X with an injective function that is both order-preserving and continuous with respect to the order topology on X and Y . The most typical example of a Debreu chain is the linearly ordered topological space (R, <, τ<) of real numbers endowed with its usual order < and the order topology τ< . The Debreu property is of fundamental importance when dealing with utility representations. Recall that a utility representation for a chain (X,≺) is a pair (U, f), where (U,<) is a chain and f : X → U is an order-preserving embedding, i.e., for each x, y ∈ X, if x ≺ y then f(x) < f(y); the linear ordering U is called the base chain of the representation, and the map f a utility function. Traditionally, the literature has been dealing with utility representations such that the base chain (Y, <) is the set R of real numbers, endowed with its natural order <. Specifically, much attention has been devoted to the the following topics: (A) the existence of an R-valued utility function; (B) the continuity of such a function. With this respect, our goals are: (A) to justify alternative types of utility representations, in particular lexicographic products of linear orderings; (B) to suggest suitable continuity properties that a utility representation should satisfy, in particular some Debreu-like properties. The topic of our research is a combination of (A) and (B), i.e., a study of a Debreu-like property of lexicographic powers. In the sequel, we briefly outline the state of the art of (A) and (B), in order to put our study into a proper perspective.