Debreu-like properties of utility representations

Traditionally the codomain of a utility function is the set of real numbers. This choice has the advantage of ensuring the existence of a continuous representation but does not allow to represent many preference structures that are relevant to utility theory. Recently, some authors have started a systematic study of utility representations that are not real-valued, introducing the notion of a Debreu chain. We continue their analysis defining two Debreu-like properties, which are connected to a local continuity of a utility representation. The classes of locally Debreu and pointwise Debreu chains here introduced enlarge the class of Debreu chains. We give several examples and analyze some properties of these two classes of chains, with particular attention to lexicographic products.