On the Ferrers property of valued interval orders

We study the relationship between the Ferrers property and the notion of interval order in the context of valued relations. Given a crisp preference structure without incomparability, the strict preference relation satisfies the Ferrers property if and only if the associated weak preference relation does. These conditions characterize a total interval order. For valued relations the Ferrers property can be written in two different and non-equivalent ways. In this work, we compare these properties by finding the kind of completeness they imply. Moreover, we study whether they still characterize a valued total interval orders.