Fuzzy interval and semi-orders

The seminal paper by Luce on intransitive indifference provoked a rich body of research on how to model such a type of indifference. Interval orders and semi-orders play a central role in the modelization of intransitive indifference. This paper is an attempt to fuzzify interval orders and semi-orders. We use an additive fuzzy preference structure to argue for a fuzzification of interval and semi-orders. We show that fuzzy interval orders are to a degree semi-orders, however, against expectations, we cannot show that fuzzy interval graphs are to a degree fuzzy indifference graphs.