Interval orders based on weak orders

AbstractOne definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b ⩽ c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “2 + 2”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “3 + 2”, “2 + N”, a six-element crown or a six-element fence