Prime Orders All of Whose Prime Suborders Are Selfdual

Let P be an order on a set V. A subset A of V is autonomous in P if every element of V not in A is either less than or greater than or incomparable to all elements of A. The empty set, the singletons from V and the set V are autonomous sets and are called trivial. Call an order prime if all its autonomous sets are trivial. We give the complete list of all finite prime orders all of whose prime suborders are selfdual.     

The Order on the Rationals has an Orthogonal Order with the Same Order Type

Two orders on the same set are orthogonal if the constant maps and the identity map are the only maps preserving both orders. We construct linear orders orthogonal to the order on the rationals.