Exponentiable linear orders need not be transitive

It is well-known that every transitive linear order is exponentiable. However, is the converse true? This question was posed in Chapter 8 of the textbook titled "Linear Orderings" by Rosenstein. We define the class CTLO of cyclically transitive linear orders that properly contains the class of transitive linear orders, and show that all discrete unbounded orders in CTLO are exponentiable, thereby providing a negative answer to the question. The class CTLO is closely related to the class of transitive cyclic orders introduced by Droste, Giraudet and Macpherson. We also discuss the closure of subclasses of CTLO under products and iterated Hausdorff condensations.
Comment: 10 pages