The First Higher Stasheff-Tamari Orders are Quotients of the Higher Bruhat Orders

We prove the conjecture that the higher Tamari orders of Dimakis and Muller-Hoissen coincide with the first higher Stasheff-Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff-Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff-Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.