Maps between Higher Bruhat Orders and Higher Stasheff-tamari Posets

We make explicit a description in terms of convex geometry of the higher Bruhat orders B(n, d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher Stasheff-Tamari poset S1(n, d). We show that the map f sketched by Kapranov and Voevodsky from B(n, d) to S1([0, n + 1], d + 1) coincides with the map constructed by Rambau, and is a surjection for d ≤ 2. We also give geometric descriptions of lk0 ◦f and lk{0,n+1} ◦f . We construct a map analogous to f from S1(n, d) to B(n − 1, d), and show that it is always a poset embedding. We also give an explicit criterion to determine if an element of B(n−1, d) is in the image of this map.