Signotopes Induce Unique Sink Orientations on Grids

A unique sink orientation (USO) is an orientation of the edges of a polytope in which every face contains a unique sink. For a product of simplices ${\mathrm{\Delta }}_{m-1}\times {\mathrm{\Delta }}_{n-1}$, Felsner, Gärtner and Tschirschnitz (2005) characterize USOs which are induced by linear functions as the USOs on a $(m\times n)$-grid that correspond to a two-colored arrangement of lines. We generalize some of their results to products ${\mathrm{\Delta }}^1\times \cdots \times {\mathrm{\Delta }}^r$ of $r$ simplices, USOs on $r$-dimensional grids and $(r+1)$-signotopes.