Definition. Sections and vertex figures [schulte2017regular, Section 2] [frap-000E]

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Let \(P\) be a abstract polytope. For faces \(F\leq G\) of ranks \(i,j\), its section \(G/F\) is the interval \(\{H:F\leq H\leq G\}\), with ranks shifted so that \(F\) has rank \(-1\). For \(F<G\), it is an abstract polytope of rank \(j-i-1\geq 0\). For \(F=G\), the singleton section is called the rank-\(-1\) polytope. These low ranks allow all intervals, including vertex figures of polygons and intervals between consecutive faces.

The section \(F_n/v\) at a vertex \(v\) is its vertex figure. Its vertices are the edges through \(v\); its faces record the incidences above \(v\). A facet \(H\) is viewed as the section \(H/F_{-1}\).