An abstract \(n\)-polytope, for \(n\geq 0\), is a
partially ordered set \(P\) of faces with a least face
\(F_{-1}\) and greatest face \(F_n\). Every chain extends to a maximal chain,
called a flag, with exactly \(n+2\) faces. Faces in each flag
have ranks \(-1,0,\ldots ,n\), consistently
across flags. Faces other than the least and greatest are proper;
vertices, edges and facets have ranks \(0,1,n-1\).
The diamond condition requires exactly two faces of rank
\(i\) between faces \(F<G\) of ranks \(i-1\) and \(i+1\), for \(0\leq i<n\).
Two flags are adjacent if they differ in exactly one proper face.
Strong flag connectivity requires any two flags to be joined
by adjacent flags, each containing their intersection.
Thus, for each \(0\leq i<n\), each flag has a unique adjacent flag
differing in rank \(i\). Strong flag connectivity restricts paths to these
adjacent flags, each containing the endpoint flags' intersection; arbitrary
paths through the least face would connect every pair without imposing an
axiom.
These conditions prescribe incidence without choosing lengths or angles.
A rank-zero polytope is a two-element chain. A rank-one polytope
has two vertices between its least and greatest faces. These low-rank
cases occur as sections of higher-rank polytopes.