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  "body": "\n## Definition\n\nLet $n \\ge -1$ be an integer and let $(\\mathcal{P}, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]], whose elements are called faces.\nA *flag* of $\\mathcal{P}$ is a maximal totally ordered subset.\nTwo flags are *adjacent* if they differ in exactly one face.\n$\\mathcal{P}$ is an *abstract polytope of rank $n$* if:\n\n1. $\\mathcal{P}$ has a least face $F_{-1}$ and a greatest face $F_n$;\n2. every flag of $\\mathcal{P}$ has exactly $n + 2$ faces;\n3. $\\mathcal{P}$ is strongly flag-connected: any two flags $\\Phi$ and $\\Psi$ are joined by flags $\\Phi = \\Phi_0, \\Phi_1, \\dots, \\Phi_k = \\Psi$.\n   Here each $\\Phi_{i-1}$ is adjacent to $\\Phi_i$, and $\\Phi \\cap \\Psi \\subseteq \\Phi_i$ for each $i$;\n4. the diamond condition holds: if $F \u003c G$ are faces whose ranks differ by $2$, then there are exactly two faces $H$ with $F \u003c H \u003c G$.\n\n## Remarks\n\nConditions 1 and 2 make $\\mathcal{P}$ a [[graded poset|1C7TKGTB2F]] with ranks $-1, 0, \\dots, n$.\nFaces of rank $0$ are vertices, faces of rank $1$ are edges, and faces of rank $n - 1$ are facets.\n\nExample: the faces of a triangle with vertices $a, b, c$, together with $\\varnothing$ as $F_{-1}$ and the triangle $T$ as $F_2$, ordered by inclusion, form an abstract polytope of rank $2$.\nEvery flag has the form $\\varnothing \u003c a \u003c ab \u003c T$, with $4 = 2 + 2$ faces.\nBetween $\\varnothing$ and an edge there are exactly two vertices, and between a vertex and $T$ there are exactly two edges, so the diamond condition holds.\nMore generally, the faces of any [[convex polytope|300K6CB1KH]] of dimension $n$, with $\\varnothing$ added, form an abstract polytope of rank $n$.\n\nNon-example: take two triangles that share one vertex $c$, with all their vertices and edges, a least face, and one greatest face of rank $2$.\nThe vertex $c$ lies on four edges, so between $c$ and the greatest face there are four faces of rank $1$, and the diamond condition fails.\n\nThe hemicube is obtained from the cube by identifying antipodal faces.\nIt is an abstract polytope of rank $3$ with $4$ vertices, $6$ edges and $3$ square faces, and it is not the face lattice of any convex polytope.\n",
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