page_title: Abstract polytope
section_title: Sections
hierarchical_section_title: Abstract polytope / Introductory concepts / Sections
caption_attribution_description: English: The graph and Hasse of a Triangular 3-prism
context_page_description: In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths. A polytope is a generalisation of polygons and polyhedra into any number of dimensions.
  An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory.
context_section_description: Any subset P' of a poset P is a poset (with the same relation <, restricted to P').
  In an abstract polytope, given any two faces F, H of P with F ≤ H, the set {G | F ≤ G ≤ H} is called a section of P, and denoted H/F. (In order theory, a section is called a closed interval of the poset and denoted [F, H].
  For example, in the prism  abcxyz (see diagram) the section xyz/ø (highlighted green) is the triangle
  {ø, x, y, z, xy, xz, yz, xyz}.
  A k-section is a section of rank k.
  A polytope that is the subset of another polytope is not necessarily a section. In the diagram, the square abcd is a subset of the tetrahedron abcd, but is not a section of it.
  P is thus a section of itself.
  This concept of section does not have the same meaning as in traditional geometry.
caption: The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (red), and a 2-section (green).
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