Editing Abstract polytope (section)

===Sections===
[[Image:Triangular 3-Prism.PNG|thumb|540px|The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (<span style="color:red;">red</span>), and a 2-section (<span style="color:green;">green</span>).]]

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 [[Partially ordered set#Intervals|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''.

P is thus a section of itself.

This concept of section ''does not'' have the same meaning as in traditional geometry.

====Facets====
The '''facet''' for a given ''j''-face ''F'' is the (''j''&minus;''1'')-section ''F''/∅, where ''F''<sub>''j''</sub> is the greatest face.

For example, in the triangle '''abc''', the facet at '''ab''' is '''ab'''/'''∅''' = {'''∅, a, b, ab'''}, which is a line segment.

The distinction between ''F'' and ''F''/∅ is not usually significant and the two are often treated as identical.

====Vertex figures====
The '''[[vertex figure]]''' at a given vertex ''V'' is the (''n''&minus;1)-section ''F''<sub>''n''</sub>/''V'', where ''F''<sub>''n''</sub> is the greatest face.

For example, in the triangle '''abc''', the vertex figure at '''b''' is '''abc'''/'''b''' = {'''b, ab, bc, abc'''}, which is a line segment.  The vertex figures of a cube are triangles.

====Connectedness====
A poset P is '''connected''' if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces

:H<sub>1</sub>, H<sub>2</sub>, ... ,H<sub>k</sub>

such that F = H<sub>1</sub>, G = H<sub>k</sub>, and each H<sub>i</sub>, i < k, is incident with its successor.

The above condition ensures that a pair of disjoint triangles '''abc''' and '''xyz''' is ''not'' a (single) polytope.

A poset P is '''strongly connected''' if every section of P (including P itself) is connected.

With this additional requirement, two pyramids that share just a vertex are also excluded. However, two square pyramids, for example, ''can'', be "glued" at their square faces - giving an octahedron. The "common face" is ''not'' then a face of the octahedron.