Editing Abstract polytope (section)

=== Rank ===
The ''rank'' of a face F is defined as (''m''&nbsp;−&nbsp;2), where ''m'' is the maximum number of faces in any [[Total order#Chains|chain]] (F',&nbsp;F",&nbsp;...&nbsp;,&nbsp;F) satisfying F'&nbsp;<&nbsp;F"&nbsp;<&nbsp;...&nbsp;<&nbsp;F. F' is always the least face, F<sub>−1</sub>.

The ''rank'' of an abstract polytope '''P''' is the maximum rank '''''n''''' of any face. It is always the rank of the greatest face F<sub>n</sub>.

The rank of a face or polytope usually corresponds to the ''dimension'' of its counterpart in traditional theory.

For some ranks, their face-types are named in the following table.
{|class=wikitable style="text-align: center;"
|- 
!! width="80" | Rank !! width="50" |  −1 !! width="50" |0 !! width="50" |1 !! width="50" |2 !! width="50" |3 !! width="30" | ... !! width="50" |''n'' − 2  !! width="50" |''n'' − 1  ||width="50" |''n'' 
|-
! Face Type 
| Least ||Vertex ||Edge ||† ||Cell ||   ||Subfacet or ridge<ref name=ARP23>{{Harvnb |McMullen |Schulte |2002 |loc=p. 23}}</ref> ||Facet<ref name=ARP23 /> ||Greatest
|}

† Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of ''any'' rank.