Adjacency

Adjacency (or incidence) loosely refers to elements or flags of an abstract polytope being near each other. It has several different definitions.

Two flags are j-adjacent if they differ by exactly one element of rank j.

Two elements of different ranks are adjacent if they are comparable, i.e. $$a \leq b$$ or $$b \leq a$$ under the abstract polytope's comparison operator.

Two elements of the same rank n > 0 are adjacent if there is an element of rank n - 1 adjacent to both of them. For example, two faces are adjacent iff they share an edge, and two edges are adjacent iff they share a vertex.

Two vertices are adjacent if there is an edge adjacent to both of them.