Incidence and adjacency

Incidence is a relation on the elements of abstract polytopes that describes whether two elements "touch". Two elements are incident iff $a\leq b$ or $b\leq a$ under the abstract polytope's comparison operator. It follows from the definition of abstract polytopes that every element is incident with itself, and two different elements can only be incident if they have different ranks (so e.g. no two vertices are incident). Many polytope-like combinatorial structures also have notions of incidence, such as incidence geometries.

Adjacency is a related and looser concept with different definitions for flags and elements:

• Two flags are j -adjacent if they differ by exactly one element of rank j .
• Two elements of different ranks are adjacent iff they are incident.
• 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 iff there is an edge adjacent to both of them.

In common usage, the "elements adjacent to e " are usually assumed not to include e  itself.