Connectivity

Connectivity or connectedness is a property of polytope like object with several closely related formulations.

Abstract Polytopes


Abstract polytopes are typically required to be strongly connected. This condition may be relaxed to allow for compounds.

Connectivity
A bounded poset is said to be connected if it is rank 1 or if for any two proper elements $F$ and $G$, there exists a sequence $$(F_1,\ldots,F_n)$$ of proper elements such that $$F=F_1$$, $$G=F_n$$, and any $$F_i,F_{i+1}$$ are comparable. The special concession that rank 1 polytopes are connected ensures that the dyad is connected and thus an abstract polytope.

Strong connectivity
A bounded poset is strongly connected iff every section is connected.

Flag connectivity
Two flags are said to be adjacent iff they differ by exactly one element. A ranked bounded poset is flag connected iff for any two flags $$\Phi$$ and $$\Psi$$, there exists a sequence of flags $$(\Phi_1,\ldots,\Phi_n)$$ such that $$\Phi=\Phi_1$$, $$\Psi=\Phi_n$$, and any pair of flags $$\Phi_i$$ and $$\Phi_{i+1}$$ are adjacent.

Strong flag connectivity
A ranked bounded poset is strongly flag connected iff every section is flag connected. On ranked bounded posets strong flag connectivity is equivalent to strong connectivity.