What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples:
- The set of all finite subsets of a (possibly infinite) set.
- The set of all finite-dimensional vector subspaces of a (possibly infinite-dimensional) vector space.
- The set of all finite-dimensional affine subspaces a (possibly infinite-dimensional) affine space.
- Any set-theoretic tree.
- Any poset that could reasonably be construed as an "abstract polytope."