# What do we call well-founded posets whose elements have a unique height?

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.”

#### Solutions Collecting From Web of "What do we call well-founded posets whose elements have a unique height?"

I think this paper gives the definition you want, if I understand you correctly:

Definition 2.9. A poset $P$ will be called locally ranked if all its principal lower ideals are ranked.