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

Question

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."

A "locally graded poset" is given here as one so that each interval $[x,y]$ is graded, which is pretty close but perhaps not equivalent to what you want. — Jair Taylor, May 20 '15 at 14:46
Actually, I think I found the right definition in another paper. See my answer below. — Jair Taylor, May 20 '15 at 14:51
"Bush, not George"? "Shrub"? "Quasi-tree"? "Well-founded poset where height is well-defined"? "Mountain"? "Gregor Clegane"? I'm starting to run out of ideas... :-) — Asaf Karagila, May 20 '15 at 16:16
@AsafKaragila, I like the first one. But I thought you did not like political jokes? For the record, I've been calling such posets "well-ranked." — goblin GONE, Jun 04 '15 at 15:17
Joking about silly former presidents is not political. Besides, well-ranked sounds reasonable. — Asaf Karagila, Jun 04 '15 at 15:37

score 1 · Accepted Answer · answered May 20 '15 at 14:50

1

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.

answered May 20 '15 at 14:50

Jair Taylor

17,307

Thanks, I think you might be right. Also, that paper looks relevant. – goblin GONE May 20 '15 at 23:54

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

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