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."
  • $\begingroup$ 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. $\endgroup$ – Jair Taylor May 20 '15 at 14:46
  • $\begingroup$ Actually, I think I found the right definition in another paper. See my answer below. $\endgroup$ – Jair Taylor May 20 '15 at 14:51
  • $\begingroup$ "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... :-) $\endgroup$ – Asaf Karagila May 20 '15 at 16:16
  • $\begingroup$ @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." $\endgroup$ – goblin Jun 4 '15 at 15:17
  • $\begingroup$ Joking about silly former presidents is not political. Besides, well-ranked sounds reasonable. $\endgroup$ – Asaf Karagila Jun 4 '15 at 15:37

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.

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

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