TITLE: Finite posets for which all intervals are atomic
QUESTION [8 upvotes]: Let $P$ be a finite poset which is a lattice with $0,1 \in P$.
An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$.
$P$ is atomic if every element is a join of atoms and coatomic if every element is a meet of coatoms.
Say that $P$ is locally branched if all intervals of length $2$ contain at least four elements. This is if, for a saturated chain $a \prec b \prec c$ in $P$, there is an element $d \neq b$ with $a < d < c$.
In a paper I am currently writing, I make use of the following equivalence:

$P$ is locally branched
Every interval in $P$ is atomic
Every interval in $P$ is coatomic

This property is not hard to prove and is very handy. For example, face lattices of polytopes have the diamond property (every interval of length $2$ contains exactly four elements), so this equivalence shows that they are atomic and coatomic.


Has the property of being locally branched been observed or even used before in the literature?
Is this equivalence already known in the literature?

REPLY [3 votes]: This struck me as a sort of opposite to thin, so googled for "thick" and found     arXiv:math/0101075 by Bayer and Hetye. In this paper a poset is called $r$-thick if every nonempty open interval contains at least $r$ elements. So $2$-thick is locally branched. I gave the paper a quick glance and do not see your equivalence. Google Scholar says the paper has 14 citations. I did not try to follow these citations, but this paper gives a start into the literature.