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?