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?