# Finite posets for which all intervals are atomic

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?
• It would seem that for finite posets which are lattices, locally branching means that there are no meet irreducible elements and no join irreducible elements. Do I have this right? If so, you might find an alternate characterization in terms of irreducibility. In which case, start from Birkhoff's Lattice Theory and a citation index and grep for irreducible. Gerhard "Fgrep Is A Useful Tool" Paseman, 2019.04.30. – Gerhard Paseman Apr 30 at 15:59
• @GerhardPaseman Having no join (meet) irreducible elements is the same as being atomig (coatomic), being locally branched means that there are no join (or meet) irreducible elements in any interval, which is exactly the equivalence I have given. Does that help me in finding it in the literature? – Christian Stump Apr 30 at 17:17

## 1 Answer

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.

• The notion we have here is not really thick but rather thin or thicker. Anyway, they indeed consider this and compute a relation on the flag vector in the case of graded posets. I cannot find, at a first glance, anything about atomic lattices in that paper or in the references. – Christian Stump Apr 30 at 13:03
• Yes, by "sort of opposite to thin" I guess I mean something that actually overlaps with thin. We do see your locally branched in the literature. However, I do not see your equivalence anywhere either. – John Machacek May 1 at 13:10