(a followup to this recent question)
I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that...):
Suppose that $z$ is covered by $x$ and $y$. Then there is a common upper bound $w$ of $x$ and $y$ such that either
- $w$ covers both $x$ and $y$, or
- $w$ covers either $x$ or $y$ (say $y$), and the other element is separated from $w$ by exactly one more element (say $a$).
(There is an example poset, computed using sage-combinat and dot2tex)
Using ASCII art, all relations are covering:
w w / \ / \ x y or a | \ / | | z x y \ / z
Does this property have some name? Could it be helpful for proving that the poset is a lattice?
Although it's rather trivial, let us note that there are non-lattices having this property:
1 / \ 2 3 |\ /| |/ \| 4 5 \ / 6
Hm, could it be that such a poset (i.e., with restricted cycle lengths) and with no occurrences of
a b a d |\ /| and |\ /| |/ \| b \/ | c d | /\ | c e
is a lattice...? No, this is not the case:
1 /|\ / | \ / | \ 2 3 4 |\ / \ /| |/ \ / \| 5 6 7 \ / \ / 8 9 \ / 0