# a poset with small “cycles”

(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

-
Given an upper bound $s$ of $x$ and $y$ (i.e., $x \le s$ and $y \le s$), do you also find a $w \le s$ with the described properties whenever $x$ and $y$ cover a common element $z$? – Someone Sep 30 '10 at 12:26
Well, no... (assuming that you mean $w$ should also be an upper bound of $x$ and $y$) – Martin Rubey Sep 30 '10 at 12:38
Yes, I was asking about the fact whether your "curious property" holds also in intervals. Is your $w$ unique for given $x$ and $y$? – Someone Sep 30 '10 at 15:49
Hm, I'm not sure I understand your question. I believe (but cannot prove yet) that $w$ is the join of $x$ and $y$, but there are other upper bounds of $x$ and $y$, too. Also, $z$ can have other covering elements beside $x$ and $y$. Finally, one can show that the posets always have a minimal and a maximal element. – Martin Rubey Sep 30 '10 at 17:23
This behaviour would be consistent with the poset having a left modular chain. (This property is not enough to guarantee that the poset has a left modular chain, but it suggests that the posets you want might have that property.) See Peter McNamara and Hugh Thomas, arxiv.org/abs/math/0211126. – Hugh Thomas Oct 3 '10 at 2:17