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(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 
share|improve this question
    
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
1  
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
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