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counterexample answering last question

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edited Sep 30 2010 at 14:18

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

add one more question

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edited Sep 30 2010 at 12:02

Martin Rubey
1,447●5●13

(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...?

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asked Sep 30 2010 at 11:02

Martin Rubey
1,447●5●13

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