3 counterexample answering last question

(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

2 add one more question

(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...?

1

# 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