a poset with small "cycles" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:47:57Z http://mathoverflow.net/feeds/question/40600 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40600/a-poset-with-small-cycles a poset with small "cycles" Martin Rubey 2010-09-30T11:02:45Z 2010-09-30T14:18:06Z <p>(a followup to <a href="http://mathoverflow.net/questions/40348/proving-that-a-poset-is-a-lattice" rel="nofollow">this recent question</a>)</p> <p>I noticed the following curious property of a poset (which I strongly believe to be a lattice, I'm still trying to prove that...):</p> <p>Suppose that \$z\$ is covered by \$x\$ and \$y\$. Then there is a common upper bound \$w\$ of \$x\$ and \$y\$ such that either</p> <ul> <li>\$w\$ covers both \$x\$ and \$y\$, or</li> <li>\$w\$ covers either \$x\$ or \$y\$ (say \$y\$), and the other element is separated from \$w\$ by exactly one more element (say \$a\$).</li> </ul> <p>(There is <a href="http://service.ifam.uni-hannover.de/~rubey/poset.pdf" rel="nofollow">an example poset</a>, computed using sage-combinat and dot2tex)</p> <p>Using ASCII art, all relations are covering:</p> <pre> w w / \ / \ x y or a | \ / | | z x y \ / z </pre> <p>Does this property have some name? Could it be helpful for proving that the poset is a lattice?</p> <p>Although it's rather trivial, let us note that there are non-lattices having this property:</p> <pre> 1 / \ 2 3 |\ /| |/ \| 4 5 \ / 6 </pre> <p>Hm, could it be that such a poset (i.e., with restricted cycle lengths) and with no occurrences of</p> <pre> a b a d |\ /| and |\ /| |/ \| b \/ | c d | /\ | c e </pre> <p>is a lattice...? No, this is not the case:</p> <pre> 1 /|\ / | \ / | \ 2 3 4 |\ / \ /| |/ \ / \| 5 6 7 \ / \ / 8 9 \ / 0 </pre>