2
$\begingroup$

I am looking for the definition of a flatness measure in lattice theory.

More generally, I am looking at finite-height lattices and I want to measure their complexity, with a perfectly flat lattice having the lowest complexity, and the highest complexity being open to the author's definition.

A perfectly flat lattice would have all (non-top, non-bottom) vertices connected only with both top and bottom and show the lowest complexity.

Despite my best efforts, I have no found such a measure. Is it known by another name?

Improve this question
$\endgroup$
5
  • $\begingroup$ Can you give some more examples: How should that flatness measure be different from just height? $\endgroup$
    – Thomas Kahle
    Commented Aug 31, 2015 at 19:01
  • $\begingroup$ For finite lattices, one could consider a ratio such as length of a maximal antichain to the number of all lattice members, or a similar ratio involving a disjoint union of all disjoint antichains of maximal length. The latter means that a chain is also very flat. Gerhard "Lay It On Its Side" Paseman, 2015.08.31 $\endgroup$
    – Gerhard Paseman
    Commented Aug 31, 2015 at 21:26
  • $\begingroup$ @ThomasKahle The difference is that a lattice composed of n antichains of size 1 and 1 complex sublattice of size s gives us a height of s. I want a measure that shows that (for a n > s), the lattice is flat. $\endgroup$
    – malaverdiere
    Commented Sep 2, 2015 at 16:25
  • $\begingroup$ @GerhardPaseman mind elaborating on 'disjoint union of all disjoint antichains of maximal length?' $\endgroup$
    – malaverdiere
    Commented Sep 2, 2015 at 16:26
  • $\begingroup$ It should be clear. M_n is flat with one maximal antichain of n elements (and ratio of n/n+2). The sublattice induced by taking the Boolean lattice of subsets of a (2k+1) element set and identifying all the subsets of size less than k, and then identifying all the subsets of size greater than k+1, is pretty flat, with all but two of its elements belonging to one of two disjoint maximal antichains, giving a measure of m/m+2 for m a quantity nearly exponential in k. You need to think about what flat "means" . Gerhard "Go And Play With It" Paseman, 2015.09.02 $\endgroup$
    – Gerhard Paseman
    Commented Sep 2, 2015 at 16:37

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .