Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra}

In the preprint of ``Pseudo-Gorenstein and Level Hibi Rings,'' Ene, Herzog, Hibi, and Saeedi Madani assert (Theorem ۴.۳) that for a regular planar lattice L with poset of join-irreducibles P, the following are equivalent:(۱) L is level;(۲) for all x,y\in P such that y\lessdot x, \height_{\hat P}(x)+\depth_{\hat P}(y)\le\rank(\hat P)+۱;(۳) for all x,y\in P such that y\lessdot x, either \depth(y)=\depth(x)+۱ or \height(x)=\height(y)+۱.They added, ``Computational evidence leads us to conjecture that the equivalent conditions given in Theorem ۴.۳ do hold for any planar lattice (without any regularity assumption).''Ene {\sl et al.} prove the equivalence of (۲) and (۳) for a regular simple planar lattice, and write, ``One may wonder whether the regularity condition ... is really needed.''We show one cannot drop the regularity condition. Ene {\sl et al.} say that ``we expect'' (۲) to imply (۱) for any finite distributive lattice L.We provide a counter-example.

کلیدواژه ها:

Distributive lattice ، (partially) ordered set ، Rank ، chain ، join-irreducible