%0 Journal Article
%T Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra}
%J Journal of Algebra and Related Topics
%I University of Guilan
%Z 2345-3931
%A Farley, J. D.
%D 2021
%\ 12/01/2021
%V 9
%N 2
%P 39-46
%! Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra}
%K Distributive lattice
%K (partially) ordered set
%K Rank
%K chain
%K join-irreducible
%R 10.22124/jart.2021.20356.1305
%X In the preprint of ``Pseudo-Gorenstein and Level Hibi Rings,'' Ene, Herzog, Hibi, and Saeedi Madani assert (Theorem 4.3) that for a regular planar lattice $L$ with poset of join-irreducibles $P$, the following are equivalent:(1) $L$ is level;(2) for all $x,y\in P$ such that $y\lessdot x$, $\height_{\hat P}(x)+\depth_{\hat P}(y)\le\rank(\hat P)+1$;(3) for all $x,y\in P$ such that $y\lessdot x$, either $\depth(y)=\depth(x)+1$ or $\height(x)=\height(y)+1$.They added, ``Computational evidence leads us to conjecture that the equivalent conditions given in Theorem 4.3 do hold for any planar lattice (without any regularity assumption).''Ene {\sl et al.} prove the equivalence of (2) and (3) 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'' (2) to imply (1) for any finite distributive lattice $L$.We provide a counter-example.
%U https://jart.guilan.ac.ir/article_5184_c9447ccc21ee53f81415d443ad81b1a9.pdf