University of GuilanJournal of Algebra and Related Topics2345-39319220211201Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra}3946518410.22124/jart.2021.20356.1305ENJ. D.FarleyDepartment of
Mathematics, Morgan State University, 1700 E. Cold
Spring Lane, Baltimore, USA.0000-0003-3247-6014Journal Article20210811In 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:<br />(1) $L$ is level;<br />(2) for all $x,y\in P$ such that $y\lessdot x$, $\height_{\hat P}(x)+\depth_{\hat P}(y)\le\rank(\hat P)+1$;<br />(3) for all $x,y\in P$ such that $y\lessdot x$, either $\depth(y)=\depth(x)+1$ or $\height(x)=\height(y)+1$.<br />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).''<br />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.''<br />We show one cannot drop the regularity condition. <br /><br />Ene {\sl et al.} say that ``we expect'' (2) to imply (1) for any finite distributive lattice $L$.<br /><br />We provide a counter-example.https://jart.guilan.ac.ir/article_5184_c9447ccc21ee53f81415d443ad81b1a9.pdf