University of Guilan Journal of Algebra and Related Topics 2345-3931 9 2 2021 12 01 Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra} 39 46 5184 10.22124/jart.2021.20356.1305 EN J. D. Farley Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, USA. 0000-0003-3247-6014 Journal Article 2021 08 11 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.

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