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

Document Type : Research Paper

Author

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, USA.

10.22124/jart.2021.20356.1305

Abstract

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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