@article {
author = {Farley, J.},
title = {Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra}},
journal = {Journal of Algebra and Related Topics},
volume = {9},
number = {2},
pages = {39-46},
year = {2021},
publisher = {University of Guilan},
issn = {2345-3931},
eissn = {2382-9877},
doi = {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.},
keywords = {Distributive lattice,(partially) ordered set,Rank,chain,join-irreducible},
url = {https://jart.guilan.ac.ir/article_5184.html},
eprint = {https://jart.guilan.ac.ir/article_5184_c9447ccc21ee53f81415d443ad81b1a9.pdf}
}