TY - JOUR
ID - 5184
TI - Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra}
JO - Journal of Algebra and Related Topics
JA - JART
LA - en
SN - 2345-3931
AU - Farley, J. D.
AD - Department of
Mathematics, Morgan State University, 1700 E. Cold
Spring Lane, Baltimore, USA.
Y1 - 2021
PY - 2021
VL - 9
IS - 2
SP - 39
EP - 46
KW - Distributive lattice
KW - (partially) ordered set
KW - Rank
KW - chain
KW - join-irreducible
DO - 10.22124/jart.2021.20356.1305
N2 - 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.
UR - https://jart.guilan.ac.ir/article_5184.html
L1 - https://jart.guilan.ac.ir/article_5184_c9447ccc21ee53f81415d443ad81b1a9.pdf
ER -