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.

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∈P such that y⋖x, \heightP^(x)+\depthP^(y)≤\rank(P^)+1;
(3) for all x,y∈P such that y⋖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

Journal of Algebra and Related Topics
Volume 9, Issue 2
December 2021
Pages 39-46

Files

Share

How to cite

Statistics