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Conjectures of Ene, Herzog, Hibi, and Saeedi Madani in the {\sl Journal of Algebra} | ||
| Journal of Algebra and Related Topics | ||
| دوره 9، شماره 2، اسفند 2021، صفحه 39-46 اصل مقاله (271.21 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2021.20356.1305 | ||
| نویسنده | ||
| J. D. Farley* | ||
| Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, USA. | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Distributive lattice؛ (partially) ordered set؛ rank؛ chain؛ join-irreducible | ||
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آمار تعداد مشاهده مقاله: 604 تعداد دریافت فایل اصل مقاله: 499 |
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