SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1261-1280, January 2020.
Since Stanley's [Discrete Comput. Geom., 1 (1986), pp. 9--23] introduction of order polytopes, their geometry has been widely used to examine (algebraic) properties of finite posets. In this paper, we follow this route to examine the levelness property of order polytopes, a property generalizing Gorensteinness. This property has been recently characterized by Miyazaki [J. Algebra, 480 (2017), pp. 215--236] for the case of order polytopes. We provide an alternative characterization using weighted digraphs. Using this characterization, we give a new infinite family of level posets and show that determining levelness is in $\operatorname{co-NP}$. Moreover, we show how a necessary condition of levelness of [J. Algebra, 431 (2015), pp. 138--161] can be restated in terms of digraphs. We then turn to the more general family of alcoved polytopes. We give a characterization for levelness of alcoved polytopes using the Minkowski sum. Then we study several cases when the product of two polytopes is level. In particular, we provide an example where the product of two level polytopes is not level.

中文翻译：

---

订单多面体的水平度

SIAM离散数学杂志，第34卷，第2期，第1261-1280页，2020年1月。
自从Stanley的[Discrete Comput。[Geom。，1（1986），pp。9--23]中介绍了有序多面体，其几何形状已广泛用于检查有限位姿的（代数）性质。在本文中，我们按照这种方法来检查有序多面体的水平性，即广义哥伦斯坦性。宫崎骏[J. 对于有序多面体的情况，《代数》，480（2017），第215--236页。我们提供了使用加权有向图的替代表征。使用此特征，我们给出了一个新的无限级水平摆线族，并表明确定水平度在$ \ operatorname {co-NP} $中。此外，我们展示了[J. [Algebra，431（2015），第138--161页]可以用有向图重新陈述。然后，我们转向凹面多面体的更一般的家族。我们使用Minkowski和求出凹凹多面体的水平度。然后，我们研究了两个多表位产物水平相等的几种情况。特别是，我们提供了一个示例，其中两个级别的多边形的乘积不是级别。