We call a poset factorable if its characteristic polynomial has all positive integer roots. Inspired by inductive and divisional freeness of a central hyperplane arrangement, we introduce and study the notion of inductive posets and their superclass of divisional posets. It then motivates us to define the so-called inductive and divisional abelian (Lie group) arrangements, whose posets of layers serve as the main examples of our posets. Our first main result is that every divisional poset is factorable. Our second main result shows that the class of inductive posets contains strictly supersolvable posets, the notion recently introduced due to Bibby and Delucchi (2022). This result can be regarded as an extension of a classical result due to Jambu and Terao (Adv. in Math. 52 (1984) 248–258), which asserts that every supersolvable hyperplane arrangement is inductively free. Our third main result is an application to toric arrangements, which states that the toric arrangement defined by an arbitrary ideal of a root system of type , or with respect to the root lattice is inductive.

中文翻译：

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归纳偏序集和除法偏序集

如果一个偏序集的特征多项式全部具有正整数根，我们称其为可分解的。受到中心超平面排列的归纳和除法自由性的启发，我们引入并研究了归纳偏序集及其超类除法偏序集的概念。然后，它促使我们定义所谓的归纳和除法阿贝尔（李群）排列，其层偏序集作为我们偏序集的主要例子。我们的第一个主要结果是每个除法偏序集都是可因式分解的。我们的第二个主要结果表明，归纳偏序集类包含严格超可解的偏序集，这一概念最近由 Bibby 和 Delucchi (2022) 引入。该结果可以被视为 Jambu 和 Terao 经典结果的扩展（Adv. in Math . 52 (1984) 248–258），该结果断言每个超可解超平面排列都是归纳自由的。我们的第三个主要结果是对环面排列的应用，它指出由类型的根系统的任意理想定义的环面排列,或者相对于根晶格是归纳的。