Abstract:In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen–Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincaré–Birkhoff–Witt basis. Furthermore, we give counter-examples to the converse.

---

中文翻译：

---

具有兼容 CL-shellable 分区偏序集的操作数承认 Poincaré-Birkhoff-Witt 基

摘要：2007 年，Vallette 通过证明歌剧是 Koszul 当且仅当关联的分区偏序集是 Cohen-Macaulay 时，在偏序集和操作集之间架起了一座桥梁。作为 Koszul 和作为 Cohen-Macaulay 的两个概念都承认不同的改进：我们在这里的目标是将这些改进中的两个联系起来。我们更精确地证明，任何（基本集）操作数，其关联的偏序组承认同构兼容的 CL 炮击，都承认 Poincaré-Birkhoff-Witt 基。此外，我们给出了相反的反例。

---