论文标题
POSET HOPF MONOIDS
Poset Hopf Monoids
论文作者
论文摘要
我们启动了一大批物种单体和共同体的研究，这些物种配备了与乘法和合并图兼容的POSET结构。我们表明，如果一个单型和共核能通过Galois连接相关，那么它们彼此之间是双重的。通过引入通过Möbius倒置构建的新基础，最好理解这种双重性。我们使用这个新的基础为固定分区，图形，超图和简单复合物的Hopf Monoids的原始词提供了统一的证明。 此外，我们表明，当且仅当HOPF MONOID是线性性，交换性和同时性时，HOPF MONOID的单体和共同体是通过Galois连接相关的。在这些情况下，我们根据对相关Poset的特征多项式的评估给出了反模型的无分组公式。这为图形，超图，设定分区和简单复合物的Hopf Monoids的抗植物提供了新的证明。
We initiate the study of a large class of species monoids and comonoids which come equipped with a poset structure that is compatible with the multiplication and comultiplication maps. We show that if a monoid and a comonoid are related through a Galois connection, then they are dual to each other. This duality is best understood by introducing a new basis constructed through Möbius inversion. We use this new basis to give uniform proofs for cofreeness and calculations of primitives for the Hopf monoids of set partitions, graphs, hypergraphs, and simplicial complexes. Further, we show that the monoid and comonoid of a Hopf monoid are related through a Galois connection if and only if the Hopf monoid is linearized, commutative, and cocommutative. In these cases, we give a grouping-free formula for the antipode in terms of an evaluation of the characteristic polynomial of a related poset. This gives new proofs for the antipodes of the Hopf monoids of graphs, hypergraphs, set partitions, and simplicial complexes.
