论文标题
套装家庭的霍夫夫
Hopf monoids of set families
论文作者
论文摘要
$ i $上的a \ textit {接地set family}是一个子集$ f \ subseteq2^i $，以便在f $中$ \ emptyset \。我们在接地套装的家族上研究了线性化的hopf monoid \ textbf {sf {sf}，其限制和收缩受抗肌瘤的相应操作的启发。许多已知的组合物种，包括简单络合物和矩形，形成\ textbf {sf}的Hopf subonoids，尽管并非总是具有“标准” Hopf结构（例如，我们的收缩操作不是矩形的通常收缩）。我们使用Aguiar和Ardila的拓扑方法来获得无有限posets阶阶阶层的HOPF subonoid的无取消对抗公式。此外，我们证明，链伙伴的秩序理想晶格的HOPF代数扩展了对称函数的HOPF代数，并且其性格组将正式幂序列的组扩展到一个变量中，并在乘数下进行恒定项1。
A \textit{grounded set family} on $I$ is a subset $F\subseteq2^I$ such that $\emptyset\in F$. We study a linearized Hopf monoid \textbf{SF} on grounded set families, with restriction and contraction inspired by the corresponding operations for antimatroids. Many known combinatorial species, including simplicial complexes and matroids, form Hopf submonoids of \textbf{SF}, although not always with the "standard" Hopf structure (for example, our contraction operation is not the usual contraction of matroids). We use the topological methods of Aguiar and Ardila to obtain a cancellation-free antipode formula for the Hopf submonoid of lattices of order ideals of finite posets. Furthermore, we prove that the Hopf algebra of lattices of order ideals of chain gangs extends the Hopf algebra of symmetric functions, and that its character group extends the group of formal power series in one variable with constant term 1 under multiplication.
