论文标题
相互作用中的三个大惊小怪的poset及其联想代数
Three Fuss-Catalan posets in interaction and their associative algebras
论文作者
论文摘要
我们引入$δ$ -Cliffs，根据范围MAP $δ$的范围的概括和增加树木的概括。我们在这些对象上定义了第一个晶格结构，并建立了有关其子框的一般结果。其中，我们描述了足够的条件，可以具有可脱壳的posets，带有算法的晶格来计算聚会和两个元素的联接，以及可通过间隔加倍构造的晶格。这些子框中的一些承认自然的几何实现。然后，我们介绍了三个子搜索家族，对于某些地图$δ$，它们的基本集被大惊小怪的 - 卡塔兰人数列出。其中，一个是史丹利晶格的概括，另一个是塔玛里晶格的概括。这三个POSET家族适合链条扩展关系，它们具有一些属性。最后，以与马尔文托 - 鲁特纳犬代数的乘积相同的方式形成了正确的弱布鲁哈特排列顺序，我们构建了代数的代数，其产品的乘积间隔是$Δ$ -Cliff的晶格。我们为$δ$提供必要和足够的条件，以具有关联，有限的呈现或自由代数。我们通过使用以前的大惊小怪的posets来定义$δ$ -Cliffs代数的商来结束这项工作。特别是，一个是loday-ronco代数的概括。
We introduce $δ$-cliffs, a generalization of permutations and increasing trees depending on a range map $δ$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $δ$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak Bruhat order of permutations, we construct algebras whose products form intervals of the lattices of $δ$-cliff. We provide necessary and sufficient conditions on $δ$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $δ$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra.
