标题： 霍普夫代数的基本和齐次基，由非对称操作构建 显示英文标题

标题： Fundamental and homogeneous bases of Hopf algebras built from nonsymmetric operads

摘要： 我们在自由非对称操作的基集上引入了新的部分序结构。 这些偏序涉及装饰的有序根树，它们的终端区间是格。 这些格不是分次的，不是自对偶的，也不是半分配的，但它们是EL-shellable的，其Möbius函数取值于$\{-1, 0, 1\}$。 它们在$m$-Fuss-Catalan对象族和树的森林族上包含子格。 这种后一种顺序结构被用来构造自由非对称操作的自然霍普夫代数的两个新基：一个基本基和一个齐次基。 结合已知的这些霍普夫代数的基本基，这就得到了一组三个基。 这种情况类似于在Malvenuto-Reutenauer、Loday-Ronco和非交换对称函数的霍普夫代数中观察到的情况，每个都表现出这样的基组以及分别涉及右弱部分序、Tamari部分序和布尔格部分序的基变换。 显示英文摘要

摘要： We introduce new partial order structures on the underlying sets of free nonsymmetric operads. These posets involve decorated ordered rooted trees, and their terminal intervals are lattices. These lattices are not graded, not self-dual, and not semi-distributive, but they are EL-shellable, and their M\"bius functions take values in $\{-1, 0, 1\}$. They admit sublattices on the families of $m$-Fuss-Catalan objects and of forests of trees. This latter order structure is used to construct two new bases for the natural Hopf algebras of free nonsymmetric operads: a fundamental basis and a homogeneous basis. Along with the already known elementary basis of these Hopf algebras, this yields a triple of bases. The situation is similar to what is observed in the Hopf algebras of Malvenuto-Reutenauer, Loday-Ronco, and noncommutative symmetric functions, each of which presents such triples of bases and basis changes involving, respectively, the right weak partial order, the Tamari partial order, and the Boolean lattice partial order.