The Malvenuto–Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto–Reutenauer algebra, both of which include the Loday–Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison, we consider a natural partial ordering on the maximal tubings of a graph and characterize those graphs for which this poset is a lattice quotient of the weak order.

Malvenuto–Reutenauer代数是一个经过精心研究的组合Hopf代数，其基数由排列索引。该代数包含各种各样有趣的子Hopf代数，尤其是Loday和Ronco引入的平面二叉树的Hopf代数。我们比较了Malvenuto-Reutenauer代数的两个子代数的一般构造，它们都包括Loday-Ronco代数。第一种是由Reading构造的，以弱阶的格商来定义，第二种是Ronco的构造，其为图associahedra。为了进行比较，我们考虑了图的最大管道上的自然偏序，并表征了那些图元是弱阶的格商的图。