The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied object in combinatorics. It is thus natural to try to extend it to other family of lattice paths. In this article, we fathom such a possibility by defining and studying an analog of the Tamari lattice on Motzkin paths. We find that the defined partial order is not a lattice, but rather a disjoint union of components, each isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also extends the structural and enumerative results to Schr\"oder paths. At the end, we discuss the relation between our work and that of Baril and Pallo (2014).

中文翻译：

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Motzkin 路径上的偏序

在加泰罗尼亚语对象（如二叉树和 Dyck 路径）上定义的 Tamari 格是组合学中一个经过充分研究的对象。因此很自然地尝试将其扩展到其他晶格路径系列。在本文中，我们通过定义和研究 Motzkin 路径上 Tamari 晶格的模拟来理解这种可能性。我们发现定义的偏序不是一个格，而是一个不相交的组件联合，每个组件都同构于经典 Tamari 格中的一个区间。有了这个结构结果，我们继续枚举我们定义的 Motzkin 路径的偏序中的组件和间隔。我们还将结构和枚举结果扩展到 Schr\"oder 路径。最后，我们讨论了我们的工作与 Baril 和 Pallo (2014) 的工作之间的关系。