装饰集团的操作
Operads of decorated cliques
摘要：
所有对角线构形多边形的向量空间被赋予一个操作数结构。这是这里介绍的函子构造$\mathsf{C}$的结果，它以酉岩浆$\mathcal{M}$为输入并产生操作数。所得到的运算涉及具有$\mathcal{M}$上标记的弧的配置的正多边形，称为$\mathcal{M}$ -装饰团，并且推广了具有对角线的配置的通常多边形。我们在这里提供了一个完整的研究操作$\mathsf{C}\mathcal{M}$。通过考虑定义的$\mathcal{M}$ -装饰团的组合子族，例如，通过限制交叉对角线的最大数目或顶点的最大度，我们得到了$\mathsf{C}\mathcal{M}$的子运算域和子代数.这导致了一个新的层次结构的操作，其中包括操作的非交叉配置，Motzkin配置，森林，解剖的多边形，和对合。我们证明了不相交组态的子域是Koszul，并展示了它的生成元和关系的表示。此外，结构$\mathsf{C}$还引出了几个运算符的替代定义，例如双色非交叉配置的运算符以及简单和双重多重波浪符的运算符。
英文摘要：
The vector space of all polygons with configurations of diagonals is endowed with an operad structure. This is the consequence of a functorial construction $\mathsf{C}$ introduced here, which takes unitary magmas $\mathcal{M}$ as input and produces operads. The obtained operads involve regular polygons with configurations of arcs labeled on $\mathcal{M}$, called $\mathcal{M}$-decorated cliques and generalizing usual polygons with configurations of diagonals. We provide here a complete study of the operads $\mathsf{C}\mathcal{M}$. By considering combinatorial subfamilies of $\mathcal{M}$-decorated cliques defined, for instance, by limiting the maximal number of crossing diagonals or the maximal degree of the vertices, we obtain suboperads and quotients of $\mathsf{C}\mathcal{M}$. This leads to a new hierarchy of operads containing, among others, operads on noncrossing configurations, Motzkin configurations, forests, dissections of polygons, and involutions. We show that the suboperad of noncrossing configurations is Koszul and exhibit its presentation by generators and relations. Besides, the construction $\mathsf{C}$ leads to alternative definitions of several operads, like the operad of bicolored noncrossing configurations and the operads of simple and double multi-tildes.
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