Operads of decorated cliques I: Construction and quotients

We introduce a functorial construction $\mathsf{C}$ which takes unitary magmas $\mathcal{M}$ as input and produces operads. The obtained operads involve configurations of chords labeled by elements of $\mathcal{M}$, called $\mathcal{M}$-decorated cliques and generalizing usual configurations of chords. 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. Besides, the construction $\mathsf{C}$ leads to alternative definitions of the operads of simple and double multi-tildes, and of the gravity operad.