Operads of decorated cliques II: Noncrossing cliques

A complete study of an operad $\mathrm{NC} \mathcal{M}$ of noncrossing configurations of chords introduced in previous work of the author is performed. This operad is defined on the linear span of all noncrossing $\mathcal{M}$-cliques. These are noncrossing configurations of chords with arcs labeled by a unitary magma $\mathcal{M}$. The magmatic product of $\mathcal{M}$ intervenes for the computation of the operadic composition of $\mathcal{M}$-cliques. We show that this operad is binary, quadratic, and Koszul by considering techniques coming from rewrite systems on trees. We also compute a presentation for its Koszul dual. Finally, we explain how $\mathrm{NC} \mathcal{M}$ allows one to obtain alternative constructions of already known operads like operads of formal fractions and the operad of bicolored noncrossing configurations.