Operads of decorated cliques II: Noncrossing cliques

A complete study of an operad NCℳ 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 ℳ-cliques. These are noncrossing configurations of chords with arcs labeled by a unitary magma ℳ. The magmatic product of ℳ intervenes for the computation of the operadic composition of 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 NCℳ allows one to obtain alternative constructions of already known operads like operads of formal fractions and the operad of bicolored noncrossing configurations.