Operads of decorated cliques I: Construction and quotients

TL;DR

This paper introduces a functorial method to construct operads from unitary magmas, generalizing configurations of chords with various combinatorial restrictions, leading to a hierarchy of new operads including noncrossing and Motzkin configurations.

Contribution

It presents a novel functorial construction of operads from unitary magmas, unifying and extending various combinatorial operads through decorated chord configurations.

Findings

Constructed operads include noncrossing, Motzkin, and dissection configurations.

Defined suboperads and quotients based on combinatorial restrictions.

Connected the construction to known operads like multi-tildes and gravity operad.

Abstract

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.