Constructing combinatorial operads from monoids

TL;DR

This paper presents a functorial method to construct set-operads from monoids, generating new operads related to various combinatorial objects and recovering known operads.

Contribution

It introduces a novel functorial construction linking monoids to set-operads, unifying and extending combinatorial operad theory.

Findings

Constructed operads include parking functions, Dyck paths, and trees.

Recovered classical operads like magmatic and associative operads.

Generated operads encompass a wide range of combinatorial structures.

Abstract

We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schr\"oder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.