Combinatorial operads from monoids

TL;DR

This paper introduces a functorial method to construct set-operads from monoids, generating new operads linked to various combinatorial objects and recovering known operads, with detailed presentations.

Contribution

It presents a novel functorial construction that produces a wide class of operads from monoids, connecting algebraic structures with combinatorial objects.

Findings

Constructed new operads from monoids involving combinatorial objects

Revealed relationships between known and new operads

Provided presentations by generators and relations for the operads

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 operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, trees with a fixed arity, Schr\"oder trees, Motzkin words, integer compositions, directed animals, and segmented integer compositions. We also recover some already known (symmetric or not) operads: the magmatic operad, the associative commutative operad, the diassociative operad, and the triassociative operad. We provide presentations by generators and relations of all constructed nonsymmetric operads.