Quotients of the magmatic operad: lattice structures and convergent rewrite systems

TL;DR

This paper explores quotients of the magmatic operad, revealing lattice structures, dimension formulas, and convergent rewriting systems, along with combinatorial models for certain quotients, advancing understanding of operad algebra.

Contribution

It introduces lattice structures for quotients of the magmatic operad, defines comb associative operads, and provides finite convergent presentations based on computer experiments.

Findings

Set of quotients admits a lattice structure.

Derived an analog of the Grassmann formula for dimensions.

Provided finite convergent rewriting systems for comb associative operads.

Abstract

We study quotients of the magmatic operad, that is the free nonsymmetric operad over one binary generator. In the linear setting, we show that the set of these quotients admits a lattice structure and we show an analog of the Grassmann formula for the dimensions of these operads. In the nonlinear setting, we define comb associative operads, that are operads indexed by nonnegative integers generalizing the associative operad. We show that the set of comb associative operads admits a lattice structure, isomorphic to the lattice of nonnegative integers equipped with the division order. Driven by computer experimentations, we provide a finite convergent presentation for the comb associative operad in correspondence with~$3$. Finally, we study quotients of the magmatic operad by one cubic relation by expressing their Hilbert series and providing combinatorial realizations.