Gr\"obner bases for operads

TL;DR

This paper introduces a new framework using Gr"obner bases for operads within a novel monoidal category, enabling algorithmic analysis of symmetric operads and their properties.

Contribution

It develops Gr"obner bases machinery for operads in a new monoidal category, providing tools for studying symmetric operads and their Koszulness.

Findings

Developed operadic versions of Diamond Lemma and Buchberger's algorithm.

Provided an effective algorithm for Hoffbeck's PBW criterion.

Enhanced understanding of symmetric operads through new computational methods.

Abstract

We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gr\"obner bases for operads, and present operadic versions of Bergman's Diamond Lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads.