Polygraphic resolutions for operated algebras

TL;DR

This paper develops a categorical framework using operated polygraphs and polyautomata to analyze rewriting systems in operated algebras, extending classical bases and constructing resolutions for various algebraic structures.

Contribution

It introduces operated polygraphs and polyautomata as new tools for studying operated algebras, generalizing existing bases, and constructing resolutions from confluent presentations.

Findings

Polyautomata extend linear polygraphs with operator structures.

Constructed polygraphic resolutions for free operated algebras.

Applied methods to Rota-Baxter, differential, and differential Rota-Baxter algebras.

Abstract

This paper introduces the structure of operated polygraphs as a categorical model for rewriting in operated algebras, generalizing Gr\"obner-Shirshov bases with non-monomial termination orders. We provide a combinatorial description of critical branchings of operated polygraphs using the structure of polyautomata that we introduce in this paper. Polyautomata extend linear polygraphs equipped with an operator structure formalized by a pushdown automaton. We show how to construct polygraphic resolutions of free operated algebras from their confluent and terminating presentations. Finally, we apply our constructions to several families of operated algebras, including Rota-Baxter algebras, differential algebras, and differential Rota-Baxter algebras.