Polygraphic resolutions for operated algebras

TL;DR

The paper presents a comprehensive framework for operated algebras using higher-dimensional rewriting via Ω-polygraphs and polyautomata, enabling termination without monomial orders and providing a coherent interpretation of operator nesting through pushdown automata. It establishes an equivalence between Ω-algebras and linear polygraphs via polyautomata and develops a construction Sq(X) that yields polygraphic resolutions from reduced convergent presentations. By classifying local and critical branchings and connecting to Gröbner–Shirshov theory, the work provides practical methods to obtain linear bases and homotopical resolutions for a wide class of operated algebras. The approach is demonstrated on fundamental examples, including free Rota-Baxter, differential, and differential Rota-Baxter algebras, with explicit reduced presentations and resolutions that generalize known GS-based techniques to higher-dimensional, operator-rich settings.

Abstract

This paper introduces the structure of operated polygraphs as a categorical model for rewriting in operated algebras, generalizing Gröbner-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.

Theorems & Definitions (47)

• Proposition 2.2.6
• proof
• Proposition 2.2.9
• Proposition 3.1.7
• proof
• Remark 3.2.5
• Lemma 3.2.6
• Theorem 3.2.7
• proof
• Proposition 3.2.8