Parafree augmented algebras and Gr\"obner-Shirshov bases for complete augmented algebras

TL;DR

This paper develops a theory of parafree augmented algebras, explores related conjectures, and introduces Gr"obner-Shirshov bases, including an example of an algebra with infinite cohomological dimension and a composition-diamond lemma.

Contribution

It introduces a new framework for parafree augmented algebras, extends the Composition-Diamond lemma to complete augmented algebras, and provides conditions for residual nilpotency.

Findings

Constructed an example of a finitely generated parafree augmented algebra with infinite cohomological dimension

Proved a version of the Composition-Diamond lemma for complete augmented algebras

Established a sufficient condition for residual nilpotency based on relations

Abstract

We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of finitely generated parafree augmented algebra of infinite cohomological dimension. Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and provide a sufficient condition for augmented algebra to be residually nilpotent on the language of its relations.