Rota's Classification Problem, rewriting systems and Gr\"obner-Shirshov bases

TL;DR

This paper reformulates Rota's Classification Problem using rewriting systems and Gr"obner-Shirshov bases, establishing relationships between these approaches and identifying new classes of operators with convergent properties.

Contribution

It introduces a new reformulation of Rota's Classification Problem in terms of rewriting systems and Gr"obner-Shirshov bases, providing effective conditions and new classes of operators.

Findings

Established relationship between rewriting systems and Gr"obner-Shirshov bases.

Provided effective conditions for Gr"obner-Shirshov operators.

Discovered a new class of Gr"obner-Shirshov operators.

Abstract

In this paper we revisit Rota's Classification Problem on classifying algebraic identities for linear operator. We reformulate Rota's Classification Problem in the contexts of rewriting systems and Gr\"obner-Shirshov bases, through which Rota's Classification Problem amounts to the classification of operators, given by their defining operator identities, that give convergent rewriting systems or Gr\"obner-Shirshov bases. Relationship is established between the reformulations in terms of rewriting systems and that of Gr\"obner-Shirshov bases. We provide an effective condition that gives Gr\"obner-Shirshov operators and obtain a new class of Gr\"obner-Shirshov operators.