New operated polynomial identities and Gr\"{o}bner-Shirshov bases

TL;DR

This paper proves that all operated polynomial identities classified by Bremner et al. are Gr"{o}bner-Shirshov, advancing the understanding of algebraic identities and their computational bases.

Contribution

It establishes that the classified identities by Bremner et al. are indeed Gr"{o}bner-Shirshov, linking classification with computational algebra tools.

Findings

All classified identities are Gr"{o}bner-Shirshov.

Connects classification of identities with Gr"{o}bner-Shirshov bases.

Enhances methods for analyzing operated polynomial identities.

Abstract

Quite recently, Bremner et al. introduced a new approach to Rota's Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified by Bremner et al. are Gr\"{o}bner-Shirshov.