# Groebner---Shirshov bases for replicated algebras

## TL;DR

This paper develops a universal method using Groebner-Shirshov bases to solve the word problem in di- and tri-algebras, extending classical algebraic results and enabling new computational approaches.

## Contribution

It introduces a universal approach to the word problem in di- and tri-algebras, applying Groebner-Shirshov bases to Lie and Leibniz algebras and proving an analogue of the PBW theorem.

## Key findings

- Solution of the word problem in di- and tri-algebras.
- Application of Groebner-Shirshov bases to Leibniz algebras.
- Proved PBW-type theorem for universal enveloping tri-algebras.

## Abstract

We establish a universal approach to solution of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows to apply Groebner---Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincare---Birkhoff---Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD^!-algebra).