The word problem for polycyclic groups and nilpotent associative algebras

TL;DR

This paper explores the decidability of the word problem in algebraic structures, focusing on polycyclic groups and nilpotent associative algebras, providing a unified approach to these problems.

Contribution

It introduces a general framework for analyzing the word problem across different algebraic structures, including polycyclic groups and nilpotent associative algebras.

Findings

Decidable word problem for polycyclic groups with consistent presentations

Framework applicable to nilpotent associative algebras

Highlights the importance of presentation types in solvability

Abstract

The word problem is an old and central problem in (computational) group theory. It is well-known that the word problem is undecidable in general, but decidable for specific types of presentations. Consistent polycyclic presentations are an important class of group presentations with solvable word problem. These presentations play a fundamental role in the algorithmic theory of polycyclic groups. Problems analogous to the word problem arise when computing with other algebraic structures. Various aspects of this topic are considered in the literature. The aim of this paper is to provide a general approach to the topic including polycyclic groups and nilpotent associative algebras as examples.