Comparing Gr\"obner bases and word reversing

TL;DR

This paper compares noncommutative Gröbner bases and word reversing methods for solving the word problem in monoids, demonstrating that they can produce fundamentally different results on certain presentations.

Contribution

It provides the first clear distinction between Gröbner bases and word reversing by constructing examples where their completion procedures diverge significantly.

Findings

Gröbner bases and word reversing can yield different completions.

The two methods are not as closely related as previously conjectured.

Explicit examples show their fundamental differences.

Abstract

Gr\"obner bases, in their noncommutative version, and word reversing are methods for solving the word problem of a presented monoid, and both rely on iteratively completing the initial list of relations. Simple examples may suggest to conjecture that both completion procedures are closely related. Here we disprove this conjecture by exhibiting families of presentations for which they radically differ.