Computing least common multiples in monoids with a finite Garside family

TL;DR

This paper analyzes the right-reversing algorithm in monoids with Garside structures, proving conditions for termination, cyclicity of non-terminating runs, and methods for computing minimal Garside families.

Contribution

It establishes theoretical results linking algorithm termination, cycles, and Garside families, enhancing understanding of monoid computations.

Findings

Non-terminating runs are necessarily cyclic.

Detecting cycles allows computation of minimal Garside families.

The correctness of right-reversing is supported under Garside theory assumptions.

Abstract

Right-reversing is an algorithm used to compute least common multiples in monoids that admit a right-complemented presentation. The algorithm can either terminate and find a result, fail, or run indefinitely. The correctness of the algorithm can be proved with additional assumptions coming from Garside theory. In the same framework, we prove that a non-terminating run of the algorithm is necessarily cyclic. Stopping when a cycle is detected provides a way of computing a minimal Garside family.