Garside Theory: a Composition--Diamond Lemma Point of View
Viktor Lopatkin
TL;DR
This paper connects Garside theory with the Composition--Diamond lemma, demonstrating how normal forms in Garside theory can be understood through confluent reductions similar to Gr"obner--Shirshov bases.
Contribution
It introduces a novel perspective on Garside theory by applying the Composition--Diamond lemma, unifying concepts of normal forms and confluent reductions.
Findings
Greedy normal form coincides with Gr"obner--Shirshov normal form in some cases
A family of a left-cancellative category is a Garside family if reductions are confluent
Provides a new framework linking Garside theory and rewriting systems
Abstract
This paper shows how to obtain the key concepts and notations of Garside theory by using the Composition--Diamond lemma. We also show in some cases the greedy normal form is exactly a Gr\"obner--Shirshov normal form and a family of a left-cancellative category is a Garside family, if and only if a suitable set of reductions is confluent up to some congruence on words.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · History and Theory of Mathematics · Advanced Topics in Algebra