This paper shows how to obtain the key concepts and notations of Garside theory by using the Composition--Diamond lemma. We also show that in some cases the greedy normal form is exactly a Grobner--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.
We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of finitely generated parafree augmented algebra of infinite cohomological dimension. Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and provide a sufficient condition for augmented algebra to be residually nilpotent on the language of its relations.
We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
