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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.
In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and double-free left modules (free modules over a free algebra). We give the Chibrikovs Composition-Diamond lemma for modules a
In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively.
In this paper, we establish Composition-Diamond lemma for tensor product $k< X> otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> otimes k< Y>$ by lifting a Groebner-Shirshov basis in
We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple.
In this contribution, we discuss the asymptotic safety scenario for quantum gravity with a functional renormalisation group approach that disentangles dynamical metric fluctuations from the background metric. We review the state of the art in pure gr