Related papers: A Normal form for HNN-extension of Dialgebras
In this paper, we define the Gr\"obner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Gr\"obner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the…
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 generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Groebner-Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.
In this paper, we establish the Composition-Diamond lemma for right-symmetric algebras. As an application, we give a Gr\"{o}bner-Shirshov basis for universal enveloping right--symmetric algebra of a Lie algebra.
In this survey article, we report some new results of Groebner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.
Let $Di\langle X\rangle$ be the free dialgebra over a field generated by a set $X$. Let $S$ be a monic subset of $Di\langle X\rangle$. A Composition-Diamond lemma for dialgebras is firstly established by Bokut, Chen and Liu in 2010…
In this paper, by using Composition-Diamond lemma for Lie algebras, we give a Gr\"obner-Shirshov basis for free partially commutative Lie algebra over a commutative ring with unit. As an application, we obtain a normal form for such a Lie…
In this survey, we formulate the Gr\"{o}bner-Shirshov bases theory for associative algebras and Lie algebras. Some new Composition-Diamond lemmas and applications are mentioned.
In this paper, we study the concept of associative $n$-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative $n$-conformal algebra. As an application, we construct…
In this survey article, we report some new results of Gr\"obner-Shirshov bases, including new Composition-Diamond lemmas and some applications of some known Composition-Diamond lemmas.
This paper gives a generic form of the diamond lemma, which includes support for additive and topological structures of the base set, and which does not require any further structure (e.g. an associative multiplication operation) to be…
Let $A$ be a brace algebra. This structure implies that $A$ is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. Using this Composition-Diamond lemma we prove that each pre-Lie algebra $L$ can…
In this paper, we firstly establish Composition-Diamond lemma for $\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $\Omega$-algebra, and then the normal form of the free…
We construct free modules over an associative conformal algebra. We establish Composition-Diamond lemma for associative conformal modules. As applications, Gr\"obner-Shirshov bases of the Virasoro conformal module and module over the…
In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential…
This paper introduces the Higman-Neumann-Neumann extension (HNN extension; for short) for Nijenhuis Lie algebras and provides an embedding theorem. To this end, we employ the theory of Gr\"obner-Shirshov basis for Lie {\Omega}-algebras in…
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…
In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gr\"obner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential…
In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gr\"obner-Shirshov bases and linear bases of free commutative Rota-Baxter algebra, free commutative…
We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gr\"obner-Shirshov bases for replicated algebras,…