Defining relations and Gröbner--Shirshov bases of Poisson algebras as of conformal modules

Averaging algebras, rewriting systems and Gr$ö$bner-Shirshov bases

In this paper, we study the averaging operator by assigning a rewriting system to it. We obtain some basic results on the kind of rewriting system we used. In particular, we obtain a sufficient and necessary condition for the confluence. We supply the relationship between rewriting systems and Grobner-Shirshov bases based on bracketed polynomials. As an application, we give a basis of the free unitary averaging algebra on a non-empty set.

Gröbner-Shirshov bases for semirings

In the paper, we establish Gröbner-Shirshov bases for semirings and commutative semirings. As applications, we obtain Gröbner-Shirshov bases and A. Blass's (1995) and M. Fiore -T. Leinster's (2004) normal forms of the semirings $\mathbb{N}[x]/(x=1+x+x^2)$ and $\mathbb{N}[x]/(x=1+x^2)$ with one generator $x$ and one defining relation, correspondingly.

Rota's Classification Problem, rewriting systems and Gröbner-Shirshov bases

In this paper we revisit Rota's Classification Problem on classifying algebraic identities for linear operator. We reformulate Rota's Classification Problem in the contexts of rewriting systems and Gröbner-Shirshov bases, through which Rota's Classification Problem amounts to the classification of operators, given by their defining operator identities, that give convergent rewriting systems or Gröbner-Shirshov bases. Relationship is established between the reformulations in terms of rewriting systems and that of Gröbner-Shirshov bases. We provide an effective condition that gives Gröbner-Shirs…

结合$n$ -共形代数的合成钻石引理

Composition-Diamond lemma for associative $n$-conformal algebras

本文研究了特征为0的域上结合$n$ -共形代数的概念，建立了自由结合$n$ -共形代数的合成Diamond引理.作为应用，我们构造了由生成元和定义关系表示的李$n$ -共形代数的Gröbner-Shirshov基.