Related papers: Polygraphic resolutions for operated algebras
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…
In this paper we use the operadic framework to find Gr\"obner-Shirshov bases for the free quadri-algebra. We perform computations using the representation of the nonsymmetric operad by planar rooted trees in a very intuitive way.…
This paper establishes a uniform procedure to split the operations in any algebraic operad, generalizing previous known notions of splitting algebraic structures from the dendriform algebra of Loday that splits the associative operation to…
We introduce, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter $\gamma$ of diassociative algebras of Loday, called $\gamma$-pluriassociative algebras. By…
Rota-Baxter operators are an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integrals in the case of the polynomial algebra $\mathbf{k}[x]$. We consider…
We examine two nonselfadjoint operator algebras: the weighted shift algebra, and the Volterra operator algebra. In both cases, the operator algebra is the norm closure of the polynomials in the operator norm. In the case of the weighted…
Quite recently, Bremner et al. introduced a new approach to Rota's Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified by Bremner et al.…
We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical $\mathbf{SK}$-algebras, linear $\mathbf{BCI}$-algebras, planar…
We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the…
The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications. This paper carries out an algebraic study of general integral operators and equations,…
Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota-Baxter operator. In applying the general framework to univariate polynomials, one is led to…
Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework…
The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We…
Bremner and Elgendy developed a classification of operated polynomial identities for linear operators on associative algebras, encompassing both classical and newly discovered cases. Within the framework of Rota's Program, each of these new…
Motivated by the recent work of Batkam-Tcheka on pointed multiplicative operads, we construct in this paper new chain complex algebras and two distinct bicomplex algebra structures on a free symmetric connected multiplicative differential…
In this paper, we compute the Gr\"obner-Shirshov bases for certain regular double extension algebras by means of an algorithm implemented in Matlab, which facilitates the underlying algebraic computations. Moreover, we establish that these…
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted…
Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the…
The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such…
Under the common theme of splitting of operations, the notions of (tri)dendriform algebras, pre-Lie algebras and post-Lie algebras have attracted sustained attention with broad applications. An important aspect of their studies is as the…