Related papers: On the Calculation of gl.dim$G^{\mathbb{N}}(A)$ an…

Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous…

The main objective of this paper is to connect the theory of $Gr\"obner$ bases to concepts of homological algebra. $Gr\"obner$ bases, an important tool in algebraic system and in linear algebra help us to understand the structure of an…

Let $A=K[a_1,\ldots,a_n]$ be a weighted $\mathbb{N}$-filtered solvable polynomial algebra with filtration $FA=\{ F_pA\}_{p\in\mathbb{N}}$, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning,…

Let $R=\oplus_{\Gamma\in\Gamma}R_{\gamma}$ be a $\Gamma$-graded $K$-algebra over a field $K$, where $\Gamma$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $\Gamma$-grading filtration $FR$ of $R$ and the…

Standard noncommutative Gr\"obner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gr\"obner basis procedures for one-sided ideals in finitely presented noncommutative…

Gr\"obner bases can be used for computing the Hilbert basis of a numerical submonoid. By using these techniques, we provide an algorithm that calculates a basis of a subspace of a finite-dimensional vector space over a finite prime field…

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of…

We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight…

It is shown that the Hecke-Kiselman algebra associated to a finite directed graph is an automaton algebra in the sense of Ufnarovskii. Consequently, its Gelfand-Kirillov dimension is an integer if it is finite. As a consequence, it is…

G-algebras, or Groebner bases algebras, were considered by Levandovsky, these algebras include very important families of algebras, like the Weyl algebras and the universal enveloping algebra of a finite dimensional Lie algebra. These…

We mainly study the growth and Gelfand-Kirillov dimension (GK-dimension) of generalized Weyl algebra (GWA) $A=D(\sigma,a)$ where $D$ is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for…

In this work we will study the universal labeling algebra A(Gamma), a related algebra B(Gamma), and their behavior as invariants of layered graphs. We will introduce the notion of an upper vertex-like basis, which allows us to recover…

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely…

We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically,…

Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X> /I$ is almost…

Polyhedral affinoid algebras have been introduced by Einsiedler, Kapranov and Lind to connect rigid analytic geometry (analytic geometry over non-archimedean fields) and tropical geometry. In this article, we present a theory of Gr{\"o}bner…

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$.…

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set…

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrow\Gamma\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free…

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the…