Related papers: The free algebra in R
We organize colored aromatic trees into a pre-Lie-Rinehart algebra (i.e. a flat torsion-free Lie-Rinehart algebra) endowed with a natural trace map, and show the freeness of this object among pre-Lie-Rinehart algebras with trace. This…
In this paper, we introduce the definition of free product of operads, following the definition of a free product of algebras. There is a given method of finding the basis and dimension of the free product of operads. By anti-commutative…
In the present paper, we describe some experiences in using programming, commands and graphical interfaces based on computer algebra systems, as tools for learning Physics and Mathematics.
Algebraic computing in relativity and gravitation dates back more than thirty years, but only relatively recently has hardware of sufficient power to tackle large scale calculations become commonplace. Whereas it is generally understood…
We continue our studies on semilattice ordered algebras. This time we accept constants in the type of algebras. We investigate identities satisfied by such algebras and describe the free objects in varieties of semilattice ordered algebras…
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…
We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows…
Over an arbitrary field $\mathbb{F}$, let $p$ and $q$ be monic polynomials with degree $2$ in $\mathbb{F}[t]$. The free Hamilton algebra of the pair $(p,q)$ is the free noncommutative algebra in two generators $a$ and $b$ subject only to…
The greed package implements the general and flexible framework of arXiv:2002.11577 for model-based clustering in the R language. Based on the direct maximization of the exact Integrated Classification Likelihood with respect to the…
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…
$f,g\colon\mathbb{R}\longrightarrow\mathbb{R}$, it is natural to define $f+g$ as the function that maps $x\in\mathbb{R}$ to $f(x) + g(x)$. However, in base R, objects of class function do not have arithmetic methods defined, so idiom such…
Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the…
Building on the foundations in our previous paper, we study Segal conditions that are given by finite products, determined by structures we call cartesian patterns. We set up Day convolution on presheaves in this setting and use it to give…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
Ordinary algebra of formal power series in one variable is convenient to study by means of the algebra of Riordan matrices and the Riordan group. In this paper we consider algebra of formal power series without constant term, isomorphic to…
A differential graded (DG for short) free algebra $\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra is $$\mathcal{A}^{\#}=\k\langle x_1,x_2,\cdots, x_n\rangle,\,\, \text{with}\,\, |x_i|=1,\,\, \forall…
This paper is the second in a series of three, the aim of which is to construct algebraic geometry over a free metabelian Lie algebra $F$. For the universal closure of free metabelian Lie algebra of finite rank $r \ge 2$ over a finite field…
We classify essential algebras whose irredundant non-refinable covers consist of primal algebras. The proof is obtained by constructing one to one correspondence between such algebras and partial orders on finite sets. Further, we prove…
The free algebra adjunction, between the category of algebras of a monad and the underlying category, induces a comonad on the category of algebras. The coalgebras of this comonad are the topic of study in this paper (following earlier…
For associative algebras in many different categories, it is possible to develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining relations for an algebra of such a category provides a "monomial replacement" of this algebra.…