Narrow Results

As it is known, the defining identities of a free Novikov algebra can be obtained from a commutative algebra with a derivation. In this paper, we consider a class of algebras obtained from the class of associative algebras with a derivation that generalizes Novikov algebras. Such objects are called noncommutative Novikov algebras. We construct a monomial basis for a free noncommutative Novikov algebra.

In this paper, we consider free transposed Poisson and free F-manifold algebras with an additional metabelian identity. We construct a linear basis for both free metabelian transposed Poisson and free metabelian F-manifold algebras.

A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.

Let be the variety of associative algebras over a field K and A = K 〈x₁,…, x_n〉 be a free associative algebra in the variety freely generated by a set X = {x₁,…, x_n}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut , where is the subcategory of finitely generated free algebras of the variety . The later result solves Problem 3.9 formulated in [17].

In universal algebraic geometry (UAG), some primary notions of classical algebraic geometry are applied to an arbitrary variety of algebras Θ and an arbitrary algebra H ∈ Θ. We consider an algebraic geometry in Θ over the distinguished algebra H and we also analyze H from the point of view of its geometric properties. This insight leads to a system of new notions and stimulates a number of new problems. They are new with respect to algebra, algebraic geometry and even with respect to the classical algebraic geometry.

In our approach, there are two main aspects: the first one is a study of the algebra H and its geometric properties, while the second is focused on studying algebraic sets and algebraic varieties over a "good", particular algebra H.

Considering the subject from the second standpoint, the main goal is to get forward as far as possible in a classification of algebraic sets over the given H. The first approach does not require such a classification which is itself an independent and extremely difficult task. We also consider some geometric relations between different H₁ and H₂ in Θ.

The present paper should be viewed as a brief review of what has been done in universal algebraic geometry. We also give a list of unsolved problems for future work.

The structure of simple and directly indecomposable members of the variety RICRL of representable idempotent commutative residuated lattices is described. These descriptions are employed to determine the size and structure of free algebras in RICRL, using existing theory on the Fraser–Horn and Apple properties. A formula for the free spectrum of RICRL is given.

For $n$ a positive integer and $K$ a finite set of finite algebras, let $L(n,K)$ denote the largest $n$-generated subdirect product whose subdirect factors are algebras in $K$. When $K$ is the set of all $n$-generated subdirectly irreducible algebras in a locally finite variety ${𝒱}$, then $L(n,K)$ is the free algebra $F_{{𝒱}}(n)$ on $n$ free generators for ${𝒱}$. For a finite algebra $A$ the algebra $L(n,\{A\})$ is the largest $n$-generated subdirect power of $A$. For every $n$ and finite $A$ we provide an upper bound on the cardinality of $L(n,\{A\})$. This upper bound depends only on $n$ and these basic parameters: the cardinality of the automorphism group of $A$, the cardinalities of the subalgebras of $A$, and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of $A$. Using this upper bound on $n$-generated subdirect powers of $A$, as $A$ ranges over the $n$-generated subdirectly irreducible algebras in ${𝒱}$, we obtain an upper bound on $|F_{{𝒱}}(n)|$. And if all the $n$-generated subdirectly irreducible algebras in ${𝒱}$ have congruence lattices that are chains, then we characterize in several ways those ${𝒱}$ for which this upper bound is obtained.

We study the structure and properties of free skew Boolean algebras (SBAs). For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements, and calculate the number of such elements. These results are used to study minimal generating sets of finite SBAs. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.

A base of the free Malcev superalgebra on one odd generator is constructed. Some corollaries for skew-symmetric functions and central elements in free Malcev and free alternative algebras are obtained.

A base of the free alternative nil-superalgebra of index 3 on one odd generator is constructed. In particular, its index of solvability is computed. We consider also the corresponding Grassmann algebra and show that the well-known Dorofeev's example of solvable non-nilpotent alternative algebra is its homomorphic image.

For a field 𝔽 and an integer d ≥ 1, we consider the universal associative 𝔽-algebra A generated by two sets of d + 1 mutually orthogonal idempotents. We display four bases for the 𝔽-vector space A that we find attractive. We determine how these bases are related to each other. We describe how the multiplication in A looks with respect to our bases. Using our bases we obtain an infinite nested sequence of two-sided ideals for A. Using our bases we obtain an infinite exact sequence involving a certain 𝔽-linear map ∂: A → A. We obtain several results concerning the kernel of ∂; for instance this kernel is a subalgebra of A that is free of rank d.

In this paper, we construct free commutative integro-differential algebras by applying the method of Gröbner–Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota–Baxter (DRB) algebras of order n. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner–Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional monomials to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.

We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation.

Jacobson investigated the structure of rings with the property that some power of each element is central, in the procedure of the study of commutativity. In this paper, we consider a class of rings in which this property occurs only for central elements, and such rings are called ppnc. We first prove that a noncommutative ppnc ring is infinite, and that a ppnc ring is commutative when it is a K-ring or a locally finite ring. We next study the structure of ppnc rings, and the relation between ppnc rings and related concepts (for example, K-rings, commutative rings and NI rings), through matrix rings, polynomial rings, right quotient rings and direct products.

Point algebras introduced by Evans are algebraic systems which capture the essence of multiplications (a,b) · (c,d)=(p,q) defined on the set of all ordered pairs of elements of a set S, where p and q are selected from among a,b,c,d by some well-defined rule. In 1961, Jonsson and Tarski gave an interesting example of a variety of algebras of type 〈2,1,1〉 for illustrating the failure of certain free algebra properties. In this paper, we show that this equational class of algebras, called the JT-variety, is a universal variety of point algebras in the sense that every variety generated by a point algebra is a reduct of the JT-variety.

Motivated by the category of probabilistic convergence spaces — a supercategory of the category of topological spaces; recently, we brought to light the categories of probabilistic convergence groups, probabilistic metric probabilistic convergence groups, probabilistic convergence transformation groups, along with their underpinning natural examples. The purpose of this paper is, first, to establish a result on the isomorphism between the categories of probabilistic metric groups, and probabilistic metric probabilistic convergence groups. Second, among others, we explore a monad in relation with probabilistic convergence groups, and probabilistic convergence spaces, and their related algebras. In so doing, we consider a product of the categories of probabilistic convergence groups and probabilistic convergence spaces in an attempt to construct a monad on it such that the corresponding category of algebras, the so-called Eilenberg–Moore category, is isomorphic to the category of probabilistic convergence transformation groups. Finally, invoking so-called Beck’s theorem on characterization of algebras, and starting with a particular adjunction, we achieve a monad. Conversely, given a monad, we obtain an adjunction which coincides with the original monad.

A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables.

In 1958 E. Marczewski introduced a general notion of independence, which contained as special cases the majority of independence notions used in various branches of mathematics. A non-empty set I of the carrier A of an algebra is called M-independent if equality of two term operations f and g of the algebra on any finite system of different elements of I implies f = g in A. There are several interesting results on this notion of independence. However the important scheme of M-independence is not wide enough to cover stochastic independence, independence in projective spaces and some others. This is why some notions weaker than the M-independence have been developed. The notion of independence with respect to a family Q of mappings (defined on subsets of A) into A, Q-independence for short, is a common way of defining almost all known notions of independences. There exists an interesting Galois correspondence between families Q of mappings and families of Q-independent sets. In our article after a brief survey of these topics we will mainly concentrate on a few easily formulated and interesting results.