Varieties of algebras as a lattice with an additional operation

arXiv (Cornell University), 2021

We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup Ω (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Lie.. .).

2000

It was recently proved that any variety of associative algebras over a field of characteristic zero has an integral exponential growth. It is known that a variety V has polynomial growth if and only if V does not contain the Grassmann algebra and the algebra of 2 × 2 upper triangular matrices. It follows that any variety with overpolynomial growth has ex- ponent at least 2. In this note we characterize varieties of exponent 2 by exhibiting a finite list of algebras playing a role similar to the one played by the two algebras above.

Sbornik: Mathematics, 2009

The following problem of Kemer is solved for prime varieties of associative algebras with unit: it is shown that over an infinite field of positive characteristic, each prime variety of associative algebras with unit is generated by an algebraic algebra of bounded algebraicity index over this field. Bibliography: 10 titles.

Proceedings of the American Mathematical Society, 2008

We develop a new method to deal with the Cancellation Conjecture of Zariski in different environments. We prove the conjecture for free associative algebras of rank two. We also produce a new proof of the conjecture for polynomial algebras of rank two over fields of zero characteristic.

Israel Journal of Mathematics

Let f be a polynomial in the free algebra over a field K, and let A be a K-algebra. We denote by SA(f), AA(f) and IA(f), respectively, the 'verbal' subspace, subalgebra, and ideal, in A, generated by the set of all f-values in A. We begin by studying the following problem: if SA(f) is finite-dimensional, is it true that AA(f) and IA(f) are also finite-dimensional? We then consider the dual to this problem for 'marginal' subspaces that are finite-codimensional in A. If f is multilinear, the marginal subspace, SA(f), of f in A is the set of all elements z in A such that f evaluates to 0 whenever any of the indeterminates in f is evaluated to z. We conclude by discussing the relationship between the finite-dimensionality of SA(f) and the finite-codimensionality of SA(f). 1. Verbal subspaces, subalgebras and ideals Throughout this paper, the term 'algebra' will be reserved for a not necessarily unital associative algebra A over a fixed base field K of characteristic p ≥ 0. We shall use A 1 to indicate its unital hull. Definition 1.1. Let A be an algebra, and let f = f (x 1 ,. .. , x n) be a polynomial in the free algebra K X on the set X = {x 1 , x 2 ,. . .}. We shall denote by S A (f), A A (f) and I A (f), respectively, the subspace, subalgebra and ideal in A generated by the set of all f-values in A: {f (a 1 ,. .. , a n) : a 1 ,. .. , a n ∈ A}. We shall call the subspace S A (f) the verbal subspace of A generated by f , and so forth.

Mathematical Notes, 1995

Proceedings of the American Mathematical Society

Let R be an algebra over a field and G a finite group of automorphisms and anti-automorphisms of R. We prove that if R satisfies an essential G-polynomial identity of degree d, then the G-codimensions of R are exponentially bounded and R satisfies a polynomial identity whose degree is bounded by an explicit function of d. As a consequence we show that if R is an algebra with involution * satisfying a *-polynomial identity of degree d, then the *-codimensions of R are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case R must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

Journal of Algebra, 2018

In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free associative algebras of finite rank over fields are elementarily equivalent if and only if their ranks are the same and the fields are equivalent in the weak second order logic. In particular, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. We prove that if an arbitrary ring B with at least one Noetherian proper centralizer is first-order equivalent to a free associative algebra of finite rank over an infinite field then B is also a free associative algebra of finite rank over a field. This solves the elementary classification problem for free associative algebras in a wide class of rings. Finally, we present a formula of the ring language which defines the set of free bases in a free associative algebra of finite rank.

Algebra Universalis, 2002

Every ordered set can be considered as an algebra in a natural way. We investigate the variety generated by order algebras. We prove, among other things, that this variety is not finitely based and, although locally finite, it is not contained in any finitely generated variety; we describe the bottom of the lattice of its subvarieties.

Transactions of the American Mathematical Society, 1959