Computing noncommutative Hilbert series

Let R be an associative ring with identity, X a set of noncommuting variables, = {αx} x ∈ X a set of automorphisms αx of R and R {X} the -twisted free associative algebra on X over R. Let Y be another set of noncommuting variables, ℬ = {βy}y∈Y a set of automorphisms βy of R {X} and S = (R{X})ℬ {Y} the ℬ-twisted free associative algebra on Y over R{X}. Next, let X1 be a set of noncommuting variables, for each l = 1,2,…. We form the free associative algebra S1 = S{X1}on Xl over S and inductively, we form the free associative algebra Sl+1 = Sl{Xl+1} on Xl+1 over Sl, l = 1,2,….

This chapter provides an introduction to the theory of regular languages and finite automata. These concepts will be used in the following chapters for describing behaviors of asynchronous circuits and for specifying abstract behaviors. The theory of regular languages and finite automata is now well established as one of the important basic tools of computer science. We present this theory in a somewhat different way than is done in most textbooks, because we feel that our approach is more general and permits us to establish the relationship between regular languages and finite automata in a very natural way. For a general introduction to regular languages and finite automata, see one of the many texts on this subject, for example [65, 77, 100, 120]; for material closer to our treatment see [16, 17, 20].

The Nowicki conjecture for relatively free algebras

Two-sided wreath product of categories

Sakoda and Sipser [Sa78] introduced a technique for constructing certain regular languages over a large alphabet and used these languages as candidates for proving lower bounds on the size increase when converting a nondeterministic finite automaton (NFA) to a deterministic one (DFA). The purpose of this note is to show that their method is quite useful in solving several problems in the theory of regular languages. In view of its intuitive appeal, we recommend it as a pedagogic aid for presenting lower bound proofs.

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

On a factorization of graded Hopf algebras using Lyndon words

Shirshov published six papers on Lie algebras in which he found the following results (in order of publication. 1953–1962):

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

The Algebra of Quasi-Symmetric Functions Is Free over the Integers

The aim of this paper is to prove that, except in caracteristic 2, the division rings considered in H. Lorenz's paper [1] do not only contain free associative algebras, but also free group algebras. The main ideas are due to L. Makar-Limanov who proved that result for D1 [2]

This result has a number of important consequences. It may be seen to imply that no simple exceptional finite dimensional Jordan algebra of characteristic not two is a homomorphic image of a special(2) Jordan algebra. But there is an exceptional simple Jordan algebra S& (of dimension 27) over a field a of characteristic not two which is generated by three of its elements. Then ID is a homomorphic image of the free Jordan algebra '3 on three generators. It follows that '3 is not special, and we also know that '3 is not isomorphic to the free special Jordan algebra 3 [x, y, z, 1 ] consisting of all reversible polynomials(') in the free associative algebra a [x, y, , 1Z] of all polynomials on the three generators x, y, z. 2. Elementary properties of 60 and certain subalgebras. We begin with a brief description of the algebra eo of our theorem. Let Z be an algebra with a unity element e over a field 0 of characteristic not two. Suppose that Z has an involution T over a; that is, an antiautomorphism

A Hopf–Galois correspondence for free algebras

Degree estimate for commutators

Gröbner bases in universal enveloping algebras of Leibniz algebras