Showing papers on "Free algebra published in 2004"
TL;DR: In this article, a duality transform for the coalgebra of the free difference quotient derivation is constructed for a free algebra of scalars. But the duality transformation is based on a generalization of the derivation to the algebra of matricial analytic functions.
Abstract: We construct a duality transform for the coalgebra of the free difference quotient derivation-comultiplication of an operator with respect to a free algebra of scalars. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.
123 citations
TL;DR: Monadic monadic MV -algebras are defined and study as pairs of MV - algebrs one of which is a special case of relatively complete subalgebra named m -relatively complete, which determines a unique monadic operator.
57 citations
01 Jan 2004
TL;DR: In this paper, the main properties of free algebras of Schreier varieties of algesbras are discussed. But the main types of free algebraic varieties of free subsets are not discussed.
Abstract: In this chapter, we consider the main properties of free algebras of Schreier varieties of algebras. A variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. In Section 11.1, we describe the main types of Schreier varieties and introduce universal multiplicative enveloping algebras of free algebras. Theorem 11.1.1 gives the main properties of the free algebras of these varieties. In Section 11.2, we expose the weak algorithm for free associative algebras and discuss Schreier’s techniques for free algebras: ranks of left ideals of free associative algebras and Schreier-type formulas for ranks of subalgebras of free algebras.
37 citations
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TL;DR: In this article, it was shown that the conic classes of divisorial ideals can be identified with the full-dimensional open cells in a decomposition of a torus naturally associated with M. This argument can be used for the computation of the Hilbert-Kunz multiplicity of R in characteristic p > 0.
Abstract: Let R=K[M] be a normal affine monoid algbera over a field K.Up to isomorphism the conic ideals are exactly the direct summands ofthe extension R^{1/n} of R. We show that the classes of the conic divisorial ideals can be identified with the full-dimensional open cells in a decomposition of a torus naturally associated with M. Furthermore, they can be characterized by the relative compactness of a certain group associated with them.
Baetica has given examples of Cohen-Macaulay divisorial ideals that are not conic. Wereview his construction and streamline the arguments somewhat.
In the last part of the paper we investigate the multiplicities ofthe conic classes in the decomposition of R^{1/n} as a function of n.This multiplicity turns out to be a quasi-polynomial for all n >= 1 counting the lattice points in the union of the interiors of certain cells of the complex mentioned above. This argument can be used for the computation of the Hilbert-Kunz multiplicity of R in characteristic p > 0. In addition it yields some assertions about the Hilbert-Kunz function of R.
26 citations
TL;DR: In this paper, the Hochschild homology of an algebra over a commutative ring K with a (possibly infinite) Grobner base G on a free algebra F is computed.
Abstract: We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Grobner base. It enables us to compute the Hochschild (co)homology of the algebra. Let A be a finitely generated algebra over a commutative ring K with a (possibly infinite) Grobner base G on a free algebra F, that is, A is the quotient F/I(G) with the ideal I(G) of F generated by G. Given a Grobner base H for an A-subbimodule L of the free A-bimodule A . X . A = A K ⊗ K . X ⊗ K A generated by a set X, we have a morphism ∂ of A-bimodules from the free A-bimodule A . H . A generated by H to A . X . A sending the generator [h] to the element h ∈ H. We construct a Grobner base C on F . H . F for the A-subbimodule Ker(∂) of A . H . A, and with this C we have the free A-bimodule A . C . A generated by C and an exact sequence A . C . A → A . H . A . A . X . A. Applying this construction inductively to the A-bimodule A itself, we have a free A-bimodule resolution of A.
26 citations
TL;DR: In this article, the structure of the endomorphism monoid of a stable basis algebra A is described, and it is shown to be an abundant monoid; the subsemigroup of endomorphisms of finite rank has a regular semigroup of left quotients.
Abstract: The structure of the endomorphism monoid of a stable basis algebra A is described. It is shown to be an abundant monoid; the subsemigroup of endomorphisms of finite rank has a regular semigroup of left quotients.
19 citations
TL;DR: In this article, it was shown that Voiculescu's non-microstate free entropy dimension is 2 for all DT-operators and that every DT-operator can generate L(F 2 ), the von Neumann algebra generated by the free group on two generators.
18 citations
TL;DR: In this article, the Gel'fand-Kirillov dimension of a relatively free algebra A over an arbitrary ground field is calculated, which is determined by the complexity type of the algebra A or by the set of semidirect products of matrix algebras over a polynomial ring contained in the variety Var(A).
Abstract: In this paper the Gel'fand-Kirillov dimension GKdim(A) is calculated for a relatively free associative algebra A over an arbitrary ground field. This dimension is determined by the complexity type of the algebra A or by the set of semidirect products of matrix algebras over a polynomial ring contained in the variety Var(A). The proof is comparatively elementary and does not use the local representability of relatively free algebras.
18 citations
TL;DR: In this paper, it was shown that the groups of tame automorphisms of a free Lie algebra of rank at least four over a field of characteristic zero admit no faithful representation by matrices over any field.
Abstract: We prove that the groups of tame automorphisms of a free Lie algebra (free associative algebra, absolutely free algebra, algebra of polynomials) of rank at least four over a field of characteristic zero admit no faithful representation by matrices over any field.
10 citations
TL;DR: In this paper, the authors give an explicit construction of a class of ∗-orderings on free associative algebras and to prove that ∗orderings from this class extend uniquely to the corresponding free skew fields.
9 citations
TL;DR: In this article, the authors deal in elementary equivalence on arbitrary structures of free algebras in a series of varieties, and show that the equivalence of these structures can be expressed as follows.
Abstract: We deal in elementary equivalence on arbitrary structures of free algebras in a series of varieties.
12 Jul 2004
TL;DR: An explicit construction of the free Kleene algebra with tests generated by a pair of sets is given and it is shown that the category KAT of Kleene algebras with tests and the categoryKAT ⊆ of Kozen’s KleeneAlgebra with tests are related by an adjunction.
Abstract: In this paper we define Kleene algebra with tests in a slightly more general way than Kozen’s definition. Then we give an explicit construction of the free Kleene algebra with tests generated by a pair of sets. We also show that the category KAT of Kleene algebras with tests and the category KAT ⊆ of Kozen’s Kleene algebras with tests are related by an adjunction. This fact shows that an infinitely-generated free Kleene algebra with tests in the sense of Kozen can be obtained as the image of our free algebra under the left adjoint from KAT to KAT ⊆ ; moreover, the image is isomorphic to itself. Therefore, our free Kleene algebra with tests is isomorphic to Kozen and Smith’s free Kleene algebra with tests if their construction available. Finally, we show that Kozen and Smith’s free Kleene algebra with tests can be presented as a coproduct of Kleene algebras. This is induced from our free construction.
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TL;DR: In this paper, it was shown that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, is the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift, which gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations.
Abstract: MV-algebras can be viewed either as the Lindenbaum algebras of Lukasiewicz infinite-valued logic, or as unit intervals [0,u] of lattice-ordered abelian groups in which a strong order unit u>0 has been fixed. They form an equational class, and the free n-generated free MV-algebra is representable as an algebra of piecewise-linear continuous functions with integer coefficients over the unit n-dimensional cube. In this paper we show that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, is the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift. This fact is noteworthy from the viewpoint of algebraic logic, since it gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations. As an ergodic theory fact, it provides explicit examples of volume-preserving homeomorphisms of the unit cube which are piecewise-linear with integer coefficients, preserve the denominators of rational points, and enjoy the Bernoulli property.
01 Jan 2004
TL;DR: The Shirshov theorem as mentioned in this paper states that finitely generated algebras behave as polynomial modules of polynomials, which generalizes commutative and finite-dimensional algeses.
Abstract: Algebras with polynomial identities generalize commutative and finite dimensional algebras. This generalization is not only formal. PI-algebras enjoy many other properties of commutative and finite dimensional algebras. In this section we shall present the Shirshov theorem which, roughly speaking, states that finitely generated algebras behave as finitely generated modules of polynomial algebras. Then we shall apply it to the positive solution of the Kurosh problem for PI-algebras.