Showing papers on "Free algebra published in 2007"
TL;DR: The fuzzy real-valued double sequence space @?Fp2 is introduced and different properties of the space like completeness, solidity, symmetry, convergence free, sequence algebra, etc are studied.
71 citations
TL;DR: In this article, it was shown that every quasivariety of positive Sugihara monoids is a variety and that every finite subdirectly irreducible algebra is a retract of a free algebra.
Abstract: It is proved that in the variety of positive Sugihara monoids, every finite subdirectly irreducible algebra is a retract of a free algebra. It follows that every quasivariety of positive Sugihara monoids is a variety, in contrast with the situation in several neighboring varieties. This result shows that when the logic R-mingle is formulated with the Ackermann constant t, then its full negation-free fragment is hereditarily structurally complete.
48 citations
TL;DR: In this article, a von Neumann regular K-algebra Q(E) is defined and the monoid of isomorphism classes of finitely generated projective right Q (E)-modules is explicitly computed.
42 citations
TL;DR: Abadie et al. as mentioned in this paper showed that the free shifts on the reduced C ∗ -algebras of RD-groups, including the free group on infinitely many generators, and amalgamated free product C ∆ -algesas, are all strictly weak mixing and not only uniquely ergodic.
35 citations
TL;DR: In this paper, the interplay between left ideals of the free ∗-algebra with n generators and their suitably defined zero sets is studied, and similarly between quadratic submodules and their positivity sets.
Abstract: We study, in the spirit of modern real algebra, the interplay between left ideals of the free ∗-algebra \(\mathbb F\) with n generators, and their suitably defined zero sets; and similarly between quadratic submodules of \(\mathbb F\) and their positivity sets.
30 citations
TL;DR: In this paper, it was shown that for a variety R ℳ of semimodules over an IBN-semiring R (an IBN is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner.
Abstract: In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ0) of automorphisms of the category Θ0 of finitely generated free algebras of Θ is of great importance. In this article, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety Rℳ of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in Plotkin (2002); in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(Rℳ0) are semi-inner.
25 citations
28 Feb 2007
TL;DR: In this article, an analog of the Bergman Centralizer Theorem for free Poisson algebras over an arbitrary field of characteristic 0 is presented. But the analog is not applicable to the case where the polynomial is fixed.
Abstract: We prove an analog of the Bergman Centralizer Theorem for free Poisson algebras over an arbitrary field of characteristic 0. Some open problems are formulated.
23 citations
TL;DR: This paper shows that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable and is able to solve the consistency problem for certain classes of single-variable equations in free inversemonoids.
Abstract: It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper, we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single-variable equations in free inverse monoids.
21 citations
TL;DR: In this paper, the authors obtained necessary conditions for skew generalized power series rings to be right (respectively left) uniserial, and proved that these conditions are also sufficient when the monoid S is commutative or totally ordered.
21 citations
TL;DR: In this paper, it was shown that the convexity of the set Dp plus the irreducibility of p imply that degree of p is at most four and that p has additional structure, which is also discussed in detail.
Abstract: A non-commutative polynomial p (x 1 ,... ,x g ) is a linear combination of words in the non-commuting variables {x 1 , ..., x g }. Such a polynomial is naturally evaluated on a tuple X = (X 1 , ...,X g ) of symmetric n x n matrices, with value p(X) an n X n matrix. The involution T on words given by sending a concatenation of letters to the same letters, but in the reverse order (for instance (x j x l ) T = x l x j ) extends naturally to such polynomials and p is itself symmetric if p T = p. In this case, p(X) is a symmetric matrix. The positivity domain D n p of a non-commutative symmetric polynomial p is the closure of the component of 0 of the set {X ∈ (R nxn sym ) g | p (X) > 0}. Here (R nxn sym ) 9 denotes the set of g-tuples of n x n real symmetric matrices. The positivity domain, Dp, is the sequence of sets {D n p }. The purpose of this paper is to prove that, under some additional hypotheses on p, the convexity of the set Dp plus the irreducibility (in an appropriate sense) of p imply that degree of p is at most four and that p has additional structure, which is also discussed in detail. This result may portend a type of noncommutative (in a free algebra) real algebraic geometry in which basic conditions on a variety V constrain V much more than occurs classically. Here an irreducible noncommutative variety (namely the boundary of D p ) with nonnegative curvature has degree no greater than four. The problem itself is motivated by linear system engineering and the vast quantity of work there on Linear Matrix Inequalities (LMIs) and Convex Matrix Inequalities. It suggests that in systems problems, whose form scales with dimension, convex situations are very heavily constrained. This paper treats the geometry of noncommutative varieties, whereas earlier work [14] [16] treats "convex" noncommutative polynomials and rational functions. Our approach here includes an analysis of non-commutative second directional derivatives p" (x) [h], a non-commutative polynomial in 2g variables, with respect to the number of positive and negative eigenvalues of p"(X)[H] for X, H ∈ (R nxn sym ) g . The analysis in the present paper is for X in the boundary of Dp, and H corresponding to directions tangent to the boundary of D p , restrictions which cause very many difficulties. The case where X is not constrained is treated in [6].
16 citations
01 Jun 2007
TL;DR: In this paper, the authors give necessary and sufficient conditions for Sn−1 to be generated by idempotents over Euclidean domains and free left T -sets of finite rank, where T is a cancellative monoid in which every finitely generated left ideal is principal.
Abstract: If A is a stable basis algebra of rank n, then the set Sn−1 of endomorphisms of rank at most n − 1 is a subsemigroup of the endomorphism monoid of A. This paper gives a number of necessary and sufficient conditions for Sn−1 to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left T -sets of finite rank, where T is cancellative monoid in which every finitely generated left ideal is principal.
TL;DR: The group of automorphisms of the semigroup End(W(X), the free commutative or a free associative algebra, is studied.
Abstract: Let W(X) be a free commutative or a free associative algebra. The group of automorphisms of the semigroup End(W(X)) is studied.
TL;DR: The class of quasi-armendariz rings is closed under some kinds of upper triangular matrix rings as mentioned in this paper, and the relation between the quasi-Baer property of a ring R and those of the monoid ring R]M] is studied in this paper.
09 Jul 2007
TL;DR: The use of equational systems as needed in modern applications are illustrated, specifically to the theory of substitution in the presence of variable binding and to models of name-passing process calculi.
Abstract: The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop a theory of free constructions for such equational systems; and to illustrate the use of equational systems as needed in modern applications, specifically to the theory of substitution in the presence of variable binding and to models of name-passing process calculi.
TL;DR: A base of the free alternative superalgebra on one odd generator is constructed and a corollary of the alternative Grassmann algebra is given, finding a new element of degree 5 from the radical of thefree alternative algebra of countable rank.
Abstract: 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.
TL;DR: It is proved that the group Aut End A is generated by semi-inner and mirror automorphisms of End A, where is the subcategory of finitely generated free algebras of the variety .
Abstract: Let be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety freely generated by a set X = {x1,…, xn}, 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].
TL;DR: In this paper, a free resolution for a partially commutative monoid is constructed, and the homological dimension of the monoid can be estimated using the free resolution of the resolution.
Abstract: We construct a free resolution for a free partially commutative monoid and, using this resolution, estimate the homological dimension of the monoid.
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20 Apr 2007
TL;DR: In this paper, it was shown that the Birman-Murakami-Wenzl algebra of type Dn is semisimple and free of rank (2^n+1)n!!-(2^(n-1)+ 1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers.
Abstract: The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.
TL;DR: In this article, it was shown that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, in 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.
TL;DR: In this paper, the so-called JF-embeddings of a set and a field in a division ring have inversion height at most two, and give examples of inversion heights one and two.
TL;DR: In this article, an extension of A. R. Kemer's results to a rather broad class of rings close to associative rings, over a field of characteristic 0 (in particular, this class includes the varieties generated by finite-dimensional alternative and Jordan rings).
Abstract: The subject of this work is an extension of A. R. Kemer’s results to a rather broad class of rings close to associative rings, over a field of characteristic 0 (in particular, this class includes the varieties generated by finite-dimensional alternative and Jordan rings). We prove the finite-basedness of systems of identities (the Specht property), the representability of finitely-generated relatively free algebras, and the rationality of their Hilbert series. For this purpose, we extend the Razymslov-Zubrilin theory to Kemer polynomials. For a rather broad class of varieties, we prove Shirshov’s theorem on height.
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TL;DR: In this article, it was shown that the Birman-Murakami-Wenzl algebra of type Dn is semisimple and free of rank (2^n+1)n!!-(2^(n-1)+ 1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers.
Abstract: The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.
TL;DR: This work works with parametrized endofunctors of Set, and introduces the concept of iterativity for algebras for the end ofunctor [email protected]?H(X,X).
TL;DR: This work shows that the standard definition of free name is not preserved under the structural congruence, and develops a fixed point approach to the set of free names and shows that it is invariant under theStructural Congruence.
01 Apr 2007
TL;DR: The integral structure of the maximal weights of the finite dimensional representations of sl2(k) imply that a Tarski type decision principle for free ∗-algebras with two or more generators does not hold as discussed by the authors.
Abstract: The integral structure of the maximal weights of the finite dimensional representations of sl2(k) imply that a Tarski type decision principle for free ∗-algebras with two or more generators does not hold.
TL;DR: For ungraded quotients of an arbitrary ℤ-graded ring, the general PBW property has been defined in this paper, which covers both the classical PBW and the N-type PBW properties studied via the n-Koszulity by several authors.
Abstract: For ungraded quotients of an arbitrary ℤ-graded ring, we define the general PBW property, that covers the classical PBW property and the N-type PBW property studied via the N-Koszulity by several authors (see [2–4]). In view of the noncommutative Grobner basis theory, we conclude that every ungraded quotient of a path algebra (or a free algebra) has the general PBW property. We remark that an earlier result of Golod [5] concerning Grobner bases can be used to give a homological characterization of the general PBW property in terms of Shafarevich complex. Examples of application are given.
01 Jan 2007
TL;DR: In this paper, all the rings that are defined over a post algebra and share the properties of the Serfati ring were determined and in the case r = 3 one of them is equivalent to the post algebra.
Abstract: Serfati (7) defined a ring structure on every Post algebra. In this pa- per we determine all the rings that are defined over a Post algebra and share the properties of the Serfati ring. In the case r = 3 one of them is equivalent to the Post algebra. This is a term equivalence and it extends the equivalence between a Boolean algebra and the Boolean ring associated with it.
TL;DR: A survey of recent results on automomorphisms of polynomial algebra K[x, y, z] and free associative algebra K over a field K can be found in this article.
Abstract: We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K 〈x, y, z〉 over a field K, i.e., automorphisms that fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K 〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms that recognize z-tame automorphisms and z-tame coordinates.
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TL;DR: In this paper, a graded version of the necklace Lie algebra equipped with the Kontsevich bracket was introduced, and the notion of a linear double Poisson tensor tensor on a quiver was introduced.
Abstract: In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson-Lichnerowicz cohomology groups for three types of (linear and non-linear) double Poisson brackets on the free algebra in two variables. This allows us to develop some useful techniques for the computation of the double Poisson-Lichnerowicz cohomology.
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TL;DR: In this paper, it was shown that all pure states of the reduced C*-algebra of the free goup on $\aleph_1$ generators are *-automorphism equivalent.
Abstract: We prove that all of the pure states of the reduced C*-algebra of the free goup on $\aleph_1$ generators are *-automorphism equivalent and extract some consequences of that fact.