Showing papers on "Free algebra published in 2000"
TL;DR: The notion of hyperdecidability has been introduced by the first author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups as discussed by the authors.
Abstract: The notion of hyperdecidability has been introduced by the rst author as a tool to prove decidability of semidirect products of pseudovarieties of semigroups. In this paper we consider some stronger notions which lead to improved decidability results allowing us in turn to establish the decidability of some iterated semidirect products. Roughly speaking, the decidability of a semidirect product follows from a mild, commonly veriied property of the rst factor plus the stronger property for all the other factors. A key role in this study is played by intermediate free semigroups (relatively free objects of expanded type lying between relatively free and relatively free proonite objects). As an application of the main results, the decidability of the Krohn-Rhodes (group) complexity is shown to follow from non-algorithmic abstract properties likely to be satissed by the pseudovariety of all nite aperiodic semigroups, thereby suggesting a new approach to answer (aarmatively) the question as to whether complexity is decidable.
82 citations
TL;DR: In this article, a finite dimensional F -algebra with involution M and its ∗-polynomial identities was constructed, and it was shown that the ∆-variety generated by M, var(M, ∆) has almost polynomial growth.
46 citations
TL;DR: In this paper, the basic structure of a free Baxter algebra is investigated, and it is shown that a Baxter algebra can be reduced to a domain or a reduced Baxter algebra when it is not reduced.
44 citations
TL;DR: In this paper, the finite dual of the polynomial ring R[x] over a noetherian ring R is shown to be a coalgebra, provided that R is noetherians and hereditary.
36 citations
TL;DR: In this paper, the authors studied generalized primitive elements of free algebras of finite ranks with the Nielsen-Schreier property and their automorphic orbits, and proved that an endomorphism preserving an automomorphic orbit of a nonzero element of a free algebra of rank two is an automorphism.
26 citations
TL;DR: The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer as mentioned in this paper, and it has been shown that the number of these minimal varieties is finite for any given exponent.
Abstract: The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand–Kirillov dimension of the corresponding relatively free algebras of finite rank.
22 citations
TL;DR: In this paper, the problem of explicitly constructing non-cyclic free groups in finite-dimensional crossed products using valuation criteria was investigated, and the results were applied to produce explicit free groups and symmetric groups in group rings of finite groups.
20 citations
01 Feb 2000
TL;DR: In this paper, it was shown that the collection of identities in two variables which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum does not have a finite equational axiomatization.
Abstract: This paper shows that the collection of identities in two variables which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum does not have a finite equational axiomatization. This gives an alternative proof of the nonexistence of a finite basis for N--a result previously obtained by the authors.
10 citations
TL;DR: The Priestley space of a quasi-Stone algebra is described and it is used to show that the class of finite quasi- Stone algebras has the amalgamation property.
Abstract: In this paper we describe the Priestley space of a quasi-Stone algebra and use it to show that the class of finite quasi-Stone algebras has the amalgamation property We also describe the Priestley space of the free quasi-Stone algebra over a finite set
9 citations
TL;DR: Another way of deriving the structure of the ring of Fricke characters of free groups and a classification of unitary and associative algebras over an arbitrary field with elements having a minimal polynomial of degree at most 2 are given in this paper.
7 citations
TL;DR: In this paper, the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities was calculated. But this was only for the case where every positive integer is a sum of four squares.
Abstract: By applying the theorem that every positive integer is a sum of four squares, we calculate the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities.
TL;DR: In this paper, the results on primitive elements of free algebras of main types of Schreier varieties of algebra have been reviewed and a series of examples of almost primitive elements have been given.
Abstract: In this article, we review results on primitive elements of free algebras of main types of Schreier varieties of algebras. A variety of linear algebras over a field is Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. A system of elements of a free algebra is primitive if it is a subset of some set of free generators of this algebra. We consider free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. We present matrix criteria for systems of elements of elements. Primitive elements distinguish automorphisms: endomorphisms sending primitive elements to primitive elements are automorphisms. We give a series of examples of almost primitive elements (an element of a free algebra is almost primitive if it is not a primitive element of the whole algebra, but it is a primitive element of any proper subalgebra which contains it). We also consider generic elements and Δ-primitive elements.
TL;DR: This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternARY algebra L, so that X freely generates L.
Abstract: A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.
TL;DR: In this article, the authors considered algebraic combinatorics and combinatorial algebra over fields of prime characteristics and proved the elimination theorem for free partially commutative color Lie p-superalgebras.
Abstract: In this article we consider several aspects of algebraic combinatorics and combinatorial algebra over fields of prime characteristics. P-super-Radford theorem gives the structure of the free associative algebra over a field of prime characteristic with the new multiplication given by the super shuffle product, we show that this algebra is isomorphic to the reduced free super commutative algebra on s-regular words. We prove the elimination theorem for free partially commutative color Lie p-superalgebras and obtain a Schreier type formula for free Lie p-superalgebras using formal power series techniques.
TL;DR: In this article, a classification of certain algebres engendrees par des "operateurs de creation and annihilation" is presented, dans lesquelles les equations d'evolution de Heisenberg for un systeme doscillateurs harmoniques sont verifiees.
Abstract: Resume Nous donnons une classification de certaines algebres engendrees par des “operateurs de creation et annihilation”, dans lesquelles les equations d'evolution de Heisenberg pour un systeme d'oscillateurs harmoniques sont verifiees.
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TL;DR: Free algebras as mentioned in this paper are universal objects: every n-generated MV-algebra A is a homomorphic image of the free MV algebra Free n over n generators; if an equation is satisfied by Free n, then the equation is automatically satisfied by all MValgebraalgeses.
Abstract: Free algebras are universal objects: every n-generated MV-algebra A is a homomorphic image of the free MV-algebra Free n over n generators; if an equation is satisfied by Free n then the equation is automatically satisfied by all MV-algebras. As a consequence of the completeness theorem, Free n is easily described as an MV-algebra of piecewise linear continuous [0,1]-valued functions defined over the cube [0, 1] n . Known as McNaughton functions, they stand to MV-algebras as {0,1}-valued functions stand to boolean algebras. Many interesting classes of MV-algebras can be described as algebras of [0, l]-valued continuous functions over some compact Hausdorff space. The various representation theorems presented in this chapter all depend on our concrete representation of free MV-algebras.
01 Jul 2000
TL;DR: This work shows how similar results can be obtained for systems of one-sided linear equations in the more general setting of monoid and group rings in the application of Gröbner bases to solve linear equations.
Abstract: One of the applications of Grobner bases in commutative polynomial rings is to solve linear equations. Here we show how similar results can be obtained for systems of one-sided linear equations in the more general setting of monoid and group rings.
TL;DR: In this paper, the elementary equivalence of lattices of subalgebras of free lattice varieties was shown to be equivalent to sets X and Y being second-order equivalent.
Abstract: A class of varieties V (including all finitely based lattice varieties) is determined for which the elementary equivalence of lattices of subalgebras of free V-algebras, Fv(X) and Fv(Y), is equivalent to sets X and Y being second-order equivalent.