Recognition of certain properties of automaton algebras

Abstract. The paper considers a new algebraic object, the completely automaton binomial algebras, which generalize certain existing classes of algebras. The author presents a classification of semigroup algebras taking into account completely automaton algebras and gives the corresponding examples. A number of standard algorithmic problems are solved for completely automaton binomial algebras: the recognition of a strict and nonstrict polynomial property, the recognition of the right and/or left finite processing, and the construction of the determining regular language for an algebra with finite processing and for monomial subalgebras of a free associative algebra and certain completely automaton algebras. For an automaton monomial algebra, the author constructs the left syzygy module of a finite system of elements and the Gröbner basis of a finitely generated left ideal; also, some algorithmic problems are solved.     

Primitivity of finitely presented monomial algebras

We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show that this result holds for the class of automaton algebras; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz.     

Symbolic powers of monomial ideals and vertex cover algebras

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.     

Finite generation of Ext for a generalization of D-Koszul algebras

In this paper we study the finite generation of Ext-algebras of a class of algebras called δ-resolution determined algebras. We characterize the δ-resolution determined algebras which are monomial algebras. If Λ is a graded algebra such that the associated monomial algebra is δ-resolution determined, we classify when the Ext-algebra of Λ is finitely generated.     

Prime and primitive algebras with prescribed growth types

Bartholdi and Smoktunowicz constructed in 2014 finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type.Moreover, using a construction of an inverse system of monomial ideals which arise from this construction, we are able to further construct finitely generated primitive algebras without further limitations on the growth type.Then, inspired by Zelmanov’s example in 1979, we show how our prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of the primitive constructions.     

On the Hilbert Series of Koszul Algebras

A family of examples is obtained which shows that, generally, it is impossible to decide for known Hilbert series of a qudratic algebra and its dual algebra whether or not this algebra has the Koszul property. The simplest example is given by two finitely generated algebras concentrated at the degrees not exceeding five; one of these algebras is monomial, while the other is not a Koszul algebra. This proves the conjecture of Positselskii.     

Primitive noetherian algebras with big centers

Recent work of Artin, Small, and Zhang extends Grothendieck's classical commutative algebra result on generic freeness to a large family of non-commutative algebras. Over such an algebra, any finitely-generated module becomes free after localization at a suitable central element. In this paper, a construction is given of primitive noetherian algebras, finitely generated over the integers or over algebraic closures of finite fields, such that the faithful, simple modules don't satisfy such a freeness condition. These algebras also fail to satisfy a non-commutative version of the Nullstellensatz.

     

A Hilbert-Nagata theorem in noncommutative invariant theory

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.

     

Field-dependent homological behavior of finite dimensional algebras

It is shown that the little finitistic dimension of a finite dimensional algebra, i.e., the supremum of the finite projective dimensions attained on finitely generated modules, is not necessarily attained on a cyclic module. In general, arbitrarily high numbers of generators are required. Moreover, it is demonstrated that this phenomenon may depend on the base fieldk. In fact, for each integerd>-3, there exists a quiver Γ with a set ρ of paths such that the little finitistic dimension of the finite dimensional algebrakΓ/<ρ> is attained on a cyclic module precisely when |k|≥d. By contrast, the global dimension of finite dimensional monomial relation algebras does not depend on the base field. This research was partially supported by a grant from the National Science Foundation.     

An Auslander-type result for Gorenstein-projective modules

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105-179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269-310].     

A proof that commutative artinian rings are noetherian

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.     

The Automata that Define Representations of Monomial Algebras

It is well known that the sets of strings that define all representations of string algebras and many representations of other quotients of path algebras form a regular set, and hence are defined by finite state automata. This short article aims to explain this connection between representation theory and automata theory in elementary terms; no technical background in either representation theory or automata theory is assumed. The article describes the structure of the set of strings of a monomial algebra as a locally testable and hence regular set, and describes explicitly the construction of the automaton, illustrating the construction with an elementary example. Hence it explains how the sets of strings and bands of a monomial algebra correspond to the sets of paths and closed (non-powered) circuits in a finite graph, and how the growth rate of the set of bands is immediately visible from that graph. Presented by C. Ringel.     

Stability of representations of effective partial algebras

An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings (theorems of Moschovakis and Hertling). To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Subalgebras of FA-presentable algebras

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. In contrast, a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.     

The structure of algebraic jordan algebras without nonzero nilpotent elements

An algebraic, linear Jordan algebra without nonzero nil-potent elements is proved to be a subdirect sum of prime Jordan algebras each of which has finite capacity or contains simple subalgebras of arbitrary capacity. If in addition the base field has nonzero character-istic or the algebra satisfies a polynomial identity, then each of the summands is determined to be simple of finite capacity. Further, it is proved that algebraic, PI Jordan algebras without nonzero nilpotent elements are locally finite in the sense that any finitely generated subalgebra has finite capacity.     

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.     

Finitely Related Clones and Algebras with Cube Terms

Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related??every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.