Proper Subvariety Research Articles

Free Modal Pseudocomplemented De Morgan Algebras

Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].

Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

Let A be a superalgebra with graded involution or superinvolution ∗ and let \(c_{n}^{*}(A)\), n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of \(\mathbb {Z}_{2}\) and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519–545 2017).

Curves in Hilbert modular varieties

We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish a Second Main Theorem following Nevanlinna theory. Finally, with a metric approach, we establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions.

Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.

Varieties of algebras with pseudoinvolution and polynomial growth

Let A be an associative algebra with pseudoinvolution over an algebraically closed field of characteristic zero and let be its sequence of -codimensions. We shall prove that such a sequence is polynomially bounded if and only if the variety generated by A does not contain five explicitly described algebras with pseudoinvolution. As a consequence, we shall classify the varieties of algebras with pseudoinvolution of almost polynomial growth, i.e. varieties of exponential growth such that any proper subvariety has polynomial growth and, along the way, we shall give also the classification of their subvarieties. Finally, we shall describe the algebras with pseudoinvolution whose -codimensions are bounded by a linear function.

An uncountable family of almost nilpotent varieties of polynomial growth

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of1) a countable family of almost nilpotent varieties of at most linear growth and2) an uncountable family of almost nilpotent varieties of at most quadratic growth.

ЧИСЛОВЫЕ ХАРАКТЕРИСТИКИ АЛГЕБР ЛЕЙБНИЦА-ПУАССОНА

The paper is survey of recent results of investigations on varieties of Leibniz-Poisson algebras. We show that a variety of Leibniz-Poisson algebras has either polynomial growth or growth with exponential not less than 2, the eld being arbitrary. We show that every variety of Leibniz-Poisson algebras of polynomial growth over a eld of characteristic zero has a nite basis for its polynomial identities.We construct a variety of Leibniz-Poisson algebras with almost polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of Leibniz-Poisson algebras over a eld of characteristic zero. We show all varieties of Leibniz-Poisson algebras with almost polynomial growth in one class of varieties. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity fx; yg fz; tg = 0, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A. We show that if the ideal of identities of a Leibniz-Poisson variety V does not contain any Leibniz polynomial identity then V has overexponential growth of the codimensions. We construct a variety of Leibniz-Poisson algebras with almost exponential growth. Let f n(V)gn1 be the sequence of proper codimension growth of a variety of Leibniz-Poisson algebras V. We give one class of minimal varieties of Leibniz-Poisson algebras of polynomial growth of the sequence f n(V)gn1, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree k, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than k.

Symmetric operators on modal pseudocomplemented De Morgan algebras

Modal pseudocomplemented De Morgan algebras are pseudocomplemented De Morgan algebras which satisfy the additional equation |$\mbox{(tm) }\, x\vee \sim x \leq x\vee x^\ast.$| There is a connection between these algebras and A. Monteiro’s tetravalent modal algebras. It is worth mentioning that these algebras have been studied previously using topological tools. In the present work, we enrich the language of modal pseudocomplemented De Morgan algebras by adding an automorphism of period |$2$|⁠, giving rise to a new algebraic structure which we call |$\cal S$|-algebras. From the logical point of view, we are enriching the original language with a new involutive unary connective. Among other things, it is proved that |$\cal S$| is a semisimple, locally finite variety. Besides, the generating algebras of this variety are determined. The structure of the free |$\cal S$|-algebra with a finite number of generators is studied, and a formula for computing its size is given. Finally, the lattice |$\Lambda({\cal S})$| is exhibited showing that the varieties of Boolean algebras, involutive Boolean algebras and three-valued Łukasiewicz algebras are proper subvarieties of |${\cal S}$|⁠.

Thom Polynomials and the Green–Griffiths–Lang Conjecture for Hypersurfaces with Polynomial Degree

Abstract Green and Griffiths [25] and Lang [29] conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves $f:{\mathbb{C}} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. Using this we show that the polynomial Green–Griffiths–Lang conjecture for a generic projective hypersurface of degree $\deg (X)>2n^{9}$ follows from a positivity conjecture for Thom polynomials of Morin singularities.

Proper T-ideals of poisson algebras with extreme properties

Let {γn(V)}n≥1 be the sequence of proper codimensions of some variety V of Poisson algebras over a field of characteristic zero. A class of minimal varieties of Poisson algebras of polynomial growth of the sequence {γn(V)}n≥1 is presented, i.e., the sequence {γn(V)}n≥1 of any such variety V grows as a polynomial of some degree k, but the sequence {γn(W)}n≥1 of any proper subvariety W in V grows as a polynomial of degree strictly less than k.

Polynomial codimension growth of algebras with involutions and superinvolutions

Let A be an associative algebra over a field F of characteristic zero endowed with a graded involution or a superinvolution ⁎ and let cn⁎(A) be its sequence of ⁎-codimensions. In [4,12] it was proved that if A is finite dimensional such sequence is polynomially bounded if and only if A generates a variety not containing a finite number of ⁎-algebras: the group algebra of Z2 and a 4-dimensional subalgebra of the 4×4 upper triangular matrices with suitable graded involutions or superinvolutions.In this paper we focus our attention on such algebras since they are the only finite dimensional ⁎-algebras, up to T2⁎-equivalence, generating varieties of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. We classify the subvarieties of such varieties by giving a complete list of generating finite dimensional ⁎-algebras. Along the way we classify all minimal varieties of polynomial growth and surprisingly we show that their number is finite for any given growth. Finally we describe the ⁎-algebras whose ⁎-codimensions are bounded by a linear function.

Unobstructed symplectic packing for tori and hyper-Kähler manifolds

Let [Formula: see text] be a closed symplectic manifold of volume [Formula: see text]. We say that the symplectic packings of [Formula: see text] by balls are unobstructed if any collection of disjoint symplectic balls (of possibly different radii) of total volume less than [Formula: see text] admits a symplectic embedding to [Formula: see text]. In 1994, McDuff and Polterovich proved that symplectic packings of Kähler manifolds by balls can be characterized in terms of the Kähler cones of their blow-ups. When [Formula: see text] is a Kähler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple), these Kähler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that for any Campana simple Kähler manifold, as well as for any manifold which is a limit of Campana simple manifolds in a smooth deformation, the symplectic packings by balls are unobstructed. This is used to show that the symplectic packings by balls of all even-dimensional tori equipped with Kähler symplectic forms and of all hyper-Kähler manifolds of maximal holonomy are unobstructed. This generalizes a previous result by Latschev–McDuff–Schlenk. We also consider symplectic packings by other shapes and show, using Ratner’s orbit closure theorem, that any even-dimensional torus equipped with a Kähler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).

Universal varieties of quasi-Stone algebras

The lattice of varieties of quasi-Stone algebras ordered by inclusion is an \({\omega+1}\) chain. It is shown that the variety \({\mathbf{Q_{2,2}}}\) (of height 13) is finite-to-finite universal (in the sense of Hedrlin and Pultr). Further, it is shown that this is sharp; namely, the variety \({\mathbf{Q_{3,1}}}\) (of height 12) is not finite-to-finite universal and, hence, no proper subvariety of \({\mathbf{Q_{2,2}}}\) is finite-to-finite universal. In fact, every proper subvariety of \({\mathbf{Q_{2,2}}}\) fails to be universal. However, \({\mathbf{Q_{1,2}}}\) (the variety of height 9) is shown to be finite-tofinite universal relative to \({\mathbf{Q_{2,1}}}\) (the variety of height 8). This too is sharp; namely, no proper subvariety of \({\mathbf{Q_{1,2}}}\) is finite-to-finite relatively universal. Consequences of these facts are discussed.

On minimal Poisson algebras

We present Poisson algebra constructions on the base of associative algebras and study the relationship between the obtained Poisson algebras and the initial associative algebras. Based on obtained constructions, we present two classes of minimal varieties of Poisson algebras of polynomial growth, i.e., varieties for which the sequence of codimensions grows as a polynomial of degree k, but the sequence of codimensions of any proper subvariety grows as a polynomial whose degree is strictly less than k. Previously the author showed that there are only two varieties of Poisson algebras of almost polynomial growth. In this paper we give a complete description of all subvarieties of these two varieties.

Polynomial growth and star-varieties

Let V be a variety of associative algebras with involution over a field F of characteristic zero and let cn⁎(V), n=1,2,…, be its ⁎-codimension sequence. Such a sequence is polynomially bounded if and only if V does not contain the commutative algebra F⊕F, endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4×4 upper triangular matrices. Such algebras generate the only varieties of ⁎-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the ⁎-varieties of almost polynomial growth by giving a complete list of finite dimensional ⁎-algebras generating them.

Almost polynomial growth: Classifying varieties of graded algebras

Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the G-graded varieties of almost polynomial growth by giving a complete list of finite-dimensional G-graded algebras generating them.

Least V-quasivarieties of MV-algebras

This paper is a contribution to the study of the lattice of all quasivarieties of MV-algebras. Given a variety V of MV-algebras, we say that a quasivariety is a V-quasivariety provided that it generates V as a variety. It turns out that every variety has a least V-quasivariety denoted by QV. Moreover, it is generated as a quasivariety by its free algebra over an infinite set of generators. We obtain a Komori's type characterization of QV for every proper subvariety of MV-algebras. We investigate the order structure of the poset of least V-quasivarieties and we find all minimal quasivarieties among all quasivarieties of MV-algebras different from the class of boolean algebras. Finally, we make some remarks on the logical interpretation of least V-quasivarieties of MV-algebras as structurally complete extensions of the infinite valued Łukasiewicz logic.

On varieties of basic algebras

It is known that the variety of commutative basic algebras, the variety of MV-algebras, the variety of orthomodular lattices and hence also the variety of Boolean algebras are proper subvarieties of the variety of basic algebras. In the paper several possible axiomatizations of the variety of basic algebras are provided. The variety of symmetric basic algebras is introduced and shown to be a proper subvariety of the variety of commutative basic algebras, and whose members have the underlying lattices distributive. It is proved that commutative basic algebras can be described as residuated $$l$$ l -groupoids.

Classifying the Minimal Varieties of Polynomial Growth

AbstractLet ν be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties ν that are minimal of polynomial growth (i.e., their sequence of codimensions grows like nk, but any proper subvariety grows like nt with t < k). These varieties are the building blocks of general varieties of polynomial growth.It turns out that for k ≤ 4 there are only a finite number of varieties of polynomial growth nk, but for each k > 4, the number of minimal varieties is at least |F|, the cardinality of the base field, and we give a recipe for their construction.

Graded cocharacters of minimal subvarieties of supervarieties of almost polynomial growth

The sequence of graded codimensions of a minimal supervariety of polynomial growth grows like nt, but any proper subvariety grows like ns with s<t. Such supervarieties are the building blocks of general supervarieties of polynomial growth. In this paper, we exhibit the decompositions of the graded cocharacters of all minimal subvarieties of supervarieties of almost polynomial growth and compute their graded colengths. Furthermore, for a finite dimensional superalgebra A such that its sequence of graded codimensions grows like nt we exhibit an upper bound for the nth graded colength of A which depends only on t and on two parameters related to A such as its dimension and the index of nilpotence of its Jacobson radical J(A).