Regular double p-algebras

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

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].

Classifying the Minimal Varieties of Polynomial Growth

Let 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.

A gap principle for dynamics

Let f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.

ALGEBRAS SATISFYING THE POLYNOMIAL IDENTITY [x1,x2][x3,x4,x5]=0

Let [Formula: see text] be a field of characteristic zero, and [Formula: see text] the variety of associative unitary algebras defined by the polynomial identity [x1,x2][x3,x4,x5]=0. This variety is one of the several minimal varieties of exponent 3 (and all proper subvarieties are of exponents 1 and 2). We describe asymptotically its proper subvarieties. More precisely, we define certain algebras ℛ2k for any k∈ℕ and show that if [Formula: see text] is a proper subvariety of [Formula: see text], then the T-ideal of its polynomial identities is asymptotically equivalent to the T-ideal of the identities of one of the algebras [Formula: see text], E, ℛ2k or ℛ2k⊕E, for a suitable k∈ℕ. We give also another description relating the T-ideals of the proper subvarieties of [Formula: see text] with the polynomial identities of upper triangular matrices of a suitable size.

*-Subvarieties of the Variety Generated by

Let be a field of characteristic zero, and * = t the transpose involution for the matrix algebra M2(). Let be a proper subvariety of the variety of algebras with involution generated by . We define two sequences of algebras with involution Rp, Sq, where p, q ∊ . Then we show that T*() and T*(Rp ⊕ Sq) are *-asymptotically equivalent for suitable p, q.

Varieties and simple groups

In her book on varieties of groups, Hanna Neumann posed the following problem [13, p. 166]: “Can a variety other than D contain an infinite number of non-isomorphic non-abelian finite simple groups?”The answer to this question does not seem to be known at present. However, in [7], Heineken and Neumann described an algorithm for determining whether or not there are any non-abelian finite simple groups satisfying a given law. They also outlined a way in which their algorithm could be used to show that “only finitely many of the known non-abelian finite simple groups can satisfy a given non-trivial law”; in this paper, we shall follow their suggestions, and prove theTHEOREM. Let g be a set of mutually non-isomorphic non-abelian finite simple groups, each of which is either an alternating group or a group of Lie type, and let g generate a proper subvariety of D. Then y is finite.

On varieties of metabelian groups of prime-power exponent

Let Un denote the variety of abelian groups of exponent dividing n, and let p be an arbitrary prime. In this paper all non-nilpotent, join-ireducible subvarieties of the product variety UpUp2 are determined. The proper subvarieties of this kind in fact form an infinite ascending chain …, and an arbitrary proper subvariety B of UpUp2 is either nilpotent or a join , where L is nilpotent and k is uniquely determined by B.

On non-Cross varieties of A-groups

The purpose of this paper is to provide a proof for a result announced in [3]. The result arose from a search for just-non-Cross varieties (recall that a Cross variety is one which can be generated by a finite group, and a just-non-Cross variety is a non-Cross variety every proper subvariety of which is Cross). For the motivation for this search, we refer the reader to [12]: for related results, see [1], [12], [13].

Cubic forms over algebraic number fields

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) > 500.