Solvable covers of the boolean variety of unital ℓ-groups

The varieties of solvable lattice-ordered groups covering the abelian variety were shown independently by Gurchenkov, Reilly, and Darnel to be the Scrimger varieties of ?-groups and the three Medvedev representable covers. In this article, the authors give a parallel characterization of varieties of solvable unital ?-groups which cover the minimal nontrivial variety of boolean unital ?-groups.     

Clean unital ℓ-groups

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ?-groups. We mention the relationship of clean unital ?-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ?-groups with distinguished weak order unit.     

Continuum many top varieties of GMV-algebras and unital ℓ-groups

We prove that there are continuum many top varieties of GMV-algebras and examine the countable lattice of varieties generated by periodic primitive unital ℓ-groups with the unit a rational translation.     

Free products of unital ℓ-groups and free products of generalized MV-algebras

We generalize Komori’s characterization of the proper subvarieties of MV-algebras. Namely, within the variety of generalized MV-algebras (GMV-algebras) such that every maximal ideal is normal, we characterize the proper top varieties. In addition, we present equational bases for these top varieties. We show that there are only countably many different proper top varieties and each of them has uncountably many subvarieties. Finally, we study coproducts and we show that the amalgamation property fails for the class of n-perfect GMV-algebras, i.e., GMV-algebras that can be split into n + 1 comparable slices. This paper has been supported by the Center of Excellence SAS -Physics of Information-I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contracts No. APVT-51-032002, APVV-0071-06, Bratislava.     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

Feebly projectable ℓ-groups

In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to lattice-ordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also investigate these properties with respect to archimedean ℓ-groups with weak order unit, as well as commutative semiprime f-rings.     

On the directional bias of the ℓbp-norm

The weighted sum of order p, the ℓ_bp-norm, is a generalization of the well-known ℓ_p-norm used in predicting distances in a transportation network. The properties of the directional bias function and the unit balls for the ℓ_bp-norm are of theoretical and practical interest. We investigate these properties and compare them with the properties of the ℓ_p-norm's directional bias function and the unit balls. We find that the ℓ_bp-norm is better at capturing the nonlinearity in a transportation network than the weighted ℓ_p-norm. It is also shown that, in contrast to the weighted ℓ_p-norm, where the optimal parameter p value is confined to the interval (1,2), for the ℓ_bp-norm the parameter p can have an optimal value greater than 2.     

Maximal fitting subclasses of the class of all finite π-groups

Call a Fitting class $\mathfrak{F}$ π-maximal if $\mathfrak{F}$ is (inclusion-)maximal in the class $\mathfrak{C}_\pi$ of all finite π-groups, where π stands for a nonempty set of primes. We establish a π-maximality criterion for a Fitting class $\mathfrak{F}$ of finite π-groups: we prove that a nontrivial Fitting class $\mathfrak{F}$ is π-maximal if and only if there is a prime p ∈ π such that, for every π-group G, the index of the $\mathfrak{F}$ -radical $G_\mathfrak{F}$ in G is equal to 1 or p. This implies Laue’s familiar result on a necessary and sufficient condition of the maximality of an arbitrary Fitting class of finite groups in the class $\mathfrak{C}$ of all finite groups. The π-maximality criterion obtained also gives a confirmation of the negative solution of Skiba’s Problem asking whether a local Fitting class has no inclusion-maximal Fitting subclasses (see Problem 13.50, The Kourovka Notebook: Unsolved Problems in Group Theory, 14th ed., Sobolev Institute of Mathematics, Novosibirsk, 1999).     

Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization

The iteratively reweighted ? ₁ minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ? ₂–? _p minimization problem with 0<p<1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.     

A topological duality for some lattice ordered algebraic structures including ℓ-groups

A topological duality is developed for a wide class of lattice ordered algebraic structures by introducing in an ordered Stone space a natural binary and continuous function. In particular, duality theorems are obtained for -groups and for abelian -groups. Dedicated to my wife Eugenia Presented by W. Taylor.This research is a part of the Doctorial Thesis that the author presented at the Universidad de Buenos Aires, under the supervision of Prof. R. Cignoli. The author wishes to thank Prof. Cignoli for his guidance. The author also wishes to thank Keith Kearnes for his careful reading of this paper and for his deep suggestions that led to a substantial improvement of the first version.     

On the Distribution of the Cardinality of Row Space for Boolean Matrices

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Syzygies of torsion bundles and the geometry of the level ℓ modular variety over \overline{\mathcal{M}}_{g}

We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ? in its Jacobian. These statements can be viewed an analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space $\mathcal{R}_{g,\ell}$ of twisted level ? curves of genus g and use this to derive results about the birational geometry of $\mathcal{R}_{g, \ell}$ . For instance, we prove that $\mathcal{R}_{g,3}$ is a variety of general type when g>11 and the Kodaira dimension of $\mathcal{R}_{11,3}$ is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.     

ℓ-Divisibility of ℓ-regular partition functions

We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b _ℓ (n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∏ _m=1^∞(1−q ^m ) ^ℓ−1.