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.     

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.     

More covers of the Boolean variety of unital ℓ-groups

Within the lattice of varieties of pseudo MV-algebras, the variety ${\mathcal{B}}$ of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover ${\mathcal{B}}$ . The authors previously identified all solvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ . We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ , show that periodically primitive u?-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean.     

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.     

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.     

New torsion theory in unital abelian <Emphasis Type=Italic>ℓ</Emphasis>-groups

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category mathfrakCmathfrak{C} which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects.     

Dedekind $$\sigma $$ σ -complete $$\ell $$ ℓ -groups and Riesz spaces as varieties

Positivity - We prove that the category of Dedekind $$\sigma $$ -complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely...     

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.      

Some results on ℚ-groups

A finite group G whose irreducible characters are rational valued is called a ℚ-group. In this paper we will be concerned with the structure of a finite ℚ-group that contains a strongly embedded subgroup and the structure of a finite ℚ-group satisfying the property that none of its sections is isomorphic to .      

Strictly unital A∞-algebras

Given a graded module over a commutative ring, we define a dg-Lie algebra whose Maurer–Cartan elements are the strictly unital ${\mathrm{A}}_{\infty }$-algebra structures on that module. We use this to generalize Positselski's result that a curvature term on the bar construction compensates for a lack of augmentation, from a field to arbitrary commutative base ring. We also use this to show that the reduced Hochschild cochains control the strictly unital deformation functor. We motivate these results by giving a full development of the deformation theory of a nonunital ${\mathrm{A}}_{\infty }$-algebra.     

A family of à n -groups

For each integern≥2 and for each prime powerq, we exhibit a group Γ which acts simply transitively on the set of vertices of the building of type à _n associated with the local field $\mathbb{F}^q ((Y))$ . This generalizes work in [2] and [8] (wheren=2) and in [1] (wheren≤4).