Countably valued lattice-ordered groups

A countably valued lattice-ordered group is a lattice-ordered group in which every element has only countably many values. Such lattice-ordered groups are proven to be normal-valued and, though not necessarily special-valued, every element in a countably valuedl-group must have a special value. The class of countably valuedl-groups forms a torsion class, and the torsion radical determined by this class is anl-ideal that is the intersection of all maximal countably valued subgroups.Countably valuedl-groups are shown to be closed with respect toeventually constant sequence extensions, and it is shown that many properties of anl-group pass naturally to its eventually constant sequence extension.Presented by M. Henriksen.     

Essential extensions and radical classes of lattice-ordered groups

In this paper the relationship between essential extensions and radical classes of lattice-ordered groups is discussed. It is proved that the polar radical classes of lattice-ordered groups are exactly essentially closed classes.Received May 7, 1999; accepted in final form September 23, 2004.     

Some Examples in the Theory of Subgroup Growth

By estimating the subgroup numbers associated with various classes of large groups, we exhibit a number of new phenomena in the theory of subgroup growth.     

Some Examples in the Theory of Subgroup Growth

By estimating the subgroup numbers associated with various classes of large groups, we exhibit a number of new phenomena in the theory of subgroup growth.     

Ordered groups with a conucleus

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a “quantale like” category of algebras.     

Arbitrary torsion classes and almost free abelian groups

Using the set theoretical principle ? for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.     

σ-Interpolation Lattice-ordered groups

In [1], Jakubík showed that the class of -interpolation lattice-ordered groups forms a radical class, but left open the question of whether the class forms a torsion class. In this paper, we show that this class does indeed form a torsion class.     

Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements

This paper axiomatizes classes of Abelian lattice-ordered groups with a finite upper bound on the number of pairwise disjoint positive elements; finds model-completions for these theories; derives corresponding Nullstellensätze; determines which model-completions eliminate quantifiers; and examines quantifier elimination in a different language and for positive formulas.     

Torsion Free Commutator Subgroups of Generalized Coxeter Groups

In this note we consider generalized Coxeter groups and we study the problem of when their commutator subgroup is torsion free. As a consequence we describe all (i) Coxeter groups, (ii) triangle groups and (iii) index two orientation preserving subgroups of the finite co-volume hyperbolic Coxeter tetrahedra, for which the commutator subgroup is torsion free.     

Canonical-type connection on almost contact manifolds with B-metric

The canonical-type connection on the almost contact manifolds with B-metric is constructed. It is proved that its torsion is invariant with respect to a subgroup of the general conformal transformations of the almost contact B-metric structure. The basic classes of the considered manifolds are characterized in terms of the torsion of the canonical-type connection.     

Two-knot groups with torsion free abelian normal subgroups of rank two

We show that a torsion free abelian normal subgroup of rank two of a two-knot group which is contained in the commutator subgroup must be free abelian, the centralizer of the abelian subgroup is not contained in the commutator subgroup, and neither of the latter two groups is finitely generated. Furthermore, we characterize algebraically the groups of 2-knots which are cyclic branched covers of twist spun torus knots.     

The <Emphasis Type="Italic">C</Emphasis>-topology on lattice-ordered groups

Let A be a lattice-ordered group. Gusi′c showed that A can be equipped with a C-topology which makes A into a topological group. We give a generalization of Gusi′c's theorem,and reveal the very nature of a "C-group" of Gusi′c in this paper. Moreover,we show that the C-topological groups are topological lattice-ordered groups,and prove that every archimedean lattice-ordered vector space is a T2 topological lattice-ordered vector space under the C-topology. An easy example shows that a C-group need not be T2....     

Intermediately fully invariant subgroups of abelian groups

Describing intermediately fully invariant subgroups of divisible and torsion groups, we show that the intermediately fully invariant subgroups are direct summands in a completely decomposable group whose every homogeneous component is decomposable. For torsion groups, we find out when all their fully invariant subgroups are intermediately fully invariant; and for torsion-free groups, this question comes down to the reduced case. Also, in a torsion group that is the sum of cyclic subgroups, its subgroup is shown to be intermediately inert if and only if it is commensurable with some intermediately fully invariant subgroup.     

Radical and Torsion Theory for Acts

After recapitulating the rudiments of the Kurosh-Amitsur radical theory of S-acts, hereditary radicals are discussed. The hereditary Hoehnke radical assignments r which designate a Rees congruence r(A) to each S-act A, are the hereditary Kurosh-Amitsur radical assignments. Then the corresponding radical as well as semisimple classes are characterized. Equivalence classes of injective S-acts determine hereditary torsion assignments t, these are just the hereditary Hoehnke radicals, but the congruence t(A) of A need not be a Rees congruence. Torsion and torsionfree classes are characterized; several hereditary torsion assignments may determine the same torsion class which is always a radical class closed under taking subacts. Examples show that a hereditary torsion assignment need not be a hereditary radical.     

Regular Unital Finite-Dimensional Lattice-Ordered Algebras

The main result in this article is to show that a regular unital finite-dimensional lattice-ordered algebra over ? with zero ?-radical is isomorphic to a finite direct sum of lattice-ordered matrix algebras of lattice-ordered group algebras of finite groups over ?.     

Special-valued subgroups of lattice-ordered groups

We prove that the intersection of all maximal special-valued subgroups of a lattice-ordered group is the special-valued quasi-torsion radical of a lattice-ordered group , which extends our earlier result that the intersection of all maximal finite-valued subgroups of a lattice-ordered group is the finite-valued torsion radical of . We also show that the class of almost finite-valued lattice-ordered groups is a quasi-torsion class, and the quasi-torsion radical of a group is equal to the intersection of the group with the lateral completion of the finite-valued torsion radical of the group.

     

格序环中的幂零元

研究了格序环的ｌ－根的一些性质,特别给出了所有幂零元的集合是ｌ－理想的几类格序环。     

Macpherson's chern classes of singular algebraic varieties

Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F₂)??? in a first-order language Lo appropriate for group theory. It is shown that in every model of Th(F₂)??? the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ?-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F₂)??? while the quasi-locally free groups are the ?-free groups with maximal Abelian subgroups elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound of finitely generated quasi-locally free groups containing nontrivial torsion in their Abelianizations. The question of whether or not almost locally free groups have torsion free Abelianization is related to a bound in a free group on the number of factors needed to express certain elements of the derived group as a product of commutators     

On torsionfree classes which are not precover classes

In the class of all exact torsion theories the torsionfree classes are cover (pre-cover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented. This research has been partially supported by the Grant Agency of the Czech Republic, grant #GAČR 201/06/0510 and also by the institutional grant MSM 0021620839.