A topology on lattice ordered groups

We show that a lattice ordered group can be topologized in a natural way. The topology depends on the choice of a set of admissible elements (-topology). If a lattice ordered group is 2-divisible and satisfies a version of Archimedes' axiom (-group), then we show that the -topology is Hausdorff. Moreover, we show that a -group with the -topology is a topological group.

     

Quasi-varieties of lattice ordered groups

Whereas there is a maximal proper variety of lattice-ordered groups, it is known that there is no maximal proper quasi-variety of lattice-ordered groups. We prove that there are 2º (the maximal possible) pairwise incomparable quasi-varieties of lattice-ordered groups containing . Some of the distributive laws of the semigroup lattice of quasi-varieties are examined and their truth (or falsity) is established. It is also shown here that the latticeL of alll-group varieties is a sublattice of the latticeQ of quasi-varieties ofl-groups but fails to be a complete sublattice.This article is a part of the author's Ph.D. dissertation which was directed by Professor A. M. W. Glass. The author wishes to express his sincere gratitude to Professor Glass for his assistance and encouragement during the writing of the dissertation and this article. He also wishes to thank Professor K. K. Hickin for his help with nilpotent material, in particular for his help in establishing Theorem 4.7.Presented by L. Fuchs.     

A generalization of a theorem of Imai and its applications to Iwasawa theory

It is proved that, if $K$ is a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field satisfying a mild condition, then any abelian variety over $K$ with potentially good reduction has finite $K(K^{1/p^\infty })$ -rational torsion subgroup. This can be used to remove certain conditions assumed in some theorems in Iwasawa theory.     

Intrinsic metrics for lattice ordered groups

Swamy and Jakubik studied the metric ¦x y¦ on lattice ordered groups, and isometries which presere it. We show the only intrinsic metrics on lattice ordered groups are the multiplesn ¦x–y ¦ of theirs, and that the triangle inequality is satisfied by such a metric iff the group is abelian. We show that there are isometries for each of these metrics, but they are rare. We give a simpler proof via permutation groups of the following augmented version of a theorem of Jakubik. IfT is an isometry of the lattice ordered groupG with respect to the metric ¦x¥¦ andT(0)=0, thenG=AB, B is abelian, andT(a+b)=a–b; conversely, any suchT is an isometry.To Paul Conrad on his 60th birthdayPresented by L. Fuchs.     

A theorem on the transcendency and its applications

A theorem on the transcendency, for a limit of a sequence of algebraic numbers, is given and used to test the transcendencies of some classes of numbers. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

Weak homogeneity of lattice ordered groups

In this paper we deal with weakly homogeneous direct factors of lattice ordered groups. The main result concerns the case when the lattice ordered groups under consideration are archimedean, projectable and conditionally orthogonally complete. Supported by VEGA grant 2/4134/24. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information.     

Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations

In this paper, we establish a mountain pass theorem in an ordered Banach space, which is a Riesz-Banach space such that the absolute value is continuous. Applying our theorem to semilinear elliptic equations with sign-changing nonlinear terms, we prove the existence of a positive solution.     

The intertwining lifting theorem for ordered groups

Sz.-Nagy's dilation theorem for a contraction on a Hilbert space has been extended by Mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group. We generalize to this case the intertwining lifting theorem of Sz.-Nagy and Foias. Some previous interpolation results on ordered groups are obtained as consequences.     

Torsion classes of Specker lattice ordered groups

In this paper we investigate the relations between torsion classes of Specker lattice ordered groups and torsion classes of generalized Boolean algebras.     

A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

Weak relatively uniform convergences on abelian lattice ordered groups

The notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G ⁺, then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex ℓ-subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.