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.     

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.     

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.

     

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.     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

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.     

On cut completions of abelian lattice ordered groups

We denote by F _a the class of all abelian lattice ordered groups H such that each disjoint subset of H is finite. In this paper we prove that if G F _a, then the cut completion of G coincides with the Dedekind completion of G.     

Relatively uniform convergences in archimedean lattice ordered groups

For an archimedean lattice ordered group G let G ^d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G ^d ∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.     

The lex property of varieties of lattice ordered groups

Presented by R. S. Pierce.     

On a type of distributivity of lattice ordered groups

Let m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r _m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.     

The logic of equilibrium and abelian lattice ordered groups

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. Truth values are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of –groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal.Funding for the first and third author has been provided by FOMEC.Funding for the second author has been provided by FONDECYT 1020621, Facultad de Ciencias Exactas, U.N. de La Plata, and FOMEC.29 November 2000     

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.