Paul F. Conrad

The Lattice of Convex l -Subgroups of a Lattice-Ordered Group

In this chapter we shall examine in some detail the lattice C(G) of convex l-subgroups of a lattice-ordered group G. Our emphasis is on determining how much information is available from a strictly lattice-theoretic consideration of C(G). Many important classes of l-groups are described in these terms, but it is impossible to so describe varieties of l-groups; in section 5.4 we shall consider examples which reveal this limitation to our approach.

Complemented lattice-ordered groups

Abstract This article introduces complemented lattice-ordered groups and studies how they interact with the space of minimal prime convex l -subgroups. Along the way the concept of rigidity turns out to be of paramount importance. Another important tool: the notion of a z -subgroup, introduced some time ago, but, until now, left largely to languish.

Generalized Boolean Algebras in Lattice-Ordered Groups

In this paper, characterizations are given for the free lattice-ordered group over a generalized Boolean algebra and the freel -module of a totally ordered integral domain with unit over a generalized Boolean algebra. Extensions of lattice-ordered groups using generalized Boolean algebras are defined and their properties studied.

LATTICE-ORDERED GROUPS WHOSE LATTICES DETERMINE THEIR ADDITIONS

In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or de- termined up to /-isomorphism. In the group of integers Z with the usual order <, 1 covers 0. From this simple fact, it is easy to see that (Z, <) is a uniquely transitive chain as defined by Ohkuma (24) and that 1 is a singular element. Either property is enough to show that, having chosen 0 to be the identity of Z, the usual addition is completely specified by the chain. In this paper, we show that these properties are sufficiently general and pow- erful enough to prove that many large and familiar classes of lattice-ordered groups also have their group operations completely determined by the lattice and the choice of an identity. In particular, we will show Theorem A. Every free abelian lattice-ordered group has a unique addition. Theorem B. If G is archimedean and if for any 0 < g e G, there exists a singular element s such that 0 < s < g, then G has a unique addition.

Subgroups and Hulls of Specker Lattice-Ordered Groups

In this article, it will be shown that every ℓ-subgroup of a Specker ℓ-group has singular elements and that the class of ℓ-groups that are ℓ-subgroups of Specker ℓ-group form a torsion class. Methods of adjoining units and bases to Specker ℓ-groups are then studied with respect to the generalized Boolean algebra of singular elements, as is the strongly projectable hull of a Specker ℓ-group.

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.

Signatures andS-discrete lattice-ordered groups

AbstractThe central theme of this article is the approximation of lattice-ordered groups (l-groups) first by Specker groups and, subsequently, by the so-calledS-discretel-groups. The sense of these approximations is made precise via the notion of a signature, defined below, and by that ofa*-subgroups. Sample result: ifG is a projectablel-group then it has anl-subgroupH which is Specker and for which the mapP→P∩H defines a boolean isomorphism between the algebras of polars ofG andH.

Very large subgroups of a lattice-ordered group

This article introduces the concept of a very large subgroup in the theory of lattice-ordered groups. The existence of a minimal very large subgroup is connected to some previously known structure theory, but it is also linked to conditions not studied before. Very large subgroups are useful in studying torsion and radical classes, and among other things, extension of lattice-ordered groups using very large kernels yields an intriguing completion operation for torsion classes. In the final section there is a new contruction which produces a lattice-ordered group in which every value is essential, having no special values.

Complementing lattice-ordered groups: The projectable case

A complemented l-group G is one in which to each a ∈ G there corresponds a b ∈ G so that |a|⋏|b|=0, while |a|⋎|b| is a unit of G. For projectable l-groups this is so precisely when the group possesses a unit.The article introduces the notion of complementation, and the situation for projectable l-groups is analyzed in some detail; in particular, it is shown that any projectable l-group having a projectable complementation in which it is convex has a unique maximal one of this kind.

On torsion classes of lattice-ordered groups

We study lattice-ordered groups whose set of branch points in the root system of regular subgroups satisfies descending chain condition. We show that these lattice-ordered groups fornl a torsion class . We also show that the class of finite-valued lattice-ordered groups that are also in is closed with respect to lattice-ordered subgroups, and such class has a unique archimedean closure.