Michael R. Darnel

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.

Special-valued l-groups and Abelian covers

This paper presents a new and independent proof of the theorem (proven first by Kopytov and Gurchenkov [7] and again by Reilly [10]) that covers of the Abelian l-variety are either representable or are Scrimger covers. The proof in this paper is based upon the l-Cauchy constructions of Ball [1]; once these are applied to the problem, the proof becomes elementary.

Special-valuedl-groups

AbstractSpecial elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an a*-extension of the originall-group.Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-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.

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of ℓ-groups G such that for each g∈G, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ℓ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some g∈G∈R. The correspondences T↦Y(T) and R↦T(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of ℓ-groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable ℓ-groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal κ, the class Y(discκ), where discκ stands for the class of all compact κ-disconnected spaces. Sample results follow. Every strongly projectable ℓ-group lies in Y(e.d.). The ℓ-group G lies in Y(e.d.) if and only if for each g∈GY(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(discκ). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean ℓ-group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P⊥+Q⊥.

Minimal Non-Metabelian Varieties of ℓ-Groups Which Contain No Nonabelian o-Groups

Martinez suggested and Scrimger proved that, when n is prime, the varieties 𝒮 n defined by what are now called Scrimger ℓ-groups S n cover the abelian variety, which is the smallest nontrivial variety of ℓ-groups. Darnel, Gurchenkov, and Reilly then independently proved that these are the only minimally nonabelian varieties which contain no nonabelian totally ordered groups. Holland and Reilly then gave a complete description of the lattice of metabelian ℓ-group varieties which contain no nonabelian totally ordered groups. This article now describes all of the minimally non-metabelian ℓ-group varieties which contain no nonabelian totally ordered groups.

A variety of lattice-ordered groups containing all representable covers of the abelian variety

A small variety of representable lattice-ordered groups is constructed, which contains all of the representable covers of the abelian variety.

The Relationship of Partial Metric Varieties and Commuting Powers Varieties

Holland et al. (Algebra Univers 67:1–18, 2012) considered varieties

Uniqueness of the group operation on the lattice of order-automorphisms of the real line

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