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

A(R) is the lattice-ordered group (l-group) of all order-automorphisms of the real lineR, with the usual pointwise order and “of course” with composition as the group operation. In fact, what other choices are there for a group operation having the same identity that would give anl-group? Composition in the reverse order would work. But there are no other choices — the group operation can be recognized in the lattice. Several classes of abelianl-groups having a unique group operation have been found by Conrad and Darnel, but this is the first non-abelian example having the minimum of two group operations. “Conversely”, Holland has shown that for the groupA(R) under composition, the only lattice orderings yielding anl-group are the pointwise order and its dual. These results also hold for the rational lineQ.     

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 -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -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 , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

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.     

Special-valuedl-groups

Special 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 -embedded into a special-valuedl-group.This paper is dedicated to the memory of Prof. Samuel Wolfenstein, who initiated the study of normal-valuedl-groups and recognized early the importance of special-valuedl-groups.Presented by L. Fuchs.     

Solvable covers of the boolean variety of unital ℓ-groups

The varieties of solvable lattice-ordered groups covering the abelian variety were shown independently by Gurchenkov, Reilly, and Darnel to be the Scrimger varieties of ?-groups and the three Medvedev representable covers. In this article, the authors give a parallel characterization of varieties of solvable unital ?-groups which cover the minimal nontrivial variety of boolean unital ?-groups.     

More covers of the Boolean variety of unital ℓ-groups

Within the lattice of varieties of pseudo MV-algebras, the variety ${\mathcal{B}}$ of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover ${\mathcal{B}}$ . The authors previously identified all solvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ . We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ , show that periodically primitive u?-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean.     

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.