Class Of Lattice-ordered Groups Research Articles

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ℓ-group G is lamron if Min(G)−1 is a Hausdorff topological space, where Min(G)−1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ℓ-groups are related to ℓ-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ℓ-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ℓ-groups.

Conrad frames

A Conrad frame is a frame which is isomorphic to the frame C ( G ) of all convex ℓ-subgroups of some lattice-ordered group G. It has long been known that Conrad frames have the disjointification property. In this paper a number of properties are considered that strengthen the disjointification property; they are referred to as the Conrad conditions. A particularly strong form of the disjointification property, the C-frame condition, is studied in detail. The class of lattice-ordered groups G for which C ( G ) is a C-frame is shown to coincide with the class of pairwise splitting ℓ-groups. The arguments are mostly frame-theoretic and Choice-free, until one tackles the question of whether C-frames are Conrad frames. They are, but the proof is decidedly not point-free. This proof actually does more: it shows that every algebraic frame with the FIP and disjointification can be coherently embedded in a C-frame. When the discussion is restricted to normal-valued lattice-ordered groups, one is able to produce examples of coherent frames having disjointification, which are not Conrad frames.

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.

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.

A Representation for a Class of Lattice Ordered Groups

A Representation for a Class of Lattice Ordered Groups