Nice Bases of QTAG-Modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

Fields generated by finite rank subgroups of ℚ¯∗

Let [Formula: see text] be a finite rank subgroup of [Formula: see text]. We prove that the multiplicative group of the field generated by all elements in the divisible hull of [Formula: see text] is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond’s generalized Lehmer conjecture is satisfied.

The Relatively Uniform Completion, Epimorphisms and Units, in Divisible Archimedean L-Groups

In the category Arch of archimedean l-groups, the r.u. completion of the divisible hull, rdA, is the maximum essential reflection and the maximum majorizing reflection (Ball-Hager, 1999). In the weak-unital subcategory W, the reflections c

The nice basis rank of a primary abelian group

An abelian p-group G has a nice basis if it is the ascending union of a sequence of nice subgroups, each of which is a direct sum of cyclic groups. It is shown that if G is any group, then G ⊕ D has a nice basis, where D is the divisible hull of pωG. This leads to a consideration of the nice basis rank of G, i.e., the smallest rank of a divisible group D such that G ⊕ D has a nice basis. This concept is used to show that there exist a reduced group G and a non-reduced group H, both without a nice basis, such that G ⊕ H has a nice basis

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.

Small decidable sheaves

Fred Richman conjectured that the following principle is not constructive:(*) If A is a decidable subset of the set N of natural numbers and if, for every decidable subset B of N, either A ⊆ B or A ⊆ N − B, then, for some n ∈ N, A ⊆ {n}.A set A of natural numbers is called decidable if ∀n(n ∈ A ∨ ⌉ (n ∈ A)) holds. In recursive models, this agrees with the recursion-theoretic meaning of decidability. In other contexts, “complemented” and “detachable” are often used.Richman's conjecture was motivated by the problem of uniqueness of divisible hulls of abelian groups in constructive algebra. Richman showed that a countable discrete abelian p-group G has a unique (up to isomorphism over G) divisible hull if the subgroup pG is decidable. He also showed that the converse implies.We confirm the nonconstructive nature of by showing (in §1) that it is not provable in intuitionistic set theory, IZF. Thus, in the models we construct, there are countable discrete abelian p-groups G whose divisible hulls are unique but whose subgroups pG are not decidable.Our models do not satisfy further conditions imposed by Richman, namely Church's Thesis and Markov's Principle, so the full conjecture remains an open problem. We do, however, show (in §2) how to embellish our first model so that the fan theorem (i.e., compactness of 2N) fails. (Church's Thesis implies the stronger statement that the negation of the fan theorem holds.)Our models will be constructed by the method of sheaf semantics [1], [3]. That is, we shall construct Grothendieck topoi in whose internal logic fails.

Minimal vector lattice covers

We show that each abelian l–group G is a large l–subgroup of a minimal vector lattice V and if G is archimedean then V is unique, in fact, V is the l–subspace of (Gd)^ that is generated by G, where Gd is the divisible hull of G and (Gd)^ is the Dedekind-MacNeille completion of Gd. If G is non-archimedean then V need not be unique, even if G is totally ordered.