A Class of Strongly Decomposable Abelian Groups

Let G be a completely decomposable torsion-free Abelian group and G= G_i, where G _i is a rank 1 group. If there exists a strongly constructive numbering of G such that (G,) has a recursively enumerable sequence of elements g _i G _i , then G is called a strongly decomposable group. Let pi, i, be some sequence of primes whose denominators are degrees of a number p _i and let . A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that for some numbers i ₁,...,i _k . We bring in the concept of a quasihyperhyperimmune set, and specify a necessary and sufficient condition on the characteristic of A subject to which the group in question is strongly decomposable. Also, it is proved that every hyperhyperimmune set is quasihyperhyperimmune, the converse being not true.     

Distributive lattices of numberings

We study into a semilattice of numberings generated by a given fixed numbering via operations of completion and taking least upper bounds. It is proved that, except for the trivial cases, this semilattice is an infinite distributive lattice every principal ideal in which is finite. The least upper and the greatest lower bounds in the semilattice are invariant under extensions in the semilattice of all numberings. Isomorphism types for the semilattices in question are in one-to-one correspondence with pairs of cardinals the first component of which is equal to the cardinality of a set of non-special elements, and the second — to the cardinality of a set of special elements, of the initial numbering. Supported by INTAS grant No. 00-429. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 83–102, January–February, 2007.     

Constructive Matrix and Orderable Groups

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GL _n(K) (general), SL _n(K) (special), and UT _n(K) (unitriangular) over K. It is proved that for n 3, a corresponding group is constructible iff so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UT _n(K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.     

Quasiresolvent Models and B-Models

Relations among classes of resolvent, quasiresolvent, intrinsically enumerable models, and B-models are established. It is proved that every linear order containing a -subset isomorphic to or to ^- is not quasiresolvent. It is stated that every model of a countably categorical theory is a B-model. And it is shown that for every B-model in a hereditarily finite admissible set, the uniformization theorem fails.     

Σ-Uniform structures and Σ-functions. I

The concept of a Σ-uniform structure is introduced. A condition is derived which is necessary and sufficient for a universal Σ-function to exist in a hereditarily finite admissible set over a Σ-uniform structure.     

On quasiresolvent periodic abelian groups

This is a continuation of [1]. We introduce the concept of a primarily quasiresolvent periodic abelian group and describe primarily quasiresolvent and 1-quasiresolvent periodic abelian groups. We construct an example of a quasiresolvent but not primarily quasiresolvent periodic abelian group. For a direct sum of primary cyclic groups we obtain criteria for a group to be quasiresolvent, 1-quasiresolvent, and resolvent, and establish relations among them. We construct a set S of primes such that the direct sum of some cyclic groups of orders p ∈ S is not a quasiresolvent group.     

Computable torsion-free nilpotent groups of finite dimension

We find criteria for the computability (constructivizability) of torsion-free nilpotent groups of finite dimension. We prove the existence of a principal computable enumeration of the class of all computable torsion-free nilpotent groups of finite dimension. An example is constructed of a subgroup in the group of all unitriangular matrices of dimension 3 over the field of rationals that is not computable but the sections of any of its central series are computable.