Fully ordered groups

Reverse Mathematics and Fully Ordered Groups

The computable sets form the minimum ω-model of RCA0 and any ω-model of RCA0 is closed under Turing reducibility. RCA0 is strong enough to prove the existence of a set of unique codes for the finite sequences of elements from any set X. We use FinX to denote this set of codes. Also, we use 〈a, b〉, or more generally 〈x0, . . . , xn〉, to denote pairs, or longer sequences, of elements of N. For an...

Ordered O-stable Groups

An ordered structure M is called o-λ-stable if for any subset A with |A| ≤ λ and for any cut in M there are at most λ 1-types over A which are consistent with the cut. It is proved in the paper that an ordered o-stable group is abelian. Also there were investigated definable subsets and unary functions of o-stable groups.

On Ordered Polycyclic Groups

It has been asserted that any (full) order on a torsion-free, finitely generated, nilpotent group is defined by some F-basis of G and that the group of o-automorphisms of such a group is itself a group of the same kind: Examples provided herein demonstrate that both of these assertions are false; however, it is proved that the group of o-automorphisms of an ordered, polycyclic group is nilpoten...

Characterizing automorphism groups of ordered abelian groups

In this short note we want to characterize the groups isomorphic to full automorphism groups of ordered abelian groups. The result will follow from classical theorems on ordered groups adding an argument from proofs used to realize rings as endomorphism rings of abelian groups, see [2]. Recall that H is a right ordered group (RO-group) if (H, ·) is a group and (H,<) is a linear order satisfying...

Reverse Mathematics and Ordered Groups