We show that if H is an effectively completely decomposable computable torsion-free abelian group, then there is a computable copy G of H such that G has computable orders but not orders of every degree.

We will study the question about decidability for Boolean algebras with first elementary characteristic one. The main problem is sufficient conditions for decidability of Boolean algebras with recursive representation for extended signature by definable predicates. We will use the base definitions on recursive and constructive models from [2, 4–6, 10, 11] but on Boolean algebras from [1, 8].

We solve a longstanding question of Rosenstein, and make progress toward solving a longstanding open problem in the area of computable linear orderings by showing that every computable ƞ-like linear ordering without an infinite strongly ƞ-like interval has a computable copy without nontrivial computable self-embedding. The precise characterization of those computable linear orderings which have computable copies without nontrivial computable self-embedding remains open.

In this paper, we solve a long-standing open question, about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of (...) independent interest. It would seem to promise many further applications. (shrink)