Turing Degrees of Isomorphism Types of Algebraic Objects

Calvert, Wesley; Harizanov, Valentina; Shlapentokh, Alexandra

Turing Degrees of Isomorphism Types of Algebraic Objects

Authors

Calvert, Wesley

Harizanov, Valentina

Shlapentokh, Alexandra

Description

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees.
17 pages