Computability of dimension groups

Abstract

We investigate the computability of the isomorphism set $\operatorname{Iso}(G_A,G_B)$ between $G_A$ and $G_B$, where $G_A$ is a subgroup of $\mathbb{Q}^n$ generated by columns of integer powers of a non-singular $n \times n$-matrix $A$ with integer entries. Assuming that the characteristic polynomial of $A$ is irreducible -- and under an additional condition when $n$ is not prime -- we prove that $\operatorname{Iso}(G_A,G_B)$ is computable; that is, there exists an algorithm that determines the structure in finitely many steps. We also present illustrative examples.