The dimension group pair is a group together with an automorphism that is associated to a subshift of finite type. It is an invariant of topological conjugacy. It is also possible to associate to a subshift of finite type an ideal in the ring \({\mathbb{Z}}[1/\lambda]\). The ideal class of this ideal is also a conjugacy invariant. This paper contains a thorough and very nice discussion of these invariants. The primary aim is to examine the relationship of these algebraic invariants (as well as some others) to various dynamical relationships between subshifts of finite type.
One of the main theorems is that there is an eventual right closing factor map from \(\Sigma_ A\) to \(\Sigma_ B\) if and only if the dimension group pair of \(\Sigma_ B\) is a quotient of the dimension group pair of \(\Sigma_ A.\)
Another theorem is that two subshifts of finite type have a common right closing extension if and only if they have the same ideal classes. They also show that if two subshifts of finite type have the same ideal class then they have a common eventual factor and relate these ideas to regular isomorphisms and stable factor maps.
There is much more contained in this paper. The main flavor is to relate these two algebraic invariants as well as some others to problems about the dynamics of subshifts of finite type and sofic systems.