  The role of generalized Bowen-Franks groups (BF-groups) as topological
conjugacy invariants for $\mathbb{T}^{n}$ automorphisms is studied. Using
algebraic number theory, a link is established between equality of BF-groups
for different automorphisms ($BF$-equivalence) and an identical position in a
finite lattice ($\mathcal{L}$-equivalence). Important cases of equivalence of
the two conditions are proved. Finally, a topological interpretation of the
classical BF-group $\mathbb{Z}% ^{n}/\mathbb{Z}^{n}(I-A)$ in this context is
presented.
