Affine Mendelsohn triple systems and the Eisenstein integers

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to beaffine, nonramified. By exhibiting a one-to-one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the work of Donovan, Griggs, McCourt, Oprsal, and Stanovsky). As a consequence, all entropic MTSs of order coprime with 3 and distributive MTS of order coprime with 3 are classified. Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being nonramified, pure, and self-orthogonal are equivalent.