On the genus of a lattice over an order of a dedekind domain

The property of mutual embeddings of index not divisible by any prime in a given finite set of primes has been used successfully in the case of finitely generated groups with finite commutator subgroup to define a group structure on the non-cancellation set of such groups. If R is a Dedekind domain and O is an Order over R, it has been proved that lattices over O belonging to the same genus have mutual embeddings. This result is formulated in this article in terms of module index and thus, allows us to define an abelian monoid structure on the genus set of such modules. We construct also some homomorphisms between genera class groups.