Class semi-groups for general orders

For a maximal order O_K of a number field, the class group is the group of all fractional O_K-ideals modded out by the subgroup of principal O_K-ideals.

There are two ways we can extend this notion to general orders A in the number field. The first is looking at all the invertible fractional A-ideals. The second is to look at all fractional A-ideals. We will show some structural and numerical results for these general class semi-groups.

An important object that arises in this study is the semi-group consisting of all fractional A-ideals that generate O_K as O_K-ideal.