Cancellation properties in ideal systems: A classification of e.a.b. semistar operations

We give a classification of e.a.b. semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all e.a.b. semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of e.a.b. semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that a.b. is really a stronger condition than e.a.b.