Cancellation properties in ideal systems: A classification of $\boldsymbol{e.a.b.}$ semistar operations

TL;DR

This paper classifies e.a.b. semistar operations into four distinct classes, analyzes their properties, and demonstrates that a.b. is strictly stronger than e.a.b., providing a comprehensive understanding of their structure.

Contribution

It introduces a new classification scheme for e.a.b. semistar operations and proves the distinctness of the classes, resolving an open problem about the strength of a.b. over e.a.b.

Findings

Four distinct classes of e.a.b. semistar operations identified.

Exactly one finite type operation per equivalence class exists.

Confirmed that a.b. is strictly stronger than e.a.b.

Abstract

We give a classification of {\texttt{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 {\texttt{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 {\texttt{e.a.b.}} semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that {\texttt{a.b.}} is really a stronger condition than {\texttt{e.a.b.}}