The derived sequence of a pre-Jaffard family

TL;DR

This paper introduces pre-Jaffard families, a weaker generalization of Jaffard families, enabling new decompositions of semistar operations and length functions, and unifies the treatment of sharp and dull degrees in Pr"ufer domains.

Contribution

It defines pre-Jaffard families, constructs derived sequences of overrings, and applies these to unify concepts in Pr"ufer domain theory.

Findings

Pre-Jaffard families generalize Jaffard families with weaker conditions.

Constructed sequences decompose semistar operations and length functions more broadly.

Unified the treatment of sharp and dull degrees in Pr"ufer domains.

Abstract

We introduce the concept of \emph{pre-Jaffard family}, a generalization of Jaffard families obtained by substituting the locally finite hypothesis with a much weaker compactness hypothesis. From any such family, we construct a sequence of overrings of the starting domain that allows to decompose stable semistar operations and singular length functions in more cases than what is allowed by Jaffard families. We also apply the concept to one-dimensional domains, unifying the treatment of sharp and dull degree of a Pr\"ufer domain.