The sets of star and semistar operations on semilocal Pr\"ufer domains

TL;DR

This paper investigates the structure of semistar and star operations on semilocal Pr"ufer domains, revealing how their properties depend on the domain's spectrum and localizations, especially in h-local cases.

Contribution

It characterizes the sets of semistar and star operations on semilocal Pr"ufer domains based on spectral and localization properties, including polynomial growth in h-local cases.

Findings

Sets depend on spectrum and localizations

Polynomial growth of semistar operations in h-local domains

Properties of localizations determine the operations set

Abstract

We study the sets of semistar and star operation on a semilocal Pr\"ufer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is $h$-local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.