Semistar operations on Dedekind domains

TL;DR

This paper characterizes the structure of all semistar operations on Dedekind domains based on their maximal ideals, providing explicit descriptions and cardinality bounds, especially for finite and infinite cases.

Contribution

It offers an explicit, constructive description of the lattice of semistar operations on Dedekind domains, including cardinality estimates and exact counts for finite maximal ideal sets.

Findings

Cardinality bounds for semistar operations when the set of maximal ideals is finite.

Exact counts of semistar operations for up to 7 maximal ideals.

Cardinality of semistar operations is a double exponential for infinite maximal ideal sets.

Abstract

We give an explicit description of the lattice $\Semistar(D)$ of all semistar operations on any Dedekind domain $D$ from its set $\Max(D)$ of maximal ideals. This descpription is constructive if $\Max(D)$ is finite. As a corollary we show that $2^{{n \choose [n/2]}} \leq |\Semistar(D)| \leq 2^{2^n}$ if $n = |\Max(D)|$ is finite; we compute $|\Semistar(D)|$ if $|\Max(D)| \leq 7$; and we show that if $\Max(D)$ is infinite then $\Semistar(D)$ has cardinality $2^{2^{|\Max(D)|}}$.