Density-Like and Generalized Density Ideals

TL;DR

This paper demonstrates the existence of uncountably many nonisomorphic density-like ideals on  that are not generalized density ideals, answering a question in the field and providing conditions for when density-like ideals are generalized.

Contribution

It constructs uncountably many nonisomorphic density-like ideals not being generalized density ideals and offers conditions under which density-like ideals are generalized.

Findings

Existence of uncountably many nonisomorphic density-like ideals

These ideals are nonpathological and not generalized density ideals

Provides sufficient conditions for density-like ideals to be generalized

Abstract

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega$ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja, Farkas, and Plebanek in [J. Symb. Log. \textbf{80} (2015), 1268--1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.