Ideals which generalize $(v^0)$

TL;DR

This paper introduces generalized ideals extending $(v^0)$, formulates analogs of Hadamard's theorem, and establishes bounds on their covering and additivity numbers using advanced set-theoretic tools.

Contribution

It generalizes the ideal $(v^0)$, develops its theoretical properties, and derives new inequalities relating covering and additivity numbers.

Findings

Established bounds on $cov(d^0(\

Derived analogs of Hadamard's theorem for generalized ideals

Applied the base tree and Kulpa-Szymański theorems to set ideal properties

Abstract

We consider ideals $d^0(\mathcal{V})$ which are generalizations of the ideal $(v^0)$. We formulate couterparts of Hadamard's theorem. Then, adopting the base tree theorem and applying Kulpa-Szyma\'nski Theorem, we obtain $ cov(d^0(\mathcal{V}))\leq add(d^0(\mathcal{V}))^+$.