P-like properties of meager ideals and cardinal invariants
Abstract
We study two particular modifications of the $\mathrm{P}$-property
of ideals and related cardinal invariants $\mathtt{cof}^{\mathcal{J}}(\mathcal{I})$ and $\mathtt{cov}^+(\mathcal{I})$. We give some results on the existence
of $\mathrm{P}\left(\mathcal{J}\right)$-ideals
or non-$\mathrm{P}\left(\mathcal{J}\right)$-ideals regarding specific classes of ideals, particularly meager ideals on $\omega$. We also provide values of the cardinal invariant $\mathtt{cof}^{\mathcal{J}}(\mathcal{I})$ describing the smallest families ensuring $\mathrm{P}(\mathcal{J})$ for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities $\mathrm{Fin}<_K\left\langle\mathcal{A}\right\rangle<_K\mathrm{Fin}\times\mathrm{Fin}$ for any {MAD} family $\A$ using the weak P-ideal notion.
of ideals and related cardinal invariants $\mathtt{cof}^{\mathcal{J}}(\mathcal{I})$ and $\mathtt{cov}^+(\mathcal{I})$. We give some results on the existence
of $\mathrm{P}\left(\mathcal{J}\right)$-ideals
or non-$\mathrm{P}\left(\mathcal{J}\right)$-ideals regarding specific classes of ideals, particularly meager ideals on $\omega$. We also provide values of the cardinal invariant $\mathtt{cof}^{\mathcal{J}}(\mathcal{I})$ describing the smallest families ensuring $\mathrm{P}(\mathcal{J})$ for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities $\mathrm{Fin}<_K\left\langle\mathcal{A}\right\rangle<_K\mathrm{Fin}\times\mathrm{Fin}$ for any {MAD} family $\A$ using the weak P-ideal notion.
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Subscribers OnlyDOI: https://doi.org/10.2478/tmmp-2023-0025