P-Like Properties of Meager Ideals and Cardinal Invariants

We study two particular modifications of the P-property of ideals and related cardinal invariants cof^𝒥 (ℐ)and cov⁺(ℐ). We give some results on the existence of P (𝒥)-ideals or non-P (𝒥)-ideals regarding specific classes of ideals, particularly meager ideals on ω. We also provide values of the cardinal invariant cof^𝒥 (ℐ) describing the smallest families ensuring P(𝒥) for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities Fin <_K 〈𝒜 〉 <_K Fin × Fin for any MAD family 𝒜 using the weak P-ideal notion.